real number system homework 3

1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

• ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
• ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• ⓐ 7 = 7 1 7 = 7 1
• ⓑ 0 = 0 1 0 = 0 1
• ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• ⓐ − 5 7 − 5 7
• ⓑ 15 5 15 5
• ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

• ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
• ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
• ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
• ⓐ 68 17 68 17
• ⓑ 8 13 8 13
• ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• ⓑ 33 9 33 9
• ⓓ 17 34 17 34
• ⓔ 0.3033033303333 … 0.3033033303333 …
• ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

• ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

• ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• ⓐ 7 77 7 77
• ⓒ 4.27027002700027 … 4.27027002700027 …
• ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• ⓐ − 10 3 − 10 3
• ⓒ − 289 − 289
• ⓓ −6 π −6 π
• ⓔ 0.615384615384 … 0.615384615384 …
• ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
• ⓑ 5 5 is positive and irrational. It lies to the right of 0.
• ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
• ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
• ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
• ⓑ −11.411411411 … −11.411411411 …
• ⓒ 47 19 47 19
• ⓓ − 5 2 − 5 2
• ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• ⓔ 3.2121121112 … 3.2121121112 …

• ⓐ − 35 7 − 35 7
• ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Step 1. Simplify any expressions within grouping symbols.
• Step 2. Simplify any expressions containing exponents or radicals.
• Step 3. Perform any multiplication and division in order, from left to right.
• Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
• ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
• ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
• ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

• ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
• ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
• ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
• ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
• ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
• ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
• ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
• ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
• ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
• ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
• ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
• ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
• ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
• ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
• ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
• ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
• ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
• ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• ⓑ 4 3 π r 3 4 3 π r 3
• ⓒ 2 m 3 n 2 2 m 3 n 2

• ⓐ 2 π r ( r + h ) 2 π r ( r + h )
• ⓑ 2( L + W )
• ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

• ⓐ x = 0 x = 0
• ⓑ x = 1 x = 1
• ⓒ x = 1 2 x = 1 2
• ⓓ x = −4 x = −4
• ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
• ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
• ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
• ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

• ⓐ y = 2 y = 2
• ⓑ y = 0 y = 0
• ⓒ y = 2 3 y = 2 3
• ⓓ y = −5 y = −5

Evaluate each expression for the given values.

• ⓐ x + 5 x + 5 for x = −5 x = −5
• ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
• ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
• ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
• ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
• ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
• ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
• ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
• ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
• ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
• ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
• ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
• ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
• ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
• ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
• ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
• ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
• ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
• ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
• ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
• ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
• ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
• ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
• ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

• Simplify an Expression.
• Evaluate an Expression 1.
• Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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Real Numbers Handout

Use this one-page handout to help students solidify their understanding of the real number system, which includes irrational numbers, rational numbers, integers, whole numbers, and natural numbers. This handy printable features a definition for each type of number, as well as a helpful graphic organizer that illustrates examples of each and how they relate to one another. Students can use this handout as a study guide or reference sheet for their unit on real numbers. For hands-on practice classifying real numbers, check out this Real Numbers Card Sort  worksheet!

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Student-Centered Math Lessons

Teaching the Real Number System

The real number system can really confuse students. I will admit, at times, I felt confused too! Let’s check out 4 strategies that will help you teach classifying real numbers, and will help your students master the concept.

Vertical Alignment

Before we jump to that, let’s take a look at the standard and how it progresses through middle school, and then take a look at some STAAR test question examples. I have highlighted some helpful pieces.

Real Number System Test Questions

Strategy #1 – Vocabulary

Vocabulary is crucial when teaching the real number system. Luckily, the content scaffolds by grade level.

• 6th Grade: whole, integer, rational
• 7th Grade: natural, whole, integer, rational
• 8th Grade: natural, whole, integer, rational, irrational

Since the progression of standards is pretty clear, each subsequent year a student must learn one brand new word. Although the vocabulary is important, I think students need to see examples more than they need to memorize the exact definition. 

Bright idea! One way to push students’ learning is to ask them to come up with definitions based on observing numbers that have already been classified. Check out the example below.

Some thought provoking questions might be:

• What differences do you see between the numbers inside integers and whole numbers? 
• What is different about the fractions classified as whole numbers versus the fractions classified as rational numbers?

Strategy #2 – Visuals

Notice that in each grade level standard, the term “visual representation” is used. In addition, in each test question example, there is a venn diagram of sorts. This means that students will not be expected to classify a number in the real number system without a venn diagram present to guide them, so make sure you are modeling with one, and students are practicing with one.

Check out the one I made! Hint: Washi tape helps with straight lines.  If you use Post-it Notes , the anchor chart can be interactive and reused each class period.

Students need to be taught how to use the venn diagram. Don’t assume (like me) that it is intuitive. I would start by using a similar venn diagram that is not related to math. You can steal this example if you would like. (Warning: You may be concerned about students’ geography if you use this example.) 

You can ask these types of questions:

• If someone is from Texas, can you assume they are also from the United States?
• If someone lives in Oklahoma, where would you place them on the diagram? 

If students are struggling to use the venn diagram to understand the relationships between sets of numbers, then try exposing them to the funnel example. Using the number 17: 17 would be dropped into the natural number funnel thus falling through the whole, integer, rational, and real number funnels. The number -17 would be dropped into the integer funnel and thus continue into rational and real numbers. Then you would explain that -17 is an integer, a rational number, and a real number, but you typically call numbers by the funnel that it is dropped in.

Strategy #3 – Simplify Before You Classify

I saw this idea on a Middle School Math Facebook group, and it is so clever and catchy! Teach students to simplify before they classify in the real number system. Fractions like 16/4 are a great example of this. If students are familiar with the definition of rational numbers, they may think, “16/4 must only be a rational number because rational numbers are numbers that can be written as fractions.” That student is technically right, 16/4 is rational, but that is not all. If you teach students to simplify before you classify, a student would simplify this to 4, thus changing its classification.

You see this with square roots too. Many irrational number definitions include the phrase “square roots,” so a student might incorrectly classify the square root of 100 as irrational. Simplify before you classify!

Strategy #4 – Make It Interactive

Since this skill requires very little computation, this is an opportunity to engage students in something hands-on. Here is what I have done:

• Flyswatter Game (ideally after students have shown mastery, so they aren’t just swatting uncontrollably)
• Post-it Notes – have students write down a bunch of different types of numbers on individual Post-it Notes and then swap with a partner. Then they have to categorize the numbers from their partner.
• Grab a 6th grade or 8th grade activity bundle that includes classifying the real number activities.

Classifying numbers in the real number system can be really engaging. It also would provide a math-win for some of my struggling students. How do you make classifying numbers engaging?

Digital Math Activities

Check out these related products from my shop.

Real Number Properties

Real Numbers have properties!

Example: Multiplying by zero

When we multiply a real number by zero we get zero:

• 0 × 0.0001 = 0

It is called the "Zero Product Property", and is listed below.

Here are the main properties of the Real Numbers

Real Numbers are Commutative, Associative and Distributive :

Commutative example

a + b = b + a 2 + 6 = 6 + 2

ab = ba 4 × 2 = 2 × 4

Associative example

(a + b) + c = a + ( b + c ) (1 + 6) + 3 = 1 + (6 + 3)

(ab)c = a(bc) (4 × 2) × 5 = 4 × (2 × 5)

Distributive example

a × (b + c) = ab + ac 3 × (6+2) = 3 × 6 + 3 × 2

(b+c) × a = ba + ca (6+2) × 3 = 6 × 3 + 2 × 3

Real Numbers are closed (the result is also a real number) under addition and multiplication:

Closure example

a+b is real 2 + 3 = 5 is real

a×b is real 6 × 2 = 12 is real

Adding zero leaves the real number unchanged, likewise for multiplying by 1:

Identity example

a + 0 = a 6 + 0 = 6

a × 1 = a 6 × 1 = 6

For addition the inverse of a real number is its negative, and for multiplication the inverse is its reciprocal :

Additive Inverse example

a + (−a ) = 0 6 + (−6) = 0

Multiplicative Inverse example

a × (1/a) = 1 6 × (1/6) = 1

But not for 0 as 1/0 is undefined

Multiplying by zero gives zero (the Zero Product Property ):

Zero Product example

If ab = 0 then a=0 or b=0, or both  

a × 0 = 0 × a = 0 5 × 0 = 0 × 5 = 0

Multiplying two negatives make a positive , and multiplying a negative and a positive makes a negative:

Negation example

−1 × (−a) = −(−a) = a −1 × (−5) = −(−5) = 5

(−a)(−b) = ab (−3)(−6) = 3 × 6 = 18

(−a)(b) = (a)(−b) = −(ab) −3 × 6 = 3 × −6 = −18

Real Number System Worksheets

Real Number System Worksheets kids will be learning about rational numbers and irrational numbers, Non-Integer Fractions, Integers , Whole Numbers, and Natural Numbers. Real numbers are the set of all numbers that can be expressed as a decimal or that are on the number line. Real numbers have certain properties and different classifications, including natural, whole, integers, rational and irrational.

Benefits of Real Number System Worksheets

Real Number System Worksheets helps kids to understand the whole concept of real numbers and they can know the properties and operations of numbers which are very important in our daily lives. Real Number System Worksheets helps kids to know about the whole numbers and the fundamental operations on them. And study on  the integers, rationals, decimals, fractions and powers in this section.

Printable PDFs for Real Number System Worksheets

Real Number System Worksheets helps kids with understanding the whole concept of the real number system. These worksheets are a helpful guide for kids as well as their parents to see and review their answer sheets. Children can download the pdf format of these easily accessible Real Number System Worksheets to practice and solve questions for free.

Download Real Number System Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

Classification of Real Numbers

How to classify real numbers.

The “stack of funnels” diagram below will help us easily classify any real numbers. But first, we need to describe what kinds of elements are included in each group of numbers. A funnel represents each group or set of numbers.

Description of Each Set of Real Numbers

Natural numbers (also known as counting numbers) are the numbers that we use to count. It begins with 1, then 2, 3, and so on.

Whole numbers are a slight “upgrade” of the natural numbers because we simply add the element zero to the current set of natural numbers. Think of whole numbers as natural numbers together with zero.

Integers include all whole numbers together with the “negatives” or opposites of the natural numbers.

Rational numbers are numbers that can be expressed as a ratio of integers. That means if we can write a given number as a fraction where the numerator and denominator are both integers; then it is a rational number.

Symbolically, we can write a rational number as:

 Caution: The denominator cannot equal zero.

Rational numbers can also appear in decimal form . If the decimal number either terminates or repeats, then it is possible to write it as a fraction with an integer numerator and denominator. Thus, it is rational as well.

Irrational numbers are all numbers that when written in decimal form do not repeat and do not terminate. In other words, it goes on forever indefinitely without having a definite pattern.

Real numbers include both rational and irrational numbers. Remember that under the set of rational numbers, we have the subcategories or subsets of integers, whole numbers, and natural numbers.

Hierarchical Format of Real Numbers

We can also express the subsets of the set of real numbers in a hierarchical presentation. This way, we can clearly see how the different sets are related and distinct.

• Real Numbers [latex]\mathbb{R}[/latex]
• 1.1.1.1.1 Natural Numbers [latex]\mathbb{N}[/latex]
• 1.2 Irrational Numbers

Classification of Real Numbers Examples

Example 1 : A natural number is also a whole number. ( True or False )

The set of whole numbers includes the number zero and all natural numbers. This is a true statement.

Example 2 : An integer is always a whole number. ( True or False )

The set of integers is composed of the number zero, natural numbers, and the “negatives” of natural numbers.  That means some integers are whole numbers, but not all.

For instance, [latex] – 2[/latex] is an integer but not a whole number. This statement is false .

Example 3 : Every rational number is also an integer. ( True or False )

The word “every” means “all”. Can you think of a rational number that is not an integer? You only need one counterexample to show that this statement is false.

The fraction [latex]\Large{1 \over 2}[/latex] is an example of a rational number that is NOT an integer. So this statement is false .

Example 4 : Every integer is a rational number. ( True or False )

This is true because every integer can be written as a fraction with a denominator of [latex]1[/latex].

Example 5 : Every natural number is a whole number, integer, and a rational number. ( True or False )

Reviewing the descriptions above, natural numbers are found within the sets of whole numbers, integers, and rational numbers. That makes it a true statement.

We can also use the diagram of funnels above to help us answer this question. If we pour water into the “funnel of natural numbers”, the water should also flow through all the funnels below it. Thus, passing through the funnels of the whole numbers, integers, and rational numbers.

Example 6 : Every whole number is a natural number, integer, and rational number. ( True or False )

Using the same “funnel” analogy; if we pour some liquid into the whole numbers’ funnel, it should pass through the funnels of integers and rational numbers as it makes its way down. Since the natural numbers’ funnel is above the set of whole numbers where we started, we cannot include this funnel in the group.

It is a false statement since whole numbers belong to the sets of integers and rational numbers, but not to the set of natural numbers.

Simply put, the number zero (0) is a counterexample since it is a whole number but not a natural number. So indeed, this is a false statement.

Example 7 : Classify the number zero, [latex]0[/latex].

Definitely not a natural number but it is a whole , an integer , a rational , and a real number. It may not be obvious that zero is also a rational number. However, writing it as a fraction with a nonzero denominator would clearly show that it is indeed a rational number.

Example 8 : Classify the number [latex]5[/latex].

This is a natural or counting number, a whole number, and an integer. Since we can write it as a fraction with a denominator of [latex]1[/latex], that is, [latex]\Large{5 \over 1}[/latex], it is also a rational number. And of course, this is a real number.

Example 9 : Classify the number [latex]0.25[/latex].

The given decimal number terminates and so we can write it as a fraction which is a characteristic of a rational number. This number is also a real number.

[latex]\Large{0.25 = {{25} \over {100}} = {1 \over 4}}[/latex]

Example 10 : Classify the number [latex]{\rm{2}}{1 \over 5}[/latex].

We can rewrite this mixed fraction as an improper fraction so that it is clear that we have a ratio of two integers.

[latex]\Large{{\rm{2}}{1 \over 5} = {{11} \over 5}}[/latex]

This number is a rational and real number.

Example 11 : Classify the number [latex]{\rm{5.241879132…}}[/latex].

The decimal number is non-terminating and non-repeating that means it is an irrational number. Of course, any irrational number is also a real number.

Example 12 : Classify the number [latex]1.7777…[/latex]

Since the decimal is repeating, it is a rational number. Any rational number must also be a real number.

Example 13 : Classify the number [latex]\sqrt 2 [/latex].

This is an irrational number because when written in decimal form, it is non-terminating and non-repeating. This is also a real number.

Example 14 : Classify the number [latex] – \sqrt {16} [/latex] .

First, we need to simplify this radical expression which gives us [latex] – \sqrt {16} = – \,4[/latex]. The number [latex] – \,4[/latex] is an integer, a rational number, and a real number.

Example 15 : Classify the number [latex] – 8.123123…[/latex].

The decimal number is nonterminating, however, the string of numbers 123 after the decimal point keeps on repeating. We can rewrite the decimal number with a “bar” on top of the repeating numbers.

This makes it a rational number. Don’t forget that it is also a real number.

Take a quiz:

• Classifying Real Numbers Quiz

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• Properties of Real Numbers
• Properties of Equality

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Watch CBS News

Hackers may have stolen your Social Security number in a massive breach. Here's what to know.

By Aimee Picchi

Edited By Anne Marie Lee

Updated on: August 16, 2024 / 3:02 PM EDT / CBS News

A new lawsuit is claiming hackers have gained access to the personal information of "billions of individuals," including their Social Security numbers, current and past addresses and the names of siblings and parents — personal data that could allow fraudsters to infiltrate financial accounts or take out loans in their names. 

The allegation arose in a lawsuit filed earlier this month by Christopher Hofmann, a California resident who claims his identity theft protection service alerted him that his personal information had been leaked to the dark web by the "nationalpublicdata.com" breach. The lawsuit was earlier reported by Bloomberg Law.

The breach allegedly occurred around April 2024, with a hacker group called USDoD exfiltrating the unencrypted personal information of billions of individuals from a company called National Public Data (NPD), a background check company, according to the lawsuit. Earlier this month, a hacker leaked a version of the stolen NPD data for free on a hacking forum, tech site Bleeping Computer reported . 

That hacker claimed the stolen files include 2.7 billion records, with each listing a person's full name, address, date of birth, Social Security number and phone number, Bleeping Computer said. While it's unclear how many people that includes, it's likely "that everyone with a Social Security number was impacted," said Cliff Steinhauer, director of information security and engagement at The National Cybersecurity Alliance, a nonprofit that promotes online safety.

"It's a reminder of the importance of protecting yourself, because clearly companies and the government aren't doing it for us," Steinhauer told CBS MoneyWatch.

In a statement posted to its website, NPD said the breach involved a "third-party bad actor that was trying to hack into data in late December 2023, with potential leaks of certain data in April 2024 and summer 2024."

The company added that it is working with law enforcement and government investigators. NPD said it "will try to notify you if there are further significant developments applicable to you."

Here's what to know about the alleged hack. 

What is National Public Data? 

National Public Data is a data company based in Coral Springs, Florida, that provides background checks for employers, investigators and other businesses that want to check people's backgrounds. Its searches include criminal records, vital records, SSN traces and more information, its website says.

There are many similar companies that scrape public data to create files on consumers, which they then sell to other businesses, Steinhauer said.

"They are data brokers that collect and sell data about people, sometimes for background check purposes," he said. "It's because there's no national privacy law in the U.S. — there is no law against them collecting this data against our consent."

What happened with the USDoD hack?

According to the new lawsuit, USDoD on April 8 posted a database called "National Public Data" on the dark web, claiming to have records for about 2.9 billion individuals. It was asking for a purchase price of $3.5 million, the lawsuit claims. 

However, Bleeping Computer reported that the file was later leaked for free on a hacker forum, as noted above. 

How many people have been impacted?

The number of people impacted by the breach is unclear. Although the lawsuit claims "billions of individuals" had their data stolen, the total population of the U.S. stands at about 330 million. The lawsuit also alleges that the data includes personal information of deceased individuals.

Bleeping Computer reports that the hacked data involves 2.7 billion records, with individuals having multiple records in the database. In other words, one individual could have separate records for each address where they've lived, which means the number of impacted people may be far lower than the lawsuit claims, the site noted.

The data may reach back at least three decades, according to law firm Schubert Jonckheer & Kolbe, which said on Monday it is investigating the breach.

Did NPD alert individuals about the hack? 

It's unclear, although the lawsuit claims that NPD "has still not provided any notice or warning" to Hoffman or other people affected by the breach. 

"In fact, upon information and belief, the vast majority of Class Members were unaware that their sensitive [personal information] had been compromised, and that they were, and continue to be, at significant risk of identity theft and various other forms of personal, social, and financial harm," the lawsuit claims. 

Information security company McAfee reported that it hasn't found any filings with state attorneys general. Some states require companies that have experienced data breaches to file reports with their AG offices. 

However, NPD posted an alert about the breach on its website, stating that it believes the information breached includes names, email addresses, phone numbers, Social Security numbers and mailing addresses.

Can you find out if your data was part of the hack?

There are tools available that will monitor what information about you is available on the dark web, noted Michael Blair, managing director of cybersecurity firm NukuDo. Commonly breached data includes your personal addresses, passwords and email, he added.

One such service is how Hofmann, who filed the lawsuit, found out that his information has been leaked as part of NPD breach.

"Make sure to use reputable companies to look that up," Blair said. 

What should I do to protect my information?

Security experts recommend that consumers put freezes on their credit files at the three big credit bureaus, Experian, Equifax and TransUnion. Freezing your credit is free, and will stop bad actors from taking out loans or opening credit cards in your name. 

"The biggest thing is to freeze your credit report, so it can't be used to open new accounts in your name and commit other fraud in your name," Steinhauer said. 

Steinhauer recommends consumers take several additional steps to protect their data and finances:

• Make sure your passwords are at least 16 characters in length, and are complex. 
• Use a password manager to save those long, complex passwords.
• Enable multifactor authentication, which Steinhauer calls "critical," because simply using a single password to access your accounts isn't enough protection against hackers. 
• Be on alert for phishing and other scams. One red flag is that the scammers will try to create a sense of urgency to manipulate their victims.
• Keep your security software updated on your computer and other devices. For instance, make sure you download the latest security updates from Microsoft or Apple onto your apps and devices. 

You can also get a tracking service that will alert you if your data appears on the dark web. 

"You should assume you have been compromised and act accordingly," Steinhauer said. 

• Data Breach
• Social Security

Aimee Picchi is the associate managing editor for CBS MoneyWatch, where she covers business and personal finance. She previously worked at Bloomberg News and has written for national news outlets including USA Today and Consumer Reports.

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About four months after a notorious hacking group claimed to have stolen an extraordinary amount of sensitive personal information from a major data broker, a member of the group has reportedly released most of it for free on an online marketplace for stolen personal data.

The breach, which includes Social Security numbers and other sensitive data, could power a raft of identity theft, fraud and other crimes, said Teresa Murray, consumer watchdog director for the U.S. Public Interest Research Group.

For the record:

2:39 p.m. Aug. 15, 2024 A previous version of this article identified Teresa Murray as the consumer watchdog director for the U.S. Public Information Research Group. She works for the U.S. Public Interest Research Group.

“If this in fact is pretty much the whole dossier on all of us, it certainly is much more concerning” than prior breaches, Murray said in an interview. “And if people weren’t taking precautions in the past, which they should have been doing, this should be a five-alarm wake-up call for them.”

According to a class-action lawsuit filed in U.S. District Court in Fort Lauderdale, Fla., the hacking group USDoD claimed in April to have stolen personal records of 2.9 billion people from National Public Data, which offers personal information to employers, private investigators, staffing agencies and others doing background checks. The group offered in a forum for hackers to sell the data, which included records from the United States, Canada and the United Kingdom, for $3.5 million , a cybersecurity expert said in a post on X.

The lawsuit was reported by Bloomberg Law .

Last week, a purported member of USDoD identified only as Felice told the hacking forum that they were offering “ the full NPD database ,” according to a screenshot taken by BleepingComputer. The information consists of about 2.7 billion records, each of which includes a person’s full name, address, date of birth, Social Security number and phone number, along with alternate names and birth dates, Felice claimed.

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National Public Data didn’t respond to a request for comment, nor has it formally notified people about the alleged breach. It has, however, been telling people who contacted it via email that “we are aware of certain third-party claims about consumer data and are investigating these issues.”

In that email, the company also said that it had “purged the entire database, as a whole, of any and all entries, essentially opting everyone out.” As a result, it said, it has deleted any “non-public personal information” about people, although it added, “We may be required to retain certain records to comply with legal obligations.”

Several news outlets that focus on cybersecurity have looked at portions of the data Felice offered and said they appear to be real people’s actual information. If the leaked material is what it’s claimed to be, here are some of the risks posed and the steps you can take to protect yourself.

The threat of ID theft

The leak purports to provide much of the information that banks, insurance companies and service providers seek when creating accounts — and when granting a request to change the password on an existing account.

A few key pieces appeared to be missing from the hackers’ haul. One is email addresses, which many people use to log on to services. Another is driver’s license or passport photos, which some governmental agencies rely on to verify identities.

Still, Murray of PIRG said that bad actors could do “all kinds of things” with the leaked information, the most worrisome probably being to try to take over someone’s accounts — including those associated with their bank, investments, insurance policies and email. With your name, Social Security number, date of birth and mailing address, a fraudster could create fake accounts in your name or try to talk someone into resetting the password on one of your existing accounts.

“For somebody who’s really suave at it,” Murray said, “the possibilities are really endless.”

It’s also possible that criminals could use information from previous data breaches to add email addresses to the data from the reported National Public Data leak. Armed with all that, Murray said, “you can cause all kinds of chaos, commit all kinds of crimes, steal all kinds of money.”

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How to protect yourself

Data breaches have been so common over the years, some security experts say sensitive information about you is almost certainly available in the dark corners of the internet. And there are a lot of people capable of finding it; VPNRanks, a website that rates virtual private network services, estimates that 5 million people a day will access the dark web through the anonymizing TOR browser, although only a portion of them will be up to no good.

If you suspect that your Social Security number or other important identifying information about you has been leaked, experts say you should put a freeze on your credit files at the three major credit bureaus, Experian , Equifax and TransUnion . You can do so for free, and it will prevent criminals from taking out loans, signing up for credit cards and opening financial accounts under your name. The catch is that you’ll need to remember to lift the freeze temporarily if you are obtaining or applying for something that requires a credit check.

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Placing a freeze can be done online or by phone, working with each credit bureau individually. PIRG cautions never to do so in response to an unsolicited email or text purporting to be from one of the credit agencies — such a message is probably the work of a scammer trying to dupe you into revealing sensitive personal information.

For more details, check out PIRG’s step-by-step guide to credit freezes .

You can also sign up for a service that monitors your accounts and the dark web to guard against identity theft, typically for a fee. If your data is exposed in a breach, the company whose network was breached will often provide one of these services for free for a year or more.

If you want to know whether you have something to worry about, multiple websites and service providers such as Google and Experian can scan the dark web for your information to see whether it’s out there. But those aren’t specific to the reported National Public Data breach. For that information, try a free tool from the cybersecurity company Pentester that offers to search for your information in the breached National Public Data files . Along with the search results, Pentester displays links to the sites where you can freeze your credit reports.

As important as these steps are to stop people from opening new accounts in your name, they aren’t much help protecting your existing accounts. Oddly enough, those accounts are especially vulnerable to identity thieves if you haven’t signed up for online access to them, Murray said — that’s because it’s easier for thieves to create a login and password while pretending to be you than it is for them to crack your existing login and password.

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Of course, having strong passwords that are different for every service and changed periodically helps. Password manager apps offer a simple way to create and keep track of passwords by storing them in the cloud, essentially requiring you to remember one master password instead of dozens of long and unpronounceable ones. These are available both for free (such as Apple’s iCloud Keychain) and for a fee .

Beyond that, experts say it’s extremely important to sign up for two-factor authentication. That adds another layer of security on top of your login and password. The second factor is usually something sent or linked to your phone, such as a text message; a more secure approach is to use an authenticator app, which will keep you secure even if your phone number is hijacked by scammers .

Yes, scammers can hijack your phone number through techniques called SIM swaps and port-out fraud , causing more identity-theft nightmares. To protect you on that front, AT&T allows you to create a passcode restricting access to your account; T-Mobile offers optional protection against your phone number being switched to a new device, and Verizon automatically blocks SIM swaps by shutting down both the new device and the existing one until the account holder weighs in with the existing device.

Your worst enemy may be you

As much or more than hacked data, scammers also rely on people to reveal sensitive information about themselves. One common tactic is to pose as your bank, employer, phone company or other service provider with whom you’ve done business and then try to hook you with a text or email message.

Banks, for example, routinely tell customers that they will not ask for their account information by phone. Nevertheless, scammers have coaxed victims into providing their account numbers, logins and passwords by posing as bank security officers trying to stop an unauthorized withdrawal or some other supposedly urgent threat.

People may even get an official-looking email purportedly from National Public Data, offering to help them deal with the reported leak, Murray said. “It’s not going to be NPD trying to help. It’s going to be some bad guy overseas” trying to con them out of sensitive information, she said.

It’s a good rule of thumb never to click on a link or call a phone number in an unsolicited text or email. If the message warns about fraud on your account and you don’t want to simply ignore it, look up the phone number for that company’s fraud department (it’s on the back of your debit and credit cards) and call for guidance.

“These bad guys, this is what they do for a living,” Murray said. They might send out tens of thousands of queries and get only one response, but that response could net them $10,000 from an unwitting victim. “Ten thousand dollars in one day for having one hit with one victim, that’s a pretty good return on investment,” she said. “That’s what motivates them.”

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