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Solved Examples of the Number System along with the Practice Questions

Number System is one of the most important topics of the Maths subject which the students are required to master. And the best way to master the chapter on the number system is by practicing the questions from the said chapter. But along with the practice students do need the solution of the same so that they can be sure about their progress. Hence Vedantu provides to all the students the practice questions of the number system, as well as solved examples of the same so that before attempting the practise questions students can see the examples and have a mental preparation

An Overview of the Number System

Before going directly into the practice questions of the number system, let us first have a brief understanding of the number system, and revise the concepts of it, so that you find it easy to solve the practice questions.

A method of expressing the numbers on the number line, using numerical symbols, is known as Number System. As you can see there are two terms in particular in this definition to understand in a better manner, which are:

Number Line: it is a straight line that real numbers at a fixed interval. All the types of numbers are included in the number line, that is to say, natural numbers, rational numbers, integers etc.

Numerical Symbols: it simply means the mathematical digits, that represents the numbers, which are from 0 to 9.

An Overview of Different Types of Numbers

All the types of numbers are represented in the number line and hence they are all part of the number system, therefore, let us have a quick review of all the types of numbers.

Natural Number: These are the numbers that we use in our day-to-day life because it is widely used in counting, therefore natural numbers are also called counting numbers. It includes all the positive numbers which are not a fraction, and also, it does not include 0. The range of the natural number is 1 to infinity.

Whole Numbers: Add zero to the natural number and you have the whole numbers , that is to say, it includes all the natural numbers and also the 0. Therefore, from 0 to infinity all the numbers are whole numbers.

Integers: It includes all the whole numbers, along with the negative numbers, that is to say, -1, -2, -3. But it does not include the fractions. From the negative infinity number to the positive number in infinity, are all Integers.

Fractions: The numbers which are written in the form of, where b is always a natural number.

Rational Numbers: These numbers can also be represented in fractional form, the difference between rational numbers and fractions is that rational numbers can be any integers except for the 0 as the denominator.

Irrational Numbers: These are the numbers that cannot be represented in a fractional manner such as the root of 2 (\[\sqrt{2}\]) and pie (π)

Real Numbers: When we combine the whole numbers, integers, and fractions are all real numbers. In a simple manner, all the integers along with the decimals and fractions are real numbers.

Prime Numbers: The numbers which only have two factors, which are 1 and the number itself are called prime numbers. For example, 37, can only be divided by 1 and by 37 itself.

The Number System is an important chapter of mathematics. A student needs to be strong in the fundamentals of the number system to solve other problems related to Maths. Some students face difficulty in solving sums of the number system. So, here in this article, we have provided some crucial sums relating to the number system. A student can practice these questions, and it would be easy for him/her to understand the chapter. In this article, we have provided various questions based on number systems such as number system questions and answers, number system practice questions, MCQs on number systems and many other important questions.

Number System Questions and Answers

1. Determine whether the numbers are rational or irrational.

\[ \sqrt{2}\]

\[\sqrt{100}\]

Ans: A rational number is a number that can be represented in the form of p/q, whereas an irrational number cannot be represented in the form of p/q. So,

\[ \sqrt{2}\] is Irrational.

1.5 is Rational.

\[\sqrt{100}\] is Rational.

3.14 is Irrational.

2. Without Actual Division, a state which of the following is a terminating decimal.

\[\frac{9}{25}\]

\[ \frac{37}{78}\]

Ans: In \[\frac{9}{25}\], the prime factors of denominator 25 are 5,5. Thus, it is a terminating decimal. 

In \[ \frac{37}{78}\], the prime factors of denominator 78 are 2, 3, and 13. Thus, it is a non-terminating decimal.

3. Express each of the following as a rational number in the form of p/q, where q ≠ 0.

\[\overline{0.6}\]

\[ \overline{0.43}\]

Ans:   1. Let x = 0.6666 …..(i)

Multiplying both side of eqn (i) by 10 we get,

10x = 6.6666…..(ii)

Now, subtracting eqn (i) from eqn (ii) we get, 

10x = 6.6666

⇒ x = 6/9 which is equal to ⅔, So the required fraction is ⅔.

2. Let x = 0.43434343….(i)

Multiplying both sides of eqn (i) by 100 we get,

100x = 43.43434343…..(ii)

Now, subtracting eqn (i) from eqn (ii) we get,

100x = 43.43434343

x = 0.43434343

⇒ x = \[\frac{43}{99}\], Hence the fraction is \[\frac{43}{99}\].

4. Find 4 rational numbers between 1 and 2.

Ans: To find 4 rational numbers between 1 and 2, we need to divide and multiply both the numbers by (4 + 1) which is 5. So we get,

\[1 \times \frac{5}{5} = \frac{5}{5}\] and \[ 2 \times \frac{5}{5}\] = \[\frac{10}{5}\], Therefore the rational numbers are:

\[\frac{5}{5}\], \[\frac{6}{5}\], \[\frac{7}{5}\], \[\frac{8}{5}\], \[\frac{9}{5}\], \[\frac{10}{5}\].

5. Compare the following numbers.

(i) 0 and \[-\frac{9}{5}\].

(ii) \[-\frac{17}{20}\] and \[-\frac{13}{20}\].

(iii) \[\frac{40}{29}\] and \[\frac{141}{29}\].

Ans: (i) We know that a negative number is always less than 0. Therefore,

0 > - \[\frac{9}{5}\].

(ii) Here the denominator is the same and we know that -17 < -13. Therefore, 

\[\frac{-17}{20}\] < \[\frac{-13}{20}\].

(iii) Here the denominator is the same and we know that 40 < 141. Therefore,

\[\frac{40}{29}\] < \[\frac{141}{29}\].

6. Write the following in decimal numbers and state what expansion it is.

(i) \[\frac{40}{100}\]  (ii) \[\frac{9}{10}\]  (iii) \[\frac{9}{37}\] (iv) \[\frac{103}{5}\]

Ans: (i) \[\frac{40}{100}\] is 0.40, and it is terminating.

(ii) \[\frac{9}{10}\] is 0.9, and it is ending.

(iii) \[\frac{9}{37}\] is 0.243243… it is non-terminating.

(iv) \[\frac{103}{5}\] is 20.6, and it is terminating.

7. Insert one rational number between 3/5 and 7/9.

Ans:   If a and b are two rational numbers, then one rational number between these two will be \[\frac{a + b}{2}\]. Hence the required rational number will be 

\[\frac{1}{2} (\frac{3}{5} + \frac{7}{9}) = \frac{1}{2} (\frac{27 + 35}{45}) = \frac{1}{2} \times \frac{62}{45} = \frac{31}{45}\]

So, the rational number is \[\frac{31}{45}\].

Questions on Number System Conversion

Here, we have provided some number system math questions which are based on number system conversion. 

1. Convert each of the following into a decimal number.

(i) \[\frac{4}{15}\]

(ii) \[2\frac{5}{12}\]

(iii) \[\frac{9}{27}\]

(iv) \[5\frac{31}{55}\]

2. Convert the following into a rational number.

(i) \[0.\overline{227}\]

(ii) \[0.\overline{2104}\]

Number System Practice Questions

As we know, practice makes everyone perfect, so for the better understanding of students, we have provided some number system important questions for practice.

1. Show the number √5 on the number line.

\[\sqrt{2}\]

\[ \sqrt{100}\]

Ans: A rational number is a number which can be represented in the form of p/q, whereas an irrational number cannot be represented in the form of p/q. So,

\[\sqrt{2}\] is Irrational.

2. Insert three rational numbers between 4 and 5.

\[ \frac{37}{78} \]

Ans: In \[ \frac{9}{25}\], the prime factors of denominator 25 are 5,5. Thus, it is a terminating decimal. 

In \[\frac{37}{78} \], the prime factors of denominator 78 are 2, 3, and 13. Thus, it is a non-terminating decimal.

3. Represent the following rational numbers in decimal form

(i) \[\frac{18}{42}\]    (ii) \[-\frac{11}{13}\]

4. Rationalise the denominator of

\[ \frac{1}{3-\sqrt{5}} \]

5. Simplify the following expression (24 - 32)a. ( 5 + 23)

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FAQs on Number System Questions

1. Give Some MCQs on the Number System.

Some important MCQs on Number System are:

1. From the following choose Co-prime numbers.

(a) 2, 3 (b) 2, 4 (c) 2, 6 (d) 2, 110

2. On adding \[2\sqrt{3}\] and \[3\sqrt{2}\] we get:

(a) \[5\sqrt{5}\] (b) \[5(\sqrt{3} + \sqrt{2})\] (c) \[ 2\sqrt{3} + 3\sqrt{2}\] (d) None of these

3. A rational number between \[\sqrt{2}\] and \[\sqrt{3}\].

(a) 1.9 (b) \[ \frac{( \sqrt{2}.\sqrt{3} )}{2}\] (c)1.5 (b) 1.8

4. Which of the following is irrational?

(a) \[ \frac{\sqrt{4}}{9} \] (b)\[ \frac{\sqrt{12}}{\sqrt{3}}\] (c) \[\sqrt{5}\] (d) \[\sqrt{81}\] 

5. The Value of (16) 3/4 is equal to:

(a) 2 (b) 4 (c) 8 (d) 16

2. What do you mean by Number System? What are its types?

The number system can be defined as the expression of numbers in a written format. These are a set of symbols and rules used to denote numbers. The number system is used to state how many objects are there in a given set. There are different types of number systems, and here we have mentioned some of the types of number systems for better knowledge of students. The following are the types of number systems:

Real Numbers.

Natural Numbers.

Whole Numbers.

Rational Number system.

Irrational Number system.

Complex Number system.

Binary Number system.

Decimal Number System.

Hexa-Decimal Number System.

Octal-Decimal Number System.

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Number System – Definition, Examples, Facts, Practice Problems

Number system, practice problems on number system, frequently asked questions on number system.

Decimal Number System: The decimal number system consists of 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 and is the most commonly used number system. We use the combination of these 10 digits to form all other numbers. The value of a digit in a number depends upon its position in the number. The place value table for the decimal number system is as:

Decimal place value chart

Each place to the left is ten times greater than the place to its right, that is, as we move from the right to the left, the place value increases ten times with each place. 

Decimal Number System 1

  • A decimal number system is also called the Base 10 system.
  • A number 49,365 is read as Forty-nine thousand three hundred sixty-five, where the value of 4 is forty thousand, 9 is nine thousand, 3 is three hundred, 6 is sixty and 5 is five. 

Binary Number System

In the binary number system, we only use two digits 0 and 1. It means a 2 number system.

Example of binary numbers: 1011; 101010; 1101101

Binary Number System

Each digit in a binary number is called a bit. So, a binary number 101 has 3 bits. 499787080

Computers and other digital devices use the binary system. The binary number system uses Base 2.

Hexadecimal Number System

The word hexadecimal comes from Hexa meaning 6, and decimal meaning 10. So, in a hexadecimal number system, there are 16 digits. It consists of digits 0 to 9 and then has first 5 letters of the alphabet as:

The table below shows numbers 1 to 20 using decimal , binary and hexadecimal numbers.

10 and 100 More than the Same Number Game

  • The decimal number system is also called the Hindu–Arabic numeral system
  • Anthropologists hypothesize that the decimal system was the most commonly used number system, due to humans having five fingers on each hand, and ten in both.

Related Worksheets

10 and 100 More than a 3-digit Number

Number Systems

Attend this Quiz & Test your knowledge.

Which number from the decimal number system does the letter A represent in the Hexadecimal Number system?

Which of the following is not used to represent numbers in the hexadecimal number system, how many unique digits does the decimal number system use to represent all the numbers, the binary system uses which two numbers from the decimal number system.

What is the most commonly used number system?

The most commonly used number system is the decimal positional numeral system.

What number systems do computers use?

Computers use decimal, binary, octal , and hexadecimal number systems.

Which number system uses letters?

The hexadecimal number system uses 6 letters (A, B, C, D, E, and F) in addition to 10 digits from 0 to 9.

What is the base of the hexadecimal number system?

The base of the hexadecimal number system is 16.

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Class 10 Mathematics Number System Assignments

We have provided below free printable Class 10 Mathematics Number System Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 10 Mathematics Number System . These Assignments for Grade 10 Mathematics Number System cover all important topics which can come in your standard 10 tests and examinations. Free printable Assignments for CBSE Class 10 Mathematics Number System , school and class assignments, and practice test papers have been designed by our highly experienced class 10 faculty. You can free download CBSE NCERT printable Assignments for Mathematics Number System Class 10 with solutions and answers. All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Number System Class 10. Students can click on the links below and download all Pdf Assignments for Mathematics Number System class 10 for free. All latest Kendriya Vidyalaya Class 10 Mathematics Number System Assignments with Answers and test papers are given below.

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Class 10 Mathematics Number System Assignments

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  • NCERT Solutions
  • NCERT Class 9
  • NCERT 9 Maths
  • Chapter 1: Number Systems

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Ncert solutions class 9 maths chapter 1 – cbse free pdf download.

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Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 1 Number Systems

Download most important questions for class 9 maths chapter – 1 number systems.

In NCERT Solutions for Class 9 Maths Chapter 1 , students are introduced to several important topics that are considered to be very crucial for those who wish to pursue Mathematics as a subject in their higher classes. Based on these NCERT Solutions , students can practise and prepare for their upcoming CBSE exams, as well as equip themselves with the basics of Class 10. These Maths Solutions of NCERT Class 9 are helpful as they are prepared with respect to the latest update on the CBSE syllabus for 2023-24 and its guidelines.

  • Chapter 1- Number Systems
  • Chapter 2 Polynomials
  • Chapter 3 Coordinate Geometry
  • Chapter 4 Linear Equations in Two Variables
  • Chapter 5 Introduction to Euclids Geometry
  • Chapter 6 Lines and Angles
  • Chapter 7 Triangles
  • Chapter 8 Quadrilaterals
  • Chapter 9 Areas of Parallelograms and Triangles
  • Chapter 10 Circles
  • Chapter 11 Constructions
  • Chapter 12 Heron’s Formula
  • Chapter 13 Surface Areas and Volumes
  • Chapter 14 Statistics
  • Chapter 15 Inroduction to Probability

NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

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ncert solutions for class 9 maths april05 chapter 1 number system 01

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Access Answers to NCERT Class 9 Maths Chapter 1 – Number Systems

Exercise 1.1 page: 5.

1. Is zero a rational number? Can you write it in the form p/q where p and q are integers and q ≠ 0?

We know that a number is said to be rational if it can be written in the form p/q , where p and q are integers and q ≠ 0.

Taking the case of ‘0’,

Zero can be written in the form 0/1, 0/2, 0/3 … as well as , 0/1, 0/2, 0/3 ..

Since it satisfies the necessary condition, we can conclude that 0 can be written in the p/q form, where q can either be positive or negative number.

Hence, 0 is a rational number.

2. Find six rational numbers between 3 and 4.

There are infinite rational numbers between 3 and 4.

As we have to find 6 rational numbers between 3 and 4, we will multiply both the numbers, 3 and 4, with 6+1 = 7 (or any number greater than 6)

i.e., 3 × (7/7) = 21/7

and, 4 × (7/7) = 28/7. The numbers in between 21/7 and 28/7 will be rational and will fall between 3 and 4.

Hence, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7 are the 6 rational numbers between 3 and 4.

3. Find five rational numbers between 3/5 and 4/5.

There are infinite rational numbers between 3/5 and 4/5.

To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5

with 5+1=6 (or any number greater than 5)

i.e., (3/5) × (6/6) = 18/30

and, (4/5) × (6/6) = 24/30

The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.

Hence,19/30, 20/30, 21/30, 22/30, 23/30 are the 5 rational numbers between 3/5 and 4/5

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

Natural numbers- Numbers starting from 1 to infinity (without fractions or decimals)

i.e., Natural numbers = 1,2,3,4…

Whole numbers – Numbers starting from 0 to infinity (without fractions or decimals)

i.e., Whole numbers = 0,1,2,3…

Or, we can say that whole numbers have all the elements of natural numbers and zero.

Every natural number is a whole number; however, every whole number is not a natural number.

(ii) Every integer is a whole number.

Integers- Integers are set of numbers that contain positive, negative and 0; excluding fractional and decimal numbers.

i.e., integers= {…-4,-3,-2,-1,0,1,2,3,4…}

Whole numbers- Numbers starting from 0 to infinity (without fractions or decimals)

i.e., Whole numbers= 0,1,2,3….

Hence, we can say that integers include whole numbers as well as negative numbers.

Every whole number is an integer; however, every integer is not a whole number.

(iii) Every rational number is a whole number.

Rational numbers- All numbers in the form p/q, where p and q are integers and q≠0.

i.e., Rational numbers = 0, 19/30 , 2, 9/-3, -12/7…

All whole numbers are rational, however, all rational numbers are not whole numbers.

Exercise 1.2 Page: 8

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Every irrational number is a real number, however, every real number is not an irrational number.

(ii) Every point on the number line is of the form √m where m is a natural number.

The statement is false since as per the rule, a negative number cannot be expressed as square roots.

E.g., √9 =3 is a natural number.

But √2 = 1.414 is not a natural number.

Similarly, we know that there are negative numbers on the number line, but when we take the root of a negative number it becomes a complex number and not a natural number.

E.g., √-7 = 7i, where i = √-1

The statement that every point on the number line is of the form √m, where m is a natural number is false.

(iii) Every real number is an irrational number.

The statement is false. Real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.

Every irrational number is a real number, however, every real number is not irrational.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

No, the square roots of all positive integers are not irrational.

For example,

√4 = 2 is rational.

√9 = 3 is rational.

Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).

3. Show how √5 can be represented on the number line.

Step 1: Let line AB be of 2 unit on a number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit.

Step 3: Join CA

Step 4: Now, ABC is a right angled triangle. Applying Pythagoras theorem,

AB 2 +BC 2 = CA 2

2 2 +1 2 = CA 2 = 5

⇒ CA = √5 . Thus, CA is a line of length √5 unit.

Step 4: Taking CA as a radius and A as a center draw an arc touching

the number line. The point at which number line get intersected by

arc is at √5 distance from 0 because it is a radius of the circle

whose center was A.

Thus, √5 is represented on the number line as shown in the figure.

Ncert solution class 9 chapter 1-1

4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP 1 of unit length (see Fig. 1.9). Now draw a line segment P 2 P 3 perpendicular to OP 2 . Then draw a line segment P 3 P 4 perpendicular to OP 3 . Continuing in Fig. 1.9 :

Ncert solution class 9 chapter 1-2

Constructing this manner, you can get the line segment P n-1 Pn by square root spiral drawing a line segment of unit length perpendicular to OP n-1 . In this manner, you will have created the points P 2 , P 3 ,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …

Ncert solution class 9 chapter 1-3

Step 1: Mark a point O on the paper. Here, O will be the center of the square root spiral.

Step 2: From O, draw a straight line, OA, of 1cm horizontally.

Step 3: From A, draw a perpendicular line, AB, of 1 cm.

Step 4: Join OB. Here, OB will be of √2

Step 5: Now, from B, draw a perpendicular line of 1 cm and mark the end point C.

Step 6: Join OC. Here, OC will be of √3

Step 7: Repeat the steps to draw √4, √5, √6….

Exercise 1.3 Page: 14

1. Write the following in decimal form and say what kind of decimal expansion each has :

NCERT Solution For Class 9 Maths Ex-1.3-1

= 0.36 (Terminating)

NCERT Solution For Class 9 Maths Ex-1.3-2

= 4.125 (Terminating)

NCERT Solution For Class 9 Maths Ex-1.3-4

(vi) 329/400

NCERT Solution For Class 9 Maths Ex-1.3-6

= 0.8225 (Terminating)

2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of 1/7 carefully.]

Ncert solution class 9 chapter 1-9

3. Express the following in the form p/q, where p and q are integers and q 0.

Ncert solution class 9 chapter 1-10

Assume that   x  = 0.666…

Then,10 x  = 6.666…

10 x  = 6 +  x

(ii) \(\begin{array}{l}0.4\overline{7}\end{array} \)

= (4/10)+(0.777/10)

Assume that  x  = 0.777…

Then, 10 x  = 7.777…

10 x  = 7 +  x

(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 )

= (36/90)+(7/90) = 43/90

Ncert solution class 9 chapter 1-14

Assume that   x  = 0.001001…

Then, 1000 x  = 1.001001…

1000 x  = 1 +  x

4. Express 0.99999…. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Assume that x  = 0.9999…..Eq (a)

Multiplying both sides by 10,

10 x  = 9.9999…. Eq. (b)

Eq.(b) – Eq.(a), we get

10 x  = 9.9999

– x  = -0.9999…

_____________

The difference between 1 and 0.999999 is 0.000001 which is negligible.

Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the answer can be justified.

5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer.

Dividing 1 by 17:

NCERT Solution For Class 9 Maths Ex-1.3-7

There are 16 digits in the repeating block of the decimal expansion of 1/17.

6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

We observe that when q is 2, 4, 5, 8, 10… Then the decimal expansion is terminating. For example:

1/2 = 0. 5, denominator q = 2 1

7/8 = 0. 875, denominator q =2 3

4/5 = 0. 8, denominator q = 5 1

We can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both.

7. Write three numbers whose decimal expansions are non-terminating non-recurring.

We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:

  • √3 = 1.732050807568
  • √26 =5.099019513592
  • √101 = 10.04987562112

8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Ncert solution class 9 chapter 1-17

Three different irrational numbers are:

  • 0.73073007300073000073…
  • 0.75075007300075000075…
  • 0.76076007600076000076…

9.  Classify the following numbers as rational or irrational according to their type:

√23 = 4.79583152331…

Since the number is non-terminating and non-recurring therefore, it is an irrational number.

√225 = 15 = 15/1

Since the number can be represented in p/q form, it is a rational number.

(iii) 0.3796

Since the number,0.3796, is terminating, it is a rational number.

(iv) 7.478478

The number,7.478478, is non-terminating but recurring, it is a rational number.

(v) 1.101001000100001…

Since the number,1.101001000100001…, is non-terminating non-repeating (non-recurring), it is an irrational number.

Exercise 1.4 Page: 18

1. Visualise 3.765 on the number line, using successive magnification.

Ncert solutions class 9 chapter 1-18

Exercise 1.5 Page: 24

1. Classify the following numbers as rational or irrational:

We know that, √5 = 2.2360679…

Here, 2.2360679…is non-terminating and non-recurring.

Now, substituting the value of √5 in 2 –√5, we get,

2-√5 = 2-2.2360679… = -0.2360679

Since the number, – 0.2360679…, is non-terminating non-recurring, 2 –√5 is an irrational number.

(ii) (3 +√23)- √23

(3 + √ 23) –√23 = 3+ √ 23–√23

Since the number 3/1 is in p/q form, ( 3 +√23)- √23 is rational.

(iii) 2√7/7√7

2√7/7√7 = ( 2/7)× (√7/√7)

We know that (√7/√7) = 1

Hence, ( 2/7)× (√7/√7) = (2/7)×1 = 2/7

Since the number, 2/7 is in p/q form, 2√7/7√7 is rational.

Multiplying and dividing numerator and denominator by √2 we get,

(1/√2) ×(√2/√2)= √2/2 ( since √2×√2 = 2)

We know that, √2 = 1.4142…

Then, √2/2 = 1.4142/2 = 0.7071..

Since the number , 0.7071..is non-terminating non-recurring, 1/√2 is an irrational number.

We know that, the value of = 3.1415

Hence, 2 = 2×3.1415.. = 6.2830…

Since the number, 6.2830…, is non-terminating non-recurring, 2 is an irrational number.

2. Simplify each of the following expressions:

(i) (3+√3)(2+√2)

(3+√3)(2+√2 )

Opening the brackets, we get, (3×2)+(3×√2)+(√3×2)+(√3×√2)

= 6+3√2+2√3+√6

(ii) (3+√3)(3-√3 )

(3+√3)(3-√3 ) = 3 2 -(√3) 2 = 9-3

(iii) (√5+√2) 2

(√5+√2) 2 = √5 2 +(2×√5×√2)+ √2 2

= 5+2×√10+2 = 7+2√10

(iv) (√5-√2)(√5+√2)

(√5-√2)(√5+√2) = (√5 2 -√2 2 ) = 5-2 = 3

3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter, (say d). That is, π =c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

There is no contradiction. When we measure a value with a scale, we only obtain an approximate value. We never obtain an exact value. Therefore, we may not realize whether c or d is irrational. The value of π is almost equal to 22/7 or 3.142857…

4. Represent (√9.3) on the number line.

Step 1: Draw a 9.3 units long line segment, AB. Extend AB to C such that BC=1 unit.

Step 2: Now, AC = 10.3 units. Let the centre of AC be O.

Step 3: Draw a semi-circle of radius OC with centre O.

Step 4: Draw a BD perpendicular to AC at point B intersecting the semicircle at D. Join OD.

Step 5: OBD, obtained, is a right angled triangle.

Here, OD 10.3/2 (radius of semi-circle), OC = 10.3/2 , BC = 1

OB = OC – BC

⟹ (10.3/2)-1 = 8.3/2

Using Pythagoras theorem,

OD 2 =BD 2 +OB 2

⟹ (10.3/2) 2 = BD 2 +(8.3/2) 2

⟹ BD 2 = (10.3/2) 2 -(8.3/2) 2

⟹ (BD) 2 = (10.3/2)-(8.3/2)(10.3/2)+(8.3/2)

⟹ BD 2  = 9.3

⟹ BD =  √9.3

Thus, the length of BD is √9.3.

Step 6: Taking BD as radius and B as centre draw an arc which touches the line segment. The point where it touches the line segment is at a distance of √9.3 from O as shown in the figure.

Ncert solutions class 9 chapter 1-21

5. Rationalize the denominators of the following:

Multiply and divide 1/√7 by √7

(1×√7)/(√7×√7) = √7/7

(ii) 1/(√7-√6)

Multiply and divide 1/(√7-√6) by (√7+√6)

= (√7+√6)/√7 2 -√6 2 [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+√6)/(7-6)

= (√7+√6)/1

(iii) 1/(√5+√2)

Multiply and divide 1/(√5+√2) by (√5-√2)

= (√5-√2)/(√5 2 -√2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√5-√2)/(5-2)

= (√5-√2)/3

(iv) 1/(√7-2)

Multiply and divide 1/(√7-2) by (√7+2)

1/(√7-2)×(√7+2)/(√7+2) = (√7+2)/(√7-2)(√7+2)

= (√7+2)/(√7 2 -2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+2)/(7-4)

Exercise 1.6 Page: 26

64 1/2 = (8×8) 1/2

= 8 1 [⸪2×1/2 = 2/2 =1]

32 1/5 = (2 5 ) 1/5

= 2 1 [⸪5×1/5 = 1]

(iii)125 1/3

(125) 1/3 = (5×5×5) 1/3

= 5 1 (3×1/3 = 3/3 = 1)

9 3/2 = (3×3) 3/2

= (3 2 ) 3/2

= 3 3 [⸪2×3/2 = 3]

(ii) 32 2/5

32 2/5 = (2×2×2×2×2) 2/5

= (2 5 ) 2⁄5

= 2 2 [⸪5×2/5= 2]

(iii)16 3/4

16 3/4 = (2×2×2×2) 3/4

= (2 4 ) 3⁄4

= 2 3 [⸪4×3/4 = 3]

(iv) 125 -1/3

125 -1/3 = (5×5×5) -1/3

= (5 3 ) -1⁄3

= 5 -1 [⸪3×-1/3 = -1]

3. Simplify :

(i) 2 2/3 ×2 1/5

2 2/3 ×2 1/5 = 2 (2/3)+(1/5) [⸪Since, a m ×a n =a m+n ____ Laws of exponents]

= 2 13/15 [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

(ii) (1/3 3 ) 7

(1/3 3 ) 7 = (3 -3 ) 7 [⸪Since,(a m ) n = a m x n ____ Laws of exponents]

(iii) 11 1/2 /11 1/4

11 1/2 /11 1/4 = 11 (1/2)-(1/4)

= 11 1/4 [⸪(1/2) – (1/4) = (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]

(iv) 7 1/2 ×8 1/2

7 1/2 ×8 1/2 = (7×8) 1/2 [⸪Since, (a m ×b m = (a×b) m ____ Laws of exponents]

As the Number System is one of the important topics in Maths, it has a weightage of 8 marks in Class 9 Maths CBSE exams. On an average three questions are asked from this unit.

  • One out of three questions in part A (1 marks).
  • One out of three questions in part B (2 marks).
  • One out of three questions in part C (3 marks).

This chapter talks about:

  • Introduction of Number Systems
  • Irrational Numbers
  • Real Numbers and their Decimal Expansions
  • Representing Real Numbers on the Number Line.
  • Operations on Real Numbers
  • Laws of Exponents for Real Numbers

List of Exercises in NCERT Solutions for Class 9 Maths Chapter 1:

Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)

Exercise 1.2 Solutions 4 Questions ( 3 long, 1 short)

Exercise 1.3 Solutions 9 Questions ( 9 long)

Exercise 1.4 Solutions 2 Questions ( 2 long)

Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)

Exercise 1.6 Solutions 3 Questions ( 3 long)

NCERT Solutions for Class 9 Maths Chapter 1- Number Systems

NCERT Solutions for Class 9 Maths Chapter 1 Number System is the first chapter of Class 9 Maths. The Number System is discussed in detail in this chapter. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers.

The chapter starts with the introduction of Number Systems in section 1.1, followed by two very important topics in sections 1.2 and 1.3

  • Irrational Numbers – The numbers which can’t be written in the form of p/q.
  • Real Numbers and their Decimal Expansions – Here, you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrational.

Next, it discusses the following topics.

  • Representing Real Numbers on the Number Line – In this, the solutions for 2 problems in Exercise 1.4.
  • Operations on Real Numbers – Here, you explore some of the operations like addition, subtraction, multiplication and division on irrational numbers.
  • Laws of Exponents for Real Numbers – Use these laws of exponents to solve the questions.

Explore more about Number Systems and learn how to solve various kinds of problems only on  NCERT Solutions For Class 9 Maths . It is also one of the best academic resources to revise for your CBSE exams.

Key Advantages of NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

  • These NCERT Solutions for Class 9 Maths help you solve and revise the whole CBSE syllabus of Class 9.
  • After going through the step-wise solutions given by our subject expert teachers, you will be able to score more marks in the board exams.
  • It follows NCERT guidelines.
  • It contains all the important questions from the examination point of view.

The faculty have curated the solutions in a lucid manner to improve the problem-solving abilities of the students. For a more clear idea about Number Systems, students can refer to the study materials available at BYJU’S.

  • RD Sharma Solutions for Class 9 Maths Number Systems

Disclaimer: 

Dropped Topics – 1.4 Representing real numbers on the number line.

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Assignment - Number System, Class 9 Math PDF Download

TRUE / FALSE TYPE QUESTION 1. The sum of two rational numbers in rational 2. The sum of two irrational numbers is irrational. 3. The product of two rational numbers is rational. 4. The product of two irrational numbers is irrational. 5. The sum of a rational number and an irrational number is irrational. 6. The product of a non zero rational number and an irrational number is a rational number. 7. Every real number is rational. 8. π is irrational and 22/7 is rational. 9. Every rational number must be a whole number. 10. The number zero is both positive and negative. 11. The sum of two prime numbers is always even. 12. The product of two odd numbers is always odd.

number system assignment

VERY SHORT ANSWER TYPE QUESTION

22. Express the following in the form of p/q.

number system assignment

23. Examine, whether the following numbers are rational or an irrational :

number system assignment

24. Write two irrational numbers between 0.2 and 0.21. 25. Write three irrational numbers between 0.202002000200002.... and 0.203003000300003.... . 26. (i) Write two irrational numbers between 0.21 and 0.2222.... (ii) Find three different irrational numbers between the rational numbers 5/7 and 5/9 27. Write three irrational numbers between √3 and √5 . 28. Find two irrational numbers between 0.5 and 0.55.

number system assignment

Express as a pure surd :-

number system assignment

Express each of the following as a mixed surd :-

number system assignment

45. Simplify :-

number system assignment

50. Find the rationalising factor of

number system assignment

SHORT ANSWER TYPE QUESTION

TRUE & FALSE TYPE QUESTIONS 1. True

FILL IN THE BALANKS

13. Real, Rational number, irrational number

14. Terminating, recurring

16. Rational

number system assignment

20. 718/1665

21. Rational, irrational

VERY SHORT ANSWERS

22. (i) 1/3   (ii) 37/99    (iii) 6/11   (iv) 5/99    (v) 4/3    (vi) 23/37      (vii) 311/99    (viii) 9/55    (ix) 49/37    (x)295/9

23. (i) Irrational (ii) Rational (iii) Irrational (iv)Irrational (v) Irrational (vi)Irrational (vii) Rational (viii) Irrational (ix) Irrational (x) Irrational (xi) Irrational

24. 0.2010010001...., 0.2020020002.... 25. 0.20201001000100001...., 0.202020020002..., 0.202030030003.... 26. (i) 0.21010010001.... , 0.21020020002..... (ii) 0.2010010001 ............., 0.2020020002 ..........., 27. 1.8010010001....., 1.9010010001..., 2.010010001...

28. 0.501001.001..., 0.5020020002..... 29. 0.101001000100001, 0.1020020002....

number system assignment

47. 33 

number system assignment

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number system assignment

 (a) 1⁸ × 3⁰ × 5³ × 2²           Ans; 500,                

(b)  4 -3 × 4⁸ ÷ 4²                Ans; 64

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(d) 3 -4 × 3 -5 ÷ 3¹⁰             Ans; 3 -19

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  • Crack Number System Assignments: Essential Concepts & Strategies

Essential Concepts and Strategies for Number Systems Assignment Success

Judith Lawrence

Embark on a journey into the heart of mathematics as we explore the essential concepts and strategies you need to conquer assignments focused on completing your number system assignment . From unraveling different number bases to deciphering prime numbers and modular arithmetic, this guide equips you with the knowledge and techniques to excel in your number system assignments.

In the realm of mathematics, few concepts are as fundamental as number systems. These systems provide the very building blocks upon which mathematics is constructed, serving as a canvas for complex calculations, equations, and problem-solving. Before embarking on an assignment centered around number systems, it's crucial to establish a solid grasp of the foundational topics that will pave the way for a successful journey. Let's delve into the key topics you should acquaint yourself with and unravel the strategies to conquer number system assignments.

Essential Concepts and Strategies for Number Systems Assignment Success

Understanding Number Bases and Notations

The very first step in tackling a number system assignment is comprehending different number bases and notations. The most familiar is the decimal system (base-10), where numbers are represented using ten digits (0-9). However, various other bases like binary (base-2), octal (base-8), and hexadecimal (base-16) are equally important. Binary is used extensively in computer science, octal is employed in some programming contexts, and hexadecimal simplifies working with large binary numbers.

Conversion Techniques between Bases

Being able to convert numbers between different bases is a crucial skill. Converting from one base to another involves understanding the place value system and performing successive division or multiplication. For instance, converting a binary number to decimal requires multiplying each digit by the corresponding power of 2 and summing the results. Practice converting numbers between bases to build confidence in this essential skill.

Prime Numbers and Divisibility Rules

Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. Understanding prime numbers and their properties is essential as they play a significant role in number theory and cryptography. Additionally, knowing divisibility rules (such as those for 2, 3, 5, 9, and 11) helps quickly identify whether a number is divisible by another, aiding in simplifying calculations.

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

GCD and LCM are fundamental concepts in number theory. GCD is the largest positive integer that divides two or more numbers without leaving a remainder. LCM, on the other hand, is the smallest multiple that is evenly divisible by two or more numbers. These concepts are essential when dealing with fractions, simplification, and various mathematical operations.

Rational and Irrational Numbers

Diving deeper into number classification, understanding rational and irrational numbers is crucial. Rational numbers can be expressed as fractions of integers, while irrational numbers cannot be expressed as fractions and have non-terminating, non-repeating decimal expansions. Familiarity with these concepts helps when dealing with real numbers and their properties.

Modular Arithmetic

Modular arithmetic involves working with remainders when dividing one number by another, often represented as "a mod n." This concept finds applications in cryptography, computer science, and various mathematical puzzles. It's vital for understanding cyclic patterns and congruences in number systems.

Strategies for Solving Number System Assignments

Armed with the foundational knowledge of number systems, you're now equipped to tackle assignments with confidence. Here are some strategies to streamline your approach:.

Read the Assignment Thoroughly

Before diving into any number system assignment, it's crucial to begin by thoroughly reading and understanding the assignment prompt. This initial step sets the tone for your entire approach. Take the time to carefully analyze each aspect of the assignment, from the problem statements to any accompanying instructions. By doing so, you ensure that you have a clear grasp of what is expected of you.

Reading the assignment thoroughly helps you identify the specific tasks and questions you need to address. It prevents the risk of misinterpreting the requirements and subsequently providing incorrect or irrelevant solutions. Additionally, understanding the scope of the assignment enables you to allocate time and effort appropriately to different sections.

Remember, a solid foundation is built upon a strong understanding of the assignment. It's the compass that guides you through the complexities of number system problems. So, take the extra moments to absorb the details, and you'll be better prepared to tackle each challenge with precision and confidence.

Organize Your Approach

Organizing your approach is akin to mapping out a journey before embarking on it. Once you've thoroughly understood the assignment, take a moment to strategize how you'll tackle the problems at hand. Start by identifying the order in which you'll address different sections or questions. This simple step can save you valuable time and prevent unnecessary confusion.

By having a clear plan, you're less likely to feel overwhelmed by the assignment's complexity. Breaking down the assignment into manageable steps allows you to allocate time effectively and avoid rushing through problems haphazardly.

Creating an organized approach also enhances the clarity of your thought process. It enables you to focus on each problem individually while keeping the big picture in mind. This approach not only reduces the chances of errors but also showcases a structured problem-solving methodology, making it easier for instructors to follow your thought process.

In essence, organizing your approach is like constructing a roadmap to success. It ensures that you navigate the assignment's challenges methodically, efficiently utilizing your skills and knowledge to yield accurate and well-presented solutions.

Review Relevant Concepts

Before delving into solving each problem in your number system assignment, take a moment to refresh your memory on the relevant concepts. This step is vital as it ensures that you are approaching the problem with a clear understanding of the underlying principles. Whether it's modular arithmetic, conversion between number bases, or identifying prime numbers, a quick review can help you avoid mistakes and confidently apply the correct methodology.

By revisiting the key concepts, you solidify your foundation and avoid pitfalls that can arise from misunderstanding or misapplying ideas. This practice also aids in recognizing patterns and connections between different problems, enabling you to adapt your knowledge to various scenarios. Moreover, it demonstrates your commitment to precision and accuracy in your work.

Incorporating this review step into your approach not only ensures accurate solutions but also enhances your problem-solving skills. As you tackle each question with a fresh perspective on the associated concepts, you'll find that your ability to connect theoretical knowledge with practical applications improves, contributing to a more thorough and insightful assignment.

Show Your Work

The adage "show your work" is a timeless piece of advice that holds immense value when tackling number system assignments. This principle encapsulates the idea that simply presenting the final answer isn't sufficient; showcasing the steps and calculations that lead to the solution is equally crucial.

When you show your work, you provide a clear roadmap of your thought process, allowing your instructor to assess not only the correctness of your answer but also your understanding of the concepts involved. This transparency aids in pinpointing any errors or misconceptions you might have encountered along the way.

Moreover, showing your work is a powerful tool for self-evaluation. As you lay out each step, you give yourself the opportunity to catch mistakes before they snowball into larger inaccuracies. Additionally, it allows you to identify areas where you might need to revisit certain concepts for a stronger grasp.

Ultimately, showing your work showcases your dedication to precision, your logical thinking, and your commitment to thoroughness. It's an embodiment of your problem-solving process, demonstrating not only your mathematical abilities but also your ability to communicate and justify your methods. This practice not only enhances the quality of your assignments but also contributes to your overall growth as a proficient mathematician.

Double-Check Calculations

Double-checking calculations is an integral step in ensuring the accuracy of your solutions when dealing with number system assignments. In mathematical endeavors, even the smallest errors can lead to vastly different outcomes, making the double-checking process invaluable. This practice involves revisiting each step of your calculations, verifying the arithmetic operations performed, and confirming the accuracy of intermediate and final results.

When double-checking calculations, focus on signs, exponents, and place values, especially when working with various number bases like binary or hexadecimal. This attention to detail helps mitigate potential errors that might arise due to the complexity of number systems. By taking the time to review your work meticulously, you not only enhance the correctness of your solutions but also demonstrate a commitment to precision and a mastery of the mathematical concepts at hand.

Seek Clarity

In the realm of number system assignments, seeking clarity is akin to shining a light on the path to understanding. When encountering complex or ambiguous problems, don't hesitate to reach out for guidance and clarification. This practice serves multiple purposes: it demonstrates your active engagement with the assignment, showcases your willingness to learn, and ensures that you're interpreting the problem correctly.

By seeking clarity, you avoid the pitfalls of misinterpretation that can lead you down the wrong path and ultimately result in incorrect solutions. Reach out to your instructor or peers to discuss any uncertainties, ensuring that you have a firm grasp of the problem's requirements before embarking on solving it. Seeking clarity not only enhances the quality of your work but also nurtures effective communication skills – a valuable asset in both academic and professional settings. Remember, asking questions is a sign of wisdom, and in the world of number systems, clarity is key to conquering complex assignments.

In the vast expanse of mathematics, number systems form the bedrock upon which countless mathematical concepts are built. As you embark on your journey into the realm of number systems, remember the importance of understanding different number bases, conversion techniques, prime numbers, divisibility rules, GCD and LCM, rational and irrational numbers, and modular arithmetic. Armed with this foundational knowledge and armed with problem-solving strategies, you'll find yourself well-prepared to conquer any number system assignment that comes your way.

So, as you sit down to tackle your next number system assignment, approach it with confidence, a clear understanding of the concepts, and a systematic problem-solving approach. With each problem you solve, you're not just completing an assignment – you're deepening your understanding of the mathematical universe.

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  • School Guide
  • Mathematics
  • Number System and Arithmetic
  • Trigonometry
  • Probability
  • Mensuration
  • Maths Formulas
  • Class 8 Maths Notes
  • Class 9 Maths Notes
  • Class 10 Maths Notes
  • Class 11 Maths Notes
  • Class 12 Maths Notes
  • Are negative decimals rational numbers?
  • Is the square root of 12 an irrational number?
  • Square Root Symbol
  • Can negative numbers be rational numbers?
  • What is a Whole Number?
  • How do you know if a radical is rational or irrational?
  • How to write a rational number as a repeating decimal?
  • How to write 1.5 billion in numbers?
  • How to know if a function is positive/negative and how to know if it's fractional?
  • How many numbers are there between 1 and 1,000 both inclusive?
  • Is the quotient of two integers always a rational number?
  • Are all integers rational numbers?
  • How many Zeros are there in 10 Crores?
  • How to subtract a fraction from a whole number?
  • Use a proof by contradiction to show that there is no rational number r for which r 3 + r + 1 = 0
  • Prove or disprove that if r is rational and s is rational, then r + s must be rational
  • Is 0 a Natural Number?
  • Adding and Subtract Rational Numbers with Negatives

Binary Number System

Binary Number System is a number system that is used to represent various numbers using only two symbols “0” and “1”. The word binary is derived from the word “bi” which means two. Hence, this number system is called Binary Number System. Thus, the binary number system is a system that has only two symbols.

There are generally various types of number systems and among them the four major ones are,

  • Binary Number System (Number system with Base 2)
  • Octal Number System (Number system with Base 8)
  • Decimal Number System (Number system with Base 10)
  • Hexadecimal Number System (Number system with Base 16)

Classification of Number System

Here, we are only going to learn about Binary Number System. This number system is very useful for explaining tasks to the computer. In the Binary Number System, we have two states “0” and “1” and these two states are represented by two states of a transistor. If the current passes through the transistor then the computer reads “1” and if the current is absent from the transistor then it read “0”. Thus, alternating the current the computer reads the binary number system. Each digit in the binary number system is called a “bit”. 

In this article, we will learn about the Binary Number System, the Conversion of the Binary Number System, the Binary Table, the Operation of Binary Numbers, Examples, and others in detail.

Binary Number System is the number system in which we use two digits “0” and “1” to perform all the necessary operations. In the Binary Number System, we have a base of 2. The base of the Binary Number System is also called the radix of the number system .

In a binary number system, we represent the number as,

In the above example, a binary number is given in which the base is 2. In a binary number system, each digit is called the “bit”. In the above example, there are 5 digits.

Binary Number Table

Binary to decimal conversion.

A binary number is converted into a decimal number by multiplying each digit of the binary number by the power of either 1 or 0 to the corresponding power of 2. Let us consider that a binary number has n digits, B = a n-1 …a 3 a 2 a 1 a 0 . Now, the corresponding decimal number is given as

D = (a n-1 × 2 n-1 ) +…+(a 3 × 2 3 ) + (a 2 × 2 2 ) + (a 1 × 2 1 ) + (a 0 × 2 0 )

Let us go through an example to understand the concept better.

Example: Convert (10011) 2 to a decimal number.

The given binary number is (10011) 2 . (10011) 2 = (1 × 2 4 ) + (0 × 2 3 ) + (0 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 ) = 16 + 0 + 0 + 2 + 1 = (19) 10 Hence, the binary number (10011) 2 is expressed as (19) 10 .

Decimal to Binary Conversion

A decimal number is converted into a binary number by dividing the given decimal number by 2 continuously until we get the quotient as 1, and we write the numbers from downwards to upwards.

Example: Convert (28) 10 into a binary number.

  Hence, (28) 10 is expressed as (11100) 2 .

Arithmetic Operations on Binary Numbers

We can easily perform various operations on Binary Numbers. Various arithmetic operations on the Binary number include,

Binary Addition

Binary subtraction, binary multiplication, binary division.

Now let’s learn about the same in detail.

The result of the addition of two binary numbers is also a binary number. To obtain the result of the addition of two binary numbers, we have to add the digit of the binary numbers by digit. The table added below shows the rule of binary addition.

The result of the subtraction of two binary numbers is also a binary number. To obtain the result of the subtraction of two binary numbers, we have to subtract the digit of the binary numbers by digit. The table added below shows the rule of binary subtraction.

The multiplication process of binary numbers is similar to the multiplication of decimal numbers. The rules for multiplying any two binary numbers are given in the table,

The division method for binary numbers is similar to that of the decimal number division method. Let us go through an example to understand the concept better.

Example: Divide (101101) 2 by (110) 2

1’s and 2’s Complement of a Binary Number

  • 1’s Complement of a Binary Number is obtained by inverting the digits of the binary number.

Example: Find the 1’s complement of (10011) 2 .

Given Binary Number is (10011) 2 Now, to find its 1’s complement, we have to invert the digits of the given number. Thus, 1’s complement of (10011) 2 is (01100) 2
  • 2’s Complement of a Binary Number is obtained by inverting the digits of the binary number and then by adding 1 to the least significant bit.

Example: Find the 2’s complement of (1011) 2 .

Given Binary Number is (1011) 2 To find the 2’s complement, first find its 1’s complement, i.e., (0100) 2 Now, by adding 1 to the least significant bit, we get (0101) 2 Hence, the 2’s complement of (1011) 2 is (0101) 2

Uses of Binary Number System

Binary Number Systems are used for various purposes and the most important use of the binary number system is,

  • Binary Number System is used in all Digital Electronics for performing various operations.
  • Programming Languages uses  Binary Number System for encoding and decoding data.
  • Binary Number System is used in Data Sciences for various purposes, etc.
Binary Formula Difference Between Decimal and Binary Number Systems Prime Number

Binary Number System Example

Example 1: Convert Decimal Number (98) 10 into Binary.

Solution: 

Thus, Binary Number for (98) 10 is equal to (1100010) 2

Example 2: Convert Binary Number (1010101) 2 to Decimal Number.

Given Binary Number, (1011101) 2 = (1 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 ) + (1 × 2 4 ) + (0 × 2 5 ) + (1 ×2 6 ) = 1 + 0 + 4 + 0 + 16 + 0 + 64 = (85) 10 Thus, Binary Number (1010101) 2 is equal to (85) 10 in decimal system.

Example 3: Divide (11110) 2 by (101) 2

Example 4: Add (11011) 2 and (10100) 2

Hence, (11011) 2 + (10100) 2 =  (101111) 2

Example 5: Subtract (11010) 2 and (10110) 2

Hence, (11010) 2 &#x2013 (10110) 2 = (00100) 2

Example 6: Multiply (1110) 2 and (1001) 2 .

Thus, (1110) 2 × (1001) 2 = (1111110) 2

FAQs on Binary Number System

1. what is a binary number system.

Binary Number System is one of the four number system that is used to represent the numbers using only two digits, “0” and “1”. In binary number system the digits are called ‘bits’. Binary Number System is used by computers to perform various calculations.

2. What is a B it?

A bit in Binary Number System is defined as a individual digits that holds the value ‘0’ or ‘1’.

3. What is a Nibble?

A group of four digits is called the Niblle.

4. What is the Binary Value of 10?

The binary value of 10 is (1010) 2

5. What are Types of Number Systems?

There are various types of number systems and some of them are, Binary Number System Octal Number System Decimal Number System Hexadecimal Number System

6. How to Calculate Binary Numbers?

Binary numbers are calculated from dicmal numbers by dividing the decimal number with 2 and writing the remainder. Then we arrange all the remainders from newest to oldest to get the binary number.

7. How to Add Binary Numbers?

Binary numbers are added by using the formulas written below, 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 (carry 1)

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Assignments For Class 9 Mathematics Number System

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Assignments for Class 9 Mathematics Number System as per CBSE NCERT pattern

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Number System: References Pages

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Following in-text citation of sources, of course, you are obliged to provide bibliographic information about your sources on a references page. Composing a references page is, for many writers, a painful process, particularly if they handled their references sloppily at the research stage. You simplify your task greatly by recording complete bibliographic information of your cited sources as you research, thus building your references page as you go. Some students wisely use notecards to keep track of their references, while others have a less formal system. As I cite sources in-text, I simply keep adding the complete bibliographic information to my references page right in my Word file for the paper; thus my references page is finished as soon as the last paragraph is.

Read up on the specifics of various citation styles, in particular MLA and APA, at the following pages:

"Research and Citation Resources" article from Purdue's Online Writing Lab (OWL)

"Citation Style for Research Papers" article from Long Island University

Mechanics of the Number System References Page

Using the number system, on your references page you typically provide the following information in the following order:

  • The number of the reference , followed by a period.
  • The first initials and last names of all authors , followed by a comma.
  • Title of the article enclosed in quotation marks, followed by a comma.
  • Title of book, magazine, or journal , underlined or italicized, with journal titles typically abbreviated, followed by a period.
  • Volume numbers or editors —if citing a journal or magazine, provide the volume number in boldface, followed by the issue number in brackets; if citing a book with editors or volume numbers, provide the names of the editors or the volume numbers.
  • Publication information —for a book or privately published document, provide the relevant page numbers, then the publisher’s name and location (all separated by commas), then the year in parentheses, followed by a period; for a journal or magazine, provide the relevant page numbers of the article being cited, then the year in parentheses, followed by a period.
  • The entire URL (if the source is a website), usually enclosed in brackets, followed by a period. Then provide either the last date the page was updated or the date that you accessed it, followed by a period. When citing a web document, typical bibliographic details, such as the page’s author, will often be unavailable. Therefore, skip the steps above as needed, but always provide the URL.

At times, some of the above information will be unavailable or sketchy, especially in relation to company brochures, maps, non-professional publications, and web sources. It is acceptable to omit unavailable information, of course, but when less information is available you might provide a short narrative description of a source for clarity.

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Original research article, structure and flexibility: systemic and explicit assignment extensions foster an inclusive learning environment.

number system assignment

  • Investigative Biology Teaching Laboratories, Department of Neurobiology and Behavior, Cornell University, Ithaca, NY, United States

Many educators strive to create inclusive classrooms where students receive not only knowledge but also empathy from their instructors. When students face unexpected challenges due to illness, academic pressure, or exhaustion, they often seek extensions on assignments. Instructors insert their own biases when they decide who is eligible for an extension. An explicitly communicated penalty-free extension system can eliminate this bias, create an inclusive learning environment, and disinter extension requests from the hidden curriculum. Students used an “extension without penalty” system (EWP) in a large introductory biology course. Mid-semester qualitative data collection helped design an end-of-the-semester quantitative survey about students’ perceived benefits. Assignment submission data, EWP use frequency and grades were directly extracted from the learning management system. Students preferred a two-tier extension system with ideal and extension due dates. The EWP system was used by 78% of the students, but half of them only used it once. Students reported benefits in stress reduction, handling of sickness and emergencies, and improved performance in other courses. Exploratory results indicate there were additional benefits in some areas for first-generation college students. Using the extension due dates did not impact student grades. This study uses evidence to debunk common misconceptions about assignment extensions.

1 Introduction

Creating an inclusive classroom where flexibility and structure can both support pedagogical decisions became the focus of the higher education landscape following the COVID-19 pandemic ( Hogan and Sathy, 2022 ; Sarvary et al., 2022 ). The cumulative stress that many students and instructors experienced during the pandemic ( Adedoyin, 2022 ; Jackson et al., 2022 ; Novick et al., 2022 ; Roberts-Grmela, 2023 ) has dramatically influenced policies and processes often referred to as the “new normal” in many higher education institutions ( Schapiro, 2021 ; Sarvary et al., 2022 ; Supiano, 2023 ). These policies responded to the broad range of impacts of the COVID-19 pandemic. In some cases, the pandemic resulted in decreased quality of education available despite the heroic efforts of many educators to transition smoothly and continue to improve an unplanned online classroom ( Dindar et al., 2022 ; Kuhfeld et al., 2022 ). It resulted in an increased mental strain on many during the difficult times brought on by a worldwide event that reached deeply into so many homes. Anxiety and depression became more commonplace ( Ma et al., 2022 ; Canet-Juric et al., 2023 ; Klussman et al., 2023 ; Sánchez-Martín et al., 2023 ). Students faced many challenges in online classrooms ( Castelli and Sarvary, 2021 ) and were more likely to feel overwhelmed and seek additional assistance to accomplish their tasks ( Gagné et al., 2022 ; Samji et al., 2022 ). It created increased fear of the future as the recognized stability of circumstances such as employment and safety took on tremulous qualities ( Yin et al., 2022 ). And finally, students lost social and societal capital due to isolation and other aspects of the pandemic that likely decreased their available resources and social network for information and direction but increased the associated costs of communicating with a professor ( Shiyuan et al., 2022 ). All of these situations that occurred during the height of the COVID-19 pandemic continue to impact the student population today ( Sarvary et al., 2022 ).

The education experience of current students is unique and different from previous generations and as such it comes with its own blind spots, pit falls, and hurdles that are difficult to anticipate and overcome ( Kennedy et al., 2022 ). When it comes to addressing students’ needs during these challenging post-pandemic times, instructors consider the students’ circumstances and personal experiences. But how can instructors decide without bias when they should exercise flexibility? Instructors who have never owned pets may not understand someone’s pain of losing one and do not consider that a valid reason for missing a class. Instructors who have never played group sports may be less inclined to give an athlete who missed an assignment because of a game a second chance. Instructors who were impacted by COVID-19, may be more lenient with students who suffer from long-term consequences of this illness than those instructors who never contracted this virus. Instructors may also have implicit bias based on students’ names, pronouns, year in school, etc. In addition, in large introductory classes, the number of these individual requests can easily overwhelm the instructors. Therefore, creating an inclusive learning environment by addressing individual needs can be challenging and the decisions are often influenced by instructor bias and available time.

1.1 Creating an inclusive classroom

Most educational institutions lay claim to having a space where all are welcomed to learn and develop. Policies are enacted to push instructors toward classrooms that bring a sense of belonging to various approaches to learning, to make learners of all nationalities, races, and socio-economic backgrounds feel welcome and to help diverse ideas flourish and grow. But how this is done within the classroom is often left up to the individual instructor ( Hogan and Sathy, 2022 ). Balancing content and proficiency or leniency and rigidity can lead to ambiguity in efforts to help students and leave them with the education they need and paid for ( Oleson, 2021 ). An inclusive classroom demands more than just policy and good intentions. It requires a robust structure that facilitates effective learning and comes with the built-in flexibility to adapt to a myriad of challenges. This will curtail bias and fatigue that can lead to decisions that unintentionally alienate students we are trying to help within the classroom ( Hogan and Sathy, 2022 ). Concepts such as the Universal Design for Learning ( Silver et al., 1998 ) and hidden curriculum ( Alsubaie, 2015 ) can help educators to understand what students need and guide them to build solutions into the course structure that reduce students’ need to request exceptions.

Universal Design for Learning (UDL) is an inclusive approach to the creation of environments and spaces that are to be usable by everyone regardless of ability or status. Its goal is to make solutions that are available to the largest portion of the spectrum of users. If used correctly, it should reduce the need to retroactively modify classroom assignments and structures to accommodate specific populations ( Silver et al., 1998 ). The proactive application of its principles should reduce students’ sense of exclusion and allow for increased participation in activities and assignments, thus increasing student retention and learning. UDL has been seen to positively impact student engagement, interactions, and attitudes ( Belch and Barricelli, 2004 ; Cumming and Rose, 2022 ; Edwards et al., 2022 ). Additionally, design can impact students’ learner-identity relating to inclusion via race ( Chita-Tegmark et al., 2012 ; Griggs and Moore, 2023 ), gender ( Couillard and Higbee, 2018 ), and disability ( Nieminen and Pesonen, 2020 ). Finally, implementation was observed to improve self-efficacy and job satisfaction in teachers ( Katz, 2015 ).

The concept of Self-Determination Theory (SDT) centers around the idea that people’s choices are driven by internal factors rather than external influence ( Deci et al., 1999 ). This theory highlights the significance of an individual’s innate resources in shaping their personality development and self-regulation. According to SDT, creating an environment that promotes autonomy, competence, and relatedness can encourage the highest quality of motivation and engagement in activities. Incorporating these principles into the classroom can yield benefits not only for current students but also for their future classrooms. By fostering intrinsic motivation and implementing effective practices to support it, we can counteract the challenges of shorter attention spans and heightened anxiety ( Ryan and Deci, 2000 ).

In this post-pandemic higher education environment, there is an increasing need to pay close attention and put in a significant amount of effort to truly understand and respond to the diverse circumstances and experiences among individuals ( Luk et al., 2023 ). This diversity consists not only in appearance and culture, but in neurodiversity and mental divergencies, in addition to resources and assistance allotted to individuals ( Cullinan et al., 2021 ; Stark et al., 2022 ). It also includes various circumstances that arise in the daily life of a student that affects the time and attention that can be put toward important tasks that need to be accomplished and can affect the hierarchy of tasks that they strive to accomplish ( Nanath et al., 2022 ; Désiron and Petko, 2023 ). During times of pressure, it can be beneficial for students to allocate some extra time toward nonacademic tasks. This can alleviate mental strain and enhance performance in activities that are meant to cultivate knowledge for future success in one’s professional life ( Mehta et al., 2017 ).

With all of the new or more readily recognized requirements and pressures to assist students, educator duress has become evident ( Bradshaw et al., 2023 ). It is difficult to know how best to reply to a student’s needs and yet those demands are placed frequently upon an instructor’s shoulders ( Agyapong et al., 2022 ; Edwards et al., 2022 ). University requests, such as the ones from Student Disability Services, do not always cover all the needs of a student and balancing affordances for varied circumstances can lead to reduced student learning, inhibition of classroom pedagogy, and the appearance of favoritism ( Watermeyer et al., 2021 ). Addressing best pedagogical practices to assist students in their efforts, while maintaining fair practices for all, should be extensively tested to optimize educational norms ( Hoffman et al., 2019 ). One of the strategies to achieve balance between flexibility and structure in an inclusive learning environment is the application of explicit extensions. Publications on the application of extensions to assignments within the classroom are rare and evidence of its benefits or drawbacks are anecdotal ( Gonzalez, 2019 ; Kuimelis, 2022 ). An application of an explicit extension system, one in which gaining access to the system was not based on request but instead was built into the curriculum, could serve to alleviate both teacher and student stress during educational obstructions.

1.2 What are extensions?

Extensions or extension deadlines are a commonly employed technique to make allowances for any difficulties that may arise during educational content to allow for the completion of goals. An allotment of additional time is given for completion of required work. Methods by which this extension is employed are varied, including by request of the institution, in response to student outreach, on a case-by-case basis, or a preset quantity by course, by assignment, or by the student ( Bosch, 2020 ). All extensions are designed to allow students to still reach learning- and content-based goals within an appropriate timeline but have the potential to introduce bias and additional negative outcomes to those who participate in the extension as well as to those who do not ( Kuimelis, 2022 ).

1.3 Framing extensions within the classroom

When creating an extension for student usage, as educators, we frequently balance the pros and cons of such a system. We worry about developing a system that fails to equip students with the necessary skills for their future. However, we also strive to show empathy and prevent students from falling behind due to overwhelming workloads or inadequate knowledge. Knowing what benefits students take from an extension and how they make use of it would help to guide our actions. Deadlines do not always lead to the desired goal of continued learning. Amabile et al. found instead that creating a deadline caused a reduction of future student attention and motivation toward the desired subject matter ( Amabile et al., 1976 ). By offering an extension due date without penalties, the policy reduces the extrinsic pressures associated with strict deadlines ( Kohn, 1999 ). This could help maintain or even boost the students’ intrinsic motivation, as they would be less likely to view the assignment as a task they are being externally forced to complete and more as something they choose to engage with at their own pace. While the study by Amabile et al. (1976) highlights the demotivating effects of strict deadlines, it does not necessarily imply that all structure is bad. If an assignment submission policy provides a structure (the ideal due date) but also offers flexibility (an extension on the assignment), it potentially offers a balanced approach that supports intrinsic motivation while still providing some guidelines to students.

Hidden curriculum refers to content and resources that are not explicitly stated within a classroom and yet benefit some students ( Bergenhenegouwen, 1987 ; Margolis, 2001 ; Alsubaie, 2015 ). When giving extensions, this comes into play as some may not feel comfortable asking for them (whether that is because they come from a school where interactions with educators were framed differently or they fear discrimination for sharing their circumstances, etc.) and others may not even know they exist (e.g., first-generation college students). In education, we want to create a system that allows access for every single student, and we want to avoid a system that advantages some over others. In this context, a systemic extension rather than a by-request extension may benefit the classroom. Establishing an explicit extension system that requires reduced communication to make use of, can fit well into the structure of Universal Design ( Silver et al., 1998 ; Higbee and Goff, 2008 ) and follow the Self-Determination Pedagogical Theory ( Ryan and Deci, 2000 ). When designing a curriculum with assignment extensions in mind, the goals of grades and learning outcomes should be addressed. Though courses are designed to elevate student knowledge and increase their ability to apply skills outside of the classroom, many students are focused on their Grade Point Average (GPA) at the end of the semester. While extensions may be created with the intent to allow students to complete their work and increase their alignment with the course goals, they may be taken advantage of to seek a higher letter grade. Students in large STEM courses have a lot of ideas and expectations ( Meaders et al., 2020 , 2021 ); therefore, the instructors’ efforts in designing an unbiased extension system should be student-centered and include the analysis of student perceptions and the system’s impact on final grades.

While the UDL, SDT, and the hidden curriculum are well studied concepts, as it is shown above, there is very little known about how unbiased and explicit assignment extensions contribute to the development of an inclusive classroom. Students submitting late work is a challenge for many instructors, and it is an often discussed issue in online forums ( Gonzalez, 2019 ) and in pedagogy workshops. However, the solutions are usually based on anecdotes rather than evidence. When instructors discuss how to handle late assignment submission and provide extensions, there are many opinions and misconceptions. There is a gap in the pedagogy literature to address these opinions and real-world problems using evidence from education research. In this paper, we discuss hypothetical opinions based on the authors’ experience using evidence from our study.

In this study, we designed an unbiased extension system that strives to maximize student learning while enabling them to achieve the desired grades. The system was designed to follow the Universal Design for Learning framework under the umbrella of the Self-Determination Theory and aims to bring extensions out of the hidden curriculum. It gives students a choice in when to submit their assignments—either by the ideal due date or the extension due date. By providing this flexibility, students can feel a greater sense of control over their learning, which can foster intrinsic motivation. Moreover, by allowing students to choose when they submit (within the given timeframe), they can work at their own pace and ensure they understand the material, thereby increasing their feeling of competence in the subject matter ( Ryan and Deci, 2000 ). The absence of penalties for using the extension due date is expected to reduce the fear of failure, which can hinder the sense of competence. Furthermore, while the policy does not directly address relatedness (the need to feel connected to others), by recognizing and accommodating diverse life situations, the instructor communicates understanding and empathy, potentially fostering a sense of connection. Using a mixed-methods analysis with qualitative and quantitative components, a survey instrument was developed to test the hypotheses that well-conducted and explicitly communicated assignment due date extensions will improve the classroom experience and create an inclusive and welcoming learning environment. By designing and employing a system with dual deadlines, we expected to see relatively few perceived negatives while reaping several positives. We also expected no impact on final grades when comparing those who used the extension and those who did not. Specifically, the following questions were explored in this study:

1. What assignment submission deadline system do students prefer?

2. What assignment submission behaviors are practiced when the Extension Without Penalty (EWP) system is deployed in a large introductory biology class?

3. Does the EWP system impact student performance?

4. What are the real and perceived benefits of the EWP system, and do they foster an inclusive learning environment?

2.1 The classroom application of “extension without penalty”

The study was conducted in a large introductory biology laboratory course with a maximum enrollment of 432 students each semester. This inquiry-based laboratory course teaches the scientific process, experimental design, science communication, and statistics using examples from the biological sciences ( Sarvary et al., 2022 ). Both the lecture and laboratory portion of the course have active-learning components ( Asgari et al., 2021 ) and related assignments, including audience participation through Poll Everywhere, an online response system ( Sarvary and Gifford, 2017 ), peer review of written assignments ( Biango-Daniels and Sarvary, 2021 ) and scaffolded exercises to build critical thinking, science literacy ( Sarvary and Ruesch, 2023 ) and transferable skills ( Deane-Coe et al., 2017 ). In Fall 2022, a syllabus was designed that involved an “extension without penalty” (EWP) for many assignments that occurred outside of class and a few that were performed in class. The syllabus explicitly identified a suggested due date by which the students were encouraged to complete the assignment and the EWP due date. The syllabus includes the following statement: “We understand that there can be circumstances when students need more time to complete their assignments. All assignments have ideal due dates, and they also have extension due dates. We highly recommend that you submit the assignments (if you can) by their suggested ideal due dates, to maintain a good rhythm of learning in the class. You can submit assignments by the extension due date without any penalty. We are providing the extension due dates so you can use them for certain times when you have other exams, sickness, or you just simply need a break and you do not want to think about an assignment.” During the early introduction of the system within the classroom, instructors were encouraged to stress the advantages of completing the work by the ideal due date but confirm that the EWP was available for flexibility.

2.2 Student sample

The enrollment count for Fall 2022 was 347 students. In the mid-semester and end-of-semester surveys students were given the option to skip demographic survey questions or choose not to disclose. Students self-reported their gender and year-in-school (First year, Sophomore, Junior, or Senior). Race and ethnicity were also queried, with responses used to categorize students as persons historically excluded from science (PEER) or non-PEER. PEER was defined by identifying as black or African American; Hispanic, Latinx, or Spanish origin; and American Indian or Alaska Native, or a mix including one of these groups ( Asai, 2020 ).

2.3 Survey development, validation, and dissemination

With limited published literature about the application of extensions, no prior surveys could be used to address our research questions. A mixed-method survey design thus began with careful deliberation by the authors to identify student perception of the EWP system and its advantages and disadvantages. Developed items for the survey were validated with the help of undergraduate teaching assistants, who having taken the course, were positioned to understand the clarity of the presented items. Questions were discussed one-by-one in a focus group and unclear questions were identified. Undergraduate teaching assistants who could not be present in person provided feedback via written communication ( Ouimet et al., 2004 ; Vogt et al., 2004 ).

First, open-ended questions for the mid-semester survey were designed to be exploratory, identifying any potential impacts. Using the online survey software Qualtrics, questions were asked of the students during Week 8 of the course, designed to get feedback on course content, student experience, and teaching. This survey included the following open-ended question: “What are your thoughts about the ‘extension without penalty’ due date system?” ( Supplementary material S.2 ). The responses to this question were emergently coded [inductive reasoning ( Saldaña, 2013 )] through several readings ( Fereday and Muir-Cochrane, 2006 ) to develop 19 categories for the end-of-the-semester course evaluation survey ( Figure 1 ). Initial coding was performed by a single party with categorical verification performed by another. Discussions led to the modification of the categories. Once categories had been finalized, all responses were coded by all parties until all classifications were agreed upon (100%). Participation in any of the surveys was performed in class and was voluntary, anonymous, with no credit given for participation.

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Figure 1 . Advantages and Disadvantages of the “Extension Without Penalty” (EWP) system as selected by those who used it. Advantages are uncolored ( n  = 234). The responses highlighted in pink are disadvantages ( n  = 128).

The end-of-semester (EOS) survey in Fall 2022 was conducted using the online survey software Qualtrics and included an initial question dividing the students into groups based on whether they made use of the EWP during the semester. Post this division, students were asked “Choose all the advantages of “extension without penalty” that apply to you” and “Choose all the disadvantages of “extension without penalty” that apply to you.” The categories in the answers were derived from the mid-semester survey ( Figure 1 ). An additional question occurred about the students’ preferences for an assignment submission system with the choices of no due date, one single due date, or a double due date with EWP ( Supplementary material S.1 ). The question about the preferences was repeated in the Spring 2023 semester. Both semesters used a dual assignment deadline system, so the answers for this one question were combined from the two semesters. This study’s proposal was granted exemption from Institutional Review Board review by the University’s Office of Research Integrity and Assurance (2109010595).

2.4 Submission dates and final grade analysis

Assignment submission dates were directly extracted from the Learning Management System (LMS). To assess the impact of EWP use on grades, the total points, points scored, and final grades (percentages) were also downloaded from the LMS. All data were accessed after the final grades were assigned, generalized and de-identified for analysis.

2.5 Statistical analysis

Statistical analyses were conducted with the support of the Cornell Statistical Consulting Unit. A generalized linear model was used to investigate the relationship between extension usage and first-generation college students, gender, PEER, and year-in-school. With extension usage being a binary variable, a logistic regression model was employed. Only primary effects were analyzed in this model. Final grades were analyzed using a linear regression model, with EWP usage, first-generation college students, gender, PEER, and year-in-school as predicter variables. Interaction of EWP usage with the other predictors was emphasized in the analysis. Final grades, EWP usage and survey responses were all analyzed in R statistical software (v.4.3.0) ( Crawley, 2012 ; R Core Team, 2023 ). Pairwise comparisons occurred for survey questions with the alpha level set at 0.05. Chi-squared tests were used to compare differences in survey responses between groups (first generation to non-first generation, men to women with non-binary, PEER to non-PEER, first-year to non-first-year). With the exploratory nature of these pairwise comparisons, we chose not correct for multiple measures ( Bender and Lange, 2001 ).

3.1 Course demographics

In the fall semester of 2022, 347 students were enrolled. Of those enrolled, 97% took the mid-semester survey ( n  = 338). The end-of-semester survey had 318 students who participated. Demographic categories were self-reported, and the choice to not disclose or self-describe could be selected. Results were divided by gender [women ( n  = 195) and men ( n  = 100) with non-binary ( n  = 5)], PEER ( n  = 111) or non-PEER ( n  = 217). Additionally, first generation college students ( n  = 88) were compared to non-first-generation ( n  = 220) and first-year ( n  = 133) and non-first-year [a group consisting of sophomore ( n  = 138), junior ( n  = 30), and seniors ( n  = 10)] were compared.

3.2 Extension without penalty system users

The data downloaded from the LMS indicated 78% usage of the EWP system by students at the end of the semester. Of those who used EWP due dates, the largest percentage made use of it a single time (41%) during the semester on major assignments that required substantial time commitment from the students. A slightly smaller percentage of students (37%) used it for more than one major assignment during the semester. Approximately 22% of the students have not used the EWP system at all and submitted all of the assignments by the ideal due date.

The open-ended mid-semester survey questions were used to create categories for the end-of-the-semester (EOS) survey. In the EOS survey with the categorical questions, the top three most frequently selected advantages for those who used the EWP system ( n  = 234) were “My stress was reduced” (94%), “This made things better for my other courses” (82%) and “I could handle things better when I was sick/had an emergency” (73%). All advantages caused by the EWP system were selected at above 50%, except “I was able to attend office hours/ask help from my TA.” (33%). A smaller proportion of students listed any disadvantages ( n  = 128). All disadvantages identified by those students who used the EWP system fell below 50%. The most frequently selected disadvantage was, “It increased my likelihood to procrastinate my work” (33%) followed by “The system was confusing to me” (25%). For the “Other” option under disadvantages (24%), most chose to self-describe with “None” or otherwise express a lack of disadvantage from the system.

3.3 Students who did not take advantage of the extension due dates

Those who chose not to use the EWP system and submitted all of their assignments by the suggested deadline, had similar selections for their perceived benefits ( n  = 60), with the top three for them being, “My stress was reduced” (78%), “I could handle things better when I was sick/had an emergency” (48%), and “This made things better for my other courses” (42%). While they did not take advantage of the EWP system, they clearly identified perceived advantages. Their perceived disadvantages ( n  = 58) were “The system was confusing to me” (47%), “My grades were returned to me more slowly” (36%), and “It increased my likelihood to procrastinate my work” (33%).

3.4 Final grades and response to extension without penalty

When analyzing the LMS data, there was no significant difference in final grades between those who used the EWP system and those who did not ( p  = 0.157), demonstrating that submitting one or more assignments after the suggested due date did not have a negative effect on student’s final grades in the class ( Figure 2 ).

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Figure 2 . The effect of the use of the Extension Without Penalty (EWP) system on final grades ( n  = 347). No significant difference in the final grade of those who used EWP at least once and those who did not use EWP is found ( p  > 0.05).

3.5 Analysis by first-generation college students, gender, persons historically excluded from science, and year-in-school

Advantages and disadvantages of the EWP system, final grades for students within the course, usage of the EWP system, and selection of their preferred hypothetical classroom were all analyzed by various groups to explore whether a diverse impact occurred among student populations. The inclusion of the demographic variables did not make a significant difference from the null model ( p  = 0.45) for the usage of the EWP system ( Supplementary material S.4 ). The model of final grades with extension usage was not significantly improved ( p  = 0.29) with the inclusion of interaction terms of the other predictors ( Supplementary material S.5 ). The occasions when there were significant differences among the groups were rare. These occasions included an increased percentage of first generation college students that selected “I was able to attend office hours/ask help from my TA” ( p  = 0.044) than non-first-generation students. This was also true of non-first-year ( p  = 0.041) and non-first-years were also more likely to choose “I was able to still complete assignments when I forgot or did not know a due date” ( p  = 0.039). First-year students were more likely to select “It increased my likelihood to procrastinate” ( p  = 0.0070) than non-first-years as a disadvantage of the system. Relatedly, no significant results were found for analysis by gender or PEER ( Supplementary material S.3 ).

3.6 Students’ preferences of no due date, one due date, or dual due date

When given three choices for assignment due dates in a hypothetical classroom (n = 563 students), only 8% of the respondents selected “No deadline for assignments. All assignments are due by the end of semester “, and another 10% chose “A single deadline for each assignment.” The overwhelming majority (82%) selected the system used in the class (normalized χ 2 test: χ 2  = 1735.6, df = 5, p  < 0.01), namely “A double deadline for each assignment, one being the ideal due date, the other being the extension deadline” ( Figure 3 ).

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Figure 3 . Selection of preferred hypothetical classroom by students during the end of semester surveys in two semesters combined. Selection was between “No deadline for assignments. All assignments are due by end of semester” (8%), “A single deadline for each assignment” (10%), and “A double deadline for each assignment, one being the ideal due date, the other being the extension deadline” (82%) ( n  = 563). The normalized Chi-squared test showed a highly significant preference for the EWP system over the other two choices ( p  < 0.01).

4 Discussion

The return to in-person teaching after the pandemic has created many challenges for both instructors and students, as they were adapting to the “new normal” in higher education ( Sarvary et al., 2022 ). While instructors aim to create inclusive classrooms ( Hogan and Sathy, 2022 ), there has been an active discussion about how much structure and flexibility will benefit student learning ( Kuimelis, 2022 ; Supiano, 2023 ). Students returned to the classroom with new expectations and looking for the same levels of empathy they received during the COVID-19 pandemic. In addition, the cumulative impact of social isolation and online learning resulted in a greater need for additional support ( Gagné et al., 2022 ; Samji et al., 2022 ). To meet the need for unbiased systemic support in a large classroom, the authors employed explicitly communicated extension due dates (EWP—Extension Without Penalty). This is a powerful tool that can be used to create an inclusive learning environment which understands students have difficulties that arise while striving for their educational goals. It was developed with the Universal Design for Learning and Assessment framework in mind ( Silver et al., 1998 ) using elements of the Self-Determination Theory ( Deci et al., 1999 ). But as is inherent to each tool, it has a way and circumstance for its employment. Understanding when and how to make use of this opportunity will benefit students and, as with many systems, student understanding and agreement with the system helps with its effectiveness and functionality ( Cavanagh et al., 2016 ). In our study we found that by the end of the semester, when deciding which type of hypothetical classroom they would prefer, 82% of students chose a classroom with a dual deadline (instead of a single deadline at the end of the semester or one per assignment) such as was employed in our classroom. Student appreciation for the system was expressed in categorical selections and their written statements. This interest in the balance of structure (there is a due date) and flexibility (assignments can be submitted late without penalty) was expected, based on the post-pandemic discussions about accommodating increased student needs ( Hogan and Sathy, 2022 ; Kuimelis, 2022 ).

Previous research has shown that educational innovations and assessments can disproportionately impact groups by gender ( Ballen et al., 2017 ; Aguillon et al., 2020 ; Robnett et al., 2022 ), first-generation status ( Busch et al., 2023 ; Metzger et al., 2023 ), race ( Aikens et al., 2017 ; Castelli and Sarvary, 2021 ) and year-in-school ( Biango-Daniels and Sarvary, 2021 ). For most student-identified benefits, the EWP system positively impacted everyone equally. However, in some cases, the EWP system had a more positive impact on certain groups. For example, the EWP system helped first-generation students to use the extra time for office hours and to seek help to complete the assignments. These positive impacts should not be overlooked, as one of the main goals of EWP was to remove bias and the hidden curriculum ( Bergenhenegouwen, 1987 ; Margolis, 2001 ; Alsubaie, 2015 ). Time management is a skill that students can develop with practice. Significantly more first-year students worried that the EWP system would increase the likelihood of their procrastination. Instructors teaching large introductory classes have a unique opportunity to assist first-year students who have recently transitioned from high school ( Meaders et al., 2020 ). The first experiences in college can have a profound impact ( Lane et al., 2021 ), and instructors should share effective learning strategies and time management techniques, which are known to be the most common concerns voiced by college students ( Meaders et al., 2021 ).

Categorical responses from students who used the EWP about the advantages of the system resulted in a high level of selection to nearly all possibilities for the perceived advantages of the system ( Figure 1 ). The top three advantages included “My stress was reduced” at 94% of the EWP users reporting it as a benefit. “This made things better for my courses” was selected by 82% while “I could handle things better when I was sick/had an emergency” was at 73%. But the quality of work improvement, time management benefits, instructor empathy, and mental health were all chosen by 60 + % of the students who took advantage of the EWP system. Those who did not use the system also reported perceived advantages, including stress reduction and improved time management. These findings align with the idea of an inclusive classroom that provides flexibility to help students when unexpected difficulty occurs ( Hogan and Sathy, 2022 ). With the increased attention to mental health issues due to the COVID-19 pandemic ( Ma et al., 2022 ; Canet-Juric et al., 2023 ; Klussman et al., 2023 ; Sánchez-Martín et al., 2023 ), these results show a promising solution to decreasing stress and improving mental health in the classroom. To analogize, the safety net created by allowing the built-in flexibility of EWP resulted in being able to perform without having to fear the fall.

The educational framework known as Universal Design for Learning exists to meet the diverse needs of all students by providing multiple methods of representation, engagement, and expression ( Silver et al., 1998 ). Our policy fits into this framework as it offers both ideal and extended due dates, recognizing that students have varied pacing needs and personal circumstances. This grants students the freedom to make choices based on their unique situations, which can increase motivation and engagement ( Amabile et al., 1976 ). Furthermore, the tactic of providing both ideal and extended due dates displays a flexible approach to learning, acknowledging that students may require different lengths of time to understand and complete assignments. Differentiation involves tailoring instruction to meet individual needs ( Westwood, 2001 ; Bromley, 2019 ), and this policy enables just that in terms of assignment completion, while also reducing unnecessary barriers. The provision of extension due dates without penalties is a policy that eliminates potential obstacles to learning. For students who may be dealing with external stressors such as illness or other exams, this approach guarantees that they are not further penalized in this course. Despite these allowances, the goal of the policy is not just to cause a relaxation of demands but to foster the metacognitive skills in students that help them to reflect on their own learning processes and self-regulate when they need to submit their assignments ( Tanner, 2012 ). This autonomy allows intrinsic motivation to play a role in their engagement with material and retention ( Deci et al., 1999 ). Acknowledging that students have a life outside of the classroom can influence the perspectives of both students and teachers, leading to a more holistic understanding. This approach enables students to feel recognized for their unique circumstances and individuality, which in turn, promotes empathy and ultimately enhances their learning experience ( Coffman, 1981 ; Arghode, 2012 ). It is important that educational innovations do not have a negative impact on students’ grades and performance ( Sarvary et al., 2022 ). Therefore, with students demonstrating a preference for having this system with its benefits, it is important to see if academic outcome is affected by the frequency of usage of the EWP. The final grade comparison of those who used the system and those who did not showed that using EWP does not negatively impact grades. The systemic application of an extension with a suggested deadline arguably allows for improved capability to achieve learning outcomes under adverse circumstances while not affecting academic achievement. While published literature on this topic is scarce, there are many anecdotes and misconceptions regarding assignment extensions. We generated eight hypothetical instructor opinions based on our experiences and would like to argue against them with evidence found in our study, if appropriate. While these opinions are not based on surveys or interviews, they do reflect scenarios encountered during the experience of the authors.

4.1 Anecdotal instructor opinion 1: extensions encourage procrastination of work

In many discussions across campuses and conferences, the importance of deadlines for both academics and the workforce has come up. Work must be completed so it can be handed off to others for their work to be completed and a final product to be created within a reasonable deadline. Time is a valuable resource that does not ever get returned ( Burkeman, 2021 ), and procrastination was the highest perceived cost to this system (33% reported by those who used EWP). Though this is true, time management is a skill that is sought after by employers in many fields ( Hochheiser, 1998 ; Mancini, 2003 ; Woody and Coleman, 2015 ) and this is quality that the system encourages. “Procrastination” in the end is just an aspect of time management wherein a student gets to choose the most important task at hand and work on it. Allowing them opportunities to prioritize other aspects of their lives or course load, while not impacting their learning, is a valuable part of their development. Instructors of introductory courses should pay close attention to their incoming first-year students, as the majority of the students who were concerned about procrastination in our study were the ones just starting their higher education.

4.2 Anecdotal instructor opinion 2: my students will just do the work at the last moment

Given the removal of a deadline or any enforcement of late work, it has been seen that students will try and complete all of their assignments at the last moment amounting to reduced quality and retention ( Kuimelis, 2022 ). It becomes important to frame the creation of the extension in the right context. Within the classroom, explaining the advantages of following the ideal deadline is vital, highlighting that the extension exists as a safety net. There is also importance in having extensions for each assignment separately defined to reinforce the value of completing assignments sequentially. In our study, 78% of the students used EWP at least once, but 41% used it only a single time, while it was available for multiple assignments. Students overwhelmingly preferred a dual due date system (82%), so they benefit from a structure where an ideal due date is presented, but an extension due date is offered as a safety net. This study showed that students will not wait until the last moment to turn in the assignments but benefit from the system by reducing stress (94%) and by better handling other courses (82%) and emergencies (73%).

4.3 Anecdotal instructor opinion 3: application of a systemic extension does not always resolve the student’s issue

Difficulties that arise in a student’s life may take more than a week’s time to resolve. As such, they will need to reach out and explain their difficulty. An instructor will then be required to respond and decide whether to give an extension, which can potentially reintroduce bias to the system. Our study showed that establishing EWP can address the majority of the challenges the students face and may reduce the total needed personal extensions. The mid-semester open-ended questions and the categories created based on those answers at the end of the semester showed that this EWP system does indeed provide a solution for a variety of issues, from stress reduction (94%) through emergencies (73%) to improving mental health (60%). While students do not always face these issues, responses from those who did not use the EWP showed that they enjoy the perceived benefits, as they know the system exists in case they need it. Only 12% of the students stated that “my need for the extension was not resolved by the extension deadline.” The EWP system allows the instructors of large courses to focus on these few cases that need extra attention, instead of being flooded by extension requests. Another solution could be the use of a “token” in addition to the EWP system. Students could use the token for one assignment that needs more extension than the EWP system allows. In that case, the students could submit that one assignment by the end of the semester or by another instructor-defined due date. This token assignment would help students with unexpected situations that cannot be resolved within the EWP timeframe.

4.4 Anecdotal instructor opinion 4: institutional extensions may be enforced on top of the EWP system

When accommodations are given to a student because of a disability that affects their ability to perform in the course and an extension is allowed to all students, the debate of whether that time should be augmented to the present extension or only laid alongside what is allotted in your course may be raised. This could affect the timelines that instructors are trying to maintain in the course learning objectives. Instructors need to work with the student disability services to make sure they follow the proper guidelines. This study provided results of an extension system that is outside of the institutionally required extensions, such as by the student disability services. Many students either do not know about those services or face challenges outside the purview of student disability. The EWP system creates an unbiased solution for those requests. It is not intended to overwrite the system created by the individual institutions.

4.5 Anecdotal instructor opinion 5: slower return times for assignments can negatively impact my students

Increased time between submissions and feedback can result in a disconnect of students’ attention to prior work as well as application of feedback ( Eckstein et al., 2020 ; Puppe et al., 2022 ). This is a loss that is built into the system. Efforts to overcome it can include verbal confirmation of when grades are posted as well as in class discussion of common pitfalls of the assignment ( Torrance and Pryor, 2001 ; Puppe et al., 2022 ). Frequent formative assessments that are addressed within class time can also help to guide students toward better understanding and corrections for erroneous knowledge ( Sari et al., 2022 ). In large biology courses graduate and undergraduate teaching assistants often support the teaching and grading efforts ( Asgari and Sarvary, 2020 ), allowing a faster return of the assignments. In this study only 12% of the students listed that the grades were returned to them more slowly. Instructors can design EWP systems with due dates that do not impact their return time, or selectively apply it to assignments that are not part of a scaffolded learning system. The broad variety of real and perceived benefits of this system allows for plasticity based on the instructors’ needs.

4.6 Anecdotal instructor opinion 6: this creates additional work for the instructors of the course

Due to the prep time to design a course that allows students leeway on assignments, the initial time put into course design may increase. Students may struggle with understanding the new system, therefore clear and repeated communication about the system will be required. Despite the instructors’ communication efforts, 25% of the students listed “the system was confusing to me” as one of the disadvantages. Instructor grading may be compressed to a smaller window of time, especially if it is desired that all assignments be returned before large summative assessments. In large courses without EWP the administrative burden of handling extension requests can be tremendous and introduce instructor bias. Prior to the implementation of the EWP system, hundreds of emails requesting extensions needed to be answered each semester. In addition, the instructors needed to decide for each request whether an extension should be granted and how long it should be. The EWP system decreased the number of these requests to only a few, freeing up time for the instructors and the academic staff. Therefore, we argue that an established EWP system should have similar or less time costs to a system with a single deadline.

4.7 Anecdotal instructor opinion 7: students feel that this system is unfair to those who complete their work on time

This was not observed by instructors or stated by students within the survey responses. Though massive use of EWP by all students could negatively impact student ability to participate in in-class activities, active learning, and group work. Therefore, instructors must make course-specific adjustments in response to said actions. This was not seen in our study, instead, students sought to manage their own time. The majority of the students (82%) expressed their preferences for the EWP system when they were asked whether they want EWP, a single deadline, or no due date at all for their assignments.

4.8 Anecdotal instructor opinion 8: dropping assignments provides flexibility in the classroom

All courses should have well-designed learning objectives, stating what students will take away from the class ( Sarvary et al., 2022 ). If assignments are designed correctly, they assess the knowledge and skills of the students. In introductory courses, students gain skills that they will build on during their college years and beyond, therefore allowing students to skip specific assignments may lead to a knowledge or skill gap that can negatively impact the student in the long term. This study showed that when a safety net of EWP is provided to the students, they submit all the assignments and meet all the learning objectives without impacting their grades. They enjoy real or perceived benefits from the EWP system without losing an opportunity to gain the skills and knowledge the course was designed to provide them.

5 Conclusion

The objective of including Universal Design in academic courses is to ensure that all students can learn and participate in an inclusive environment without the need for extra assignments or presentations to accommodate those who struggle with traditional course material ( Higbee and Goff, 2008 ). An explicitly communicated and systemically applied extension without penalty framework can be integrated readily into such a design as it inherently allows for flexibility when difficulties arise, no matter their reasons. By providing students with more control over when they hand in assignments, one can augment student engagement, self-regulation, and motivation. Similarly, a reduction in the need to reach out to instructors for difficulties can help to eliminate aspects of the hidden curriculum while also cutting out opportunities for bias in allowances for extensions. When selection for a reduction in points (late grade) is done based on the educator’s ability to understand the need for the extension, it inherently introduces bias. Evasion of this through an explicit extension is arguably beneficial.

The “Extension Without Penalty” system was well received by students, did not negatively impact their grades, and promoted inclusivity. In our efforts to develop an inclusive classroom environment, we should seek evidence-based practices that build up our students while not creating excessive additional work that fatigues educators and reinforces bad behaviors. This system, with the appropriate advanced planning, can be employed in a variety of classrooms and has potential in these efforts. Developing a system with explicitly communicated extension due dates brings equity to the classroom while balancing structure and flexibility.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics statement

The studies involving humans were approved by Cornell Institutional Review Board for Human Participants, Protocol ID 2109010595. The studies were conducted in accordance with the local legislation and institutional requirements. The participants provided their written informed consent to participate in this study.

Author contributions

JR: Conceptualization, Data curation, Formal analysis, Validation, Visualization, Writing – original draft, Investigation, Methodology, Writing – review & editing. MS: Conceptualization, Methodology, Writing – review & editing, Funding acquisition, Investigation, Project administration, Supervision, Validation, Visualization, Writing – original draft.

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Acknowledgments

We would like to acknowledge Chelsea Maceda for her help in survey development and analysis. We thank the Investigative Biology graduate and undergraduate teaching assistants for their valuable feedback on the Extension Without Penalty system and with the validation of the survey instrument. We also wish to acknowledge Matt Thomas from Cornell Statistical Consulting Unit for his assistance in coding and statistics. The authors are also grateful to the Cornell College of Agriculture and Life Sciences teaching seminar series, and online forums such as “Pandemic pedagogy” and #AcademicChatter, as in-person and online discussions about this emerging topic have greatly contributed to the development of this manuscript’s narrative.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2024.1324506/full#supplementary-material

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Keywords: undergraduate, extensions, inclusion, structure, deadline, flexibility, universal design for learning, self-determination theory

Citation: Ruesch JM and Sarvary MA (2024) Structure and flexibility: systemic and explicit assignment extensions foster an inclusive learning environment. Front. Educ . 9:1324506. doi: 10.3389/feduc.2024.1324506

Received: 19 October 2023; Accepted: 07 March 2024; Published: 21 March 2024.

Reviewed by:

Copyright © 2024 Ruesch and Sarvary. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Joseph M. Ruesch, [email protected]

IMAGES

  1. Number System

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  2. The Real Number System

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  3. Introduction to Number System

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  4. Number Systems

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  5. NUMBER SYSTEM (PART 1)

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VIDEO

  1. NUMBER SYSTEM

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  5. NUMBER SYSTEM (संख्या पद्धति) PART-14

  6. Class 7TH || Maths Chapter 1 Number system (part 1)

COMMENTS

  1. Number System (Definition, Types, Conversion & Examples)

    There are various types of number systems in mathematics. The four most common number system types are: Decimal number system (Base- 10) Binary number system (Base- 2) Octal number system (Base-8) Hexadecimal number system (Base- 16) Now, let us discuss the different types of number systems with examples.

  2. Class 9 Mathematics Number System Assignments

    We have provided below free printable Class 9 Mathematics Number System Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 9 Mathematics Number System.These Assignments for Grade 9 Mathematics Number System cover all important topics which can come in your standard 9 tests and examinations.Free printable Assignments for CBSE Class 9 ...

  3. PDF Number Systems and Number Representation

    Learn the basics of number systems and number representation in this lecture from Princeton University's COS 217 course. Topics include binary, hexadecimal, and octal systems, unsigned and signed integers, and rational numbers. This lecture is from spring 2015 and may differ slightly from other versions of the course.

  4. CBSE Class 9 Maths Worksheet Chapter 1 Number System

    The Maths assignment for Class 9 Number System list of questions and answers provide thorough insights on the topic's resources and offers easy tricks to identify quicker ways to solve the questions faster while also being more aware and making sure students don't go wrong or commit any silly mistakes in their solutions.

  5. Number System Questions

    Number System Questions and Answers. 1. Determine whether the numbers are rational or irrational. \ [ \sqrt {2}\] 1.5. \ [\sqrt {100}\] 3.14. Ans: A rational number is a number that can be represented in the form of p/q, whereas an irrational number cannot be represented in the form of p/q.

  6. What is Number System? Definition, Types, Example, Facts

    In the binary number system, we only use two digits 0 and 1. It means a 2 number system. Example of binary numbers: 1011; 101010; 1101101. Each digit in a binary number is called a bit. So, a binary number 101 has 3 bits. 499787080. Computers and other digital devices use the binary system. The binary number system uses Base 2.

  7. PDF Number Systems, Base Conversions, and Computer Data Representation

    Binary Coded Decimal Numbers Another number system that is encountered occasionally is Binary Coded Decimal. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number. For example: The decimal number 136 would be represented in BCD as follows: 136 = 0001 0011 0110 1 3 6

  8. PDF Assignment 1: Computer Number Systems

    In this assignment, we are using a 32-bit unsigned fixed point number system U(16,16) to store an unsigned real number, i.e., there are sixteen bits of precision in the fractional part of the number, and the radix point is sixteen bits to the right of the most significant bit stored.

  9. A Guide to Mastering Number Systems: Rational, Irrational, and

    Embarking on the journey of doing your number system assignment might initially appear as a challenging task, especially if concepts like rational, irrational, and imaginary numbers seem like uncharted territory. Fear not, for this comprehensive guide is here to illuminate your path to success. We will unravel the mysteries of number systems, providing you with the knowledge and strategies ...

  10. Numerals and numeral systems

    In its pure form a simple grouping system is an assignment of special names to the small numbers, the base b, and its powers b 2, ... These Ionic, or alphabetical, numerals, were simply a cipher system in which nine Greek letters were assigned to the numbers 1-9, nine more to the numbers 10, …, 90, and nine more to 100, …, 900. Thousands ...

  11. Class 10 Mathematics Number System Assignments

    We have provided below free printable Class 10 Mathematics Number System Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 10 Mathematics Number System.These Assignments for Grade 10 Mathematics Number System cover all important topics which can come in your standard 10 tests and examinations. ...

  12. NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

    The Number System is discussed in detail in this chapter. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers. The chapter starts with the introduction of Number Systems in section 1.1, followed by two very important topics in sections 1.2 and 1.3 ...

  13. Assignment

    The "Assignment - Number System, Class 9 Math Class 9 Questions" guide is a valuable resource for all aspiring students preparing for the Class 9 exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive ...

  14. Math Assignment Class IX Ch -1| Number System

    Math Assignment for Class IX Ch -1, Number System strictly according to the CBSE syllabus. Extra questions based on the topic Number System. MATHEMATICS ASSIGNMENT CLASS IX NUMBER SYSTEM Q1- Insert 5 rational and 5 irrational numbers between (a) 7 & 8, (b) 2 & 3.2, (c) 2.7 & 6.32,

  15. What is a number system?

    The writing system for denoting numbers using digits or symbols in a logical manner is defined as a Number system. The numeral system Represents a useful set of numbers, reflects the arithmetic and algebraic structure of a number, and Provides standard representation. The digits from 0 to 9 can be used to form all the numbers.

  16. Crack Number System Assignments: Essential Concepts & Strategies

    The very first step in tackling a number system assignment is comprehending different number bases and notations. The most familiar is the decimal system (base-10), where numbers are represented using ten digits (0-9). However, various other bases like binary (base-2), octal (base-8), and hexadecimal (base-16) are equally important.

  17. Binary Number System

    Binary Number System is a number system that is used to represent various numbers using only two symbols "0" and "1". The word binary is derived from the word "bi" which means two. Hence, this number system is called Binary Number System. Thus, the binary number system is a system that has only two symbols.

  18. Assignments For Class 9 Mathematics Number System

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  19. Number System: References Pages

    Using the number system, on your references page you typically provide the following information in the following order: The number of the reference, followed by a period. The first initials and last names of all authors, followed by a comma. Title of the article enclosed in quotation marks, followed by a comma.

  20. Frontiers

    Both semesters used a dual assignment deadline system, so the answers for this one question were combined from the two semesters. This study's proposal was granted exemption from Institutional Review Board review by the University's Office of Research Integrity and Assurance (2109010595). ... The EWP system decreased the number of these ...

  21. Hardware and Software Specifications

    This list represents a typical system used at Epic, providing a reasonable guideline for developing games with Unreal Engine 5: Windows 10 64-bit (Version 20H2) 64 GB RAM. 256 GB SSD (OS Drive) 2 TB SSD (Data Drive) NVIDIA GeForce RTX 2080 SUPER. Xoreax Incredibuild (Dev Tools Package) Six-Core Xeon E5-2643 @ 3.4GHz