homework 1 real numbers & properties

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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1-4 Properties of Real Numbers

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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, cattlemen, and tradesmen 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 A.D. 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 A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

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[latex]\,\left\{1,2,3,...\right\}\,[/latex], 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:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the opposites of the natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex] 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.

[latex]\begin{array}{lllll}\stackrel{\text{negative integers}}{\stackrel{}{\dots ,-3,-2,-1,}}\hfill & \hfill & \stackrel{\text{zero}}{\stackrel{}{0,}}\hfill & \hfill & \stackrel{\text{positive integers}}{\stackrel{}{1,2,3,\cdots }}\hfill \end{array}[/latex]

The set of rational numbers is written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.\,[/latex]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 in decimal form. Any rational number can be represented as either:

a terminating decimal:[latex]\,\frac{15}{8}=1.875,[/latex] or
a repeating decimal:[latex]\,\frac{4}{11}=0.36363636\dots =0.\overline{36}[/latex]

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.

[latex]7=\frac{7}{1}[/latex]
[latex]0=\frac{0}{1}[/latex]
[latex]-8=-\frac{8}{1}[/latex]
[latex]\frac{11}{1}[/latex]
[latex]\frac{3}{1}[/latex]
[latex]-\frac{4}{1}[/latex]

Identifying Rational Numbers

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

[latex]-\frac{5}{7}[/latex]
[latex]\frac{15}{5}[/latex]
[latex]\frac{13}{25}[/latex]

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

[latex]-\frac{5}{7}=-0.\stackrel{\text{———}}{714285},[/latex] a repeating decimal
[latex]\frac{15}{5}=3\,[/latex](or 3.0), a terminating decimal
[latex]\frac{13}{25}=0.52,[/latex] a terminating decimal
[latex]\frac{68}{17}[/latex]
[latex]\frac{8}{13}[/latex]
[latex]-\frac{17}{20}[/latex]
4 (or 4.0), terminating;
[latex]0.\overline{615384},[/latex]repeating;
–0.85, terminating

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[latex]\,\frac{3}{2},[/latex]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.

[latex]\sqrt{25}[/latex]
[latex]\frac{33}{9}[/latex]
[latex]\sqrt{11}[/latex]
[latex]\frac{17}{34}[/latex]
[latex]0.3033033303333\dots[/latex]
[latex]\sqrt{25}:\,[/latex]This can be simplified as[latex]\,\sqrt{25}=5.\,[/latex]Therefore,[latex]\sqrt{25}\,[/latex]is rational.

So,[latex]\,\frac{33}{9}\,[/latex]is rational and a repeating decimal.

[latex]\sqrt{11}:\,[/latex]This cannot be simplified any further. Therefore,[latex]\,\sqrt{11}\,[/latex]is an irrational number.

So,[latex]\,\frac{17}{34}\,[/latex]is rational and a terminating decimal.

[latex]0.3033033303333\dots \,[/latex]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.
[latex]\frac{7}{77}[/latex]
[latex]\sqrt{81}[/latex]
[latex]4.27027002700027\dots[/latex]
[latex]\frac{91}{13}[/latex]
[latex]\sqrt{39}[/latex]
rational and repeating;
rational and terminating;
irrational;

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?

[latex]-\frac{10}{3}[/latex]
[latex]\sqrt{5}[/latex]
[latex]-\sqrt{289}[/latex]
[latex]-6\pi[/latex]
[latex]0.615384615384\dots[/latex]
[latex]-\frac{10}{3}\,[/latex]is negative and rational. It lies to the left of 0 on the number line.
[latex]\sqrt{5}\,[/latex]is positive and irrational. It lies to the right of 0.
[latex]-\sqrt{289}=-\sqrt{{17}^{2}}=-17\,[/latex]is negative and rational. It lies to the left of 0.
[latex]-6\pi \,[/latex]is negative and irrational. It lies to the left of 0.
[latex]0.615384615384\dots \,[/latex]is a repeating decimal so it is rational and positive. It lies to the right of 0.
[latex]\sqrt{73}[/latex]
[latex]-11.411411411\dots[/latex]
[latex]\frac{47}{19}[/latex]
[latex]-\frac{\sqrt{5}}{2}[/latex]
[latex]6.210735[/latex]
positive, irrational; right
negative, rational; left
positive, rational; right
negative, irrational; left

Sets of Numbers as Subsets

A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3… N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: …, -3, -2, -1 I. The outermost circle contains: m/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q´.

Sets of Numbers

The set of natural numbers includes the numbers used for counting:[latex]\,\left\{1,2,3,...\right\}.[/latex]

The set of whole numbers is the set of natural numbers plus zero:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the negative natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex]

The set of rational numbers includes fractions written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.[/latex]

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating:[latex]\,\left\{h|h\text{ is not a rational number}\right\}.[/latex]

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′ ).

[latex]\sqrt{36}[/latex]
[latex]\frac{8}{3}[/latex]

[latex]-6[/latex]

[latex]3.2121121112\dots[/latex]
[latex]-\frac{35}{7}[/latex]
[latex]0[/latex]
[latex]\sqrt{169}[/latex]
[latex]\sqrt{24}[/latex]
[latex]4.763763763\dots[/latex]

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,[latex]\,{4}^{2}=4\cdot 4=16.\,[/latex]We can raise any number to any power. In general, the exponential notation[latex]\,{a}^{n}\,[/latex]means that the number or variable[latex]\,a\,[/latex]is used as a factor[latex]\,n\,[/latex]times.

In this notation,[latex]\,{a}^{n}\,[/latex]is read as the n th power of[latex]\,a,\,[/latex]where[latex]\,a\,[/latex]is called the base and[latex]\,n\,[/latex]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,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}\,[/latex]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.

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[latex]\,{4}^{2}\,[/latex]as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}=12.[/latex]

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.

Simplify any expressions within grouping symbols.
Simplify any expressions containing exponents or radicals.
Perform any multiplication and division in order, from left to right.
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.

[latex]{\left(3\cdot 2\right)}^{2}-4\left(6+2\right)[/latex]
[latex]\frac{{5}^{2}-4}{7}-\sqrt{11-2}[/latex]
[latex]6-|5-8|+3\left(4-1\right)[/latex]
[latex]\frac{14-3\cdot 2}{2\cdot 5-{3}^{2}}[/latex]
[latex]7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1[/latex]
[latex]\begin{array}{cccc}\hfill{\left(3\cdot 2\right)}^{2}-4\left(6+2\right) =& {\left(6\right)}^{2}-4\left(8\right)\phantom{\rule{1em}{0ex}} \text{Simplify parentheses}\end{array}[/latex] [latex]\begin{array}\\ =& 36-4\left(8\right) & \phantom{\rule{2em}{0ex}}\text{Simplify exponent}\end{array}[/latex] [latex]\begin{array}\\ =& 36-32 & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\ =& 4 & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex]

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.

[latex]\begin{array}{cccc}{ 6-|5-8|+3\left(4-1\right)} \end{array}[/latex] [latex]\begin{array}= 6-|-3|+3\left(3\right) \phantom{\rule{2em}{0ex}}\text{Simplify inside grouping symbols} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+3\left(3\right)\phantom{\rule{2em}{0ex}}\text{Simplify absolute value} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\&= 3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex] [latex]\begin{array}\\&= 12\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify addition} \end{array}[/latex]

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

[latex]\begin{array}{cccc}\hfill 7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1& =& 7\left(15\right)-2\left[\left(3\right)-{4}^{2}\right]+1\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify inside parentheses}\hfill \\ & =& 7\left(15\right)-2\left(3-16\right)+1\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify exponent}\hfill \\ & =& 7\left(15\right)-2\left(-13\right)+1\hfill & \phantom{\rule{2em}{0ex}}\text{Subtract}\hfill \\ & =& 105+26+1\hfill & \phantom{\rule{2em}{0ex}}\text{Multiply}\hfill \\ & =& 132\hfill & \phantom{\rule{2em}{0ex}}\text{Add}\hfill \end{array}[/latex]
[latex]\sqrt{{5}^{2}-{4}^{2}}+7{\left(5-4\right)}^{2}[/latex]
[latex]1+\frac{7\cdot 5-8\cdot 4}{9-6}[/latex]
[latex]|1.8-4.3|+0.4\sqrt{15+10}[/latex]
[latex]\frac{1}{2}\left[5\cdot {3}^{2}-{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}[/latex]
[latex]\left[{\left(3-8\right)}^{2}-4\right]-\left(3-8\right)[/latex]

Using Properties of Real Numbers

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,[latex]\,17-5\,[/latex]is not the same as[latex]\,5-17.\,[/latex]Similarly,[latex]\,20÷5\ne 5÷20.[/latex]

Associative Properties

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.

The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.

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[latex]\,12-\left(5+3\right).\,[/latex]We can rewrite the difference of the two terms 12 and[latex]\,\left(5+3\right)\,[/latex]by turning the subtraction expression into addition of the opposite. So instead of subtracting[latex]\,\left(5+3\right),[/latex]we add the opposite.

Now, distribute[latex]\,-1\,[/latex]and simplify the result.

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[latex]\,\left(-6\right)+0=-6\,[/latex]and[latex]\,23\cdot 1=23.\,[/latex]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− a , that, when added to the original number, results in the additive identity, 0.

For example, if[latex]\,a=-8,[/latex]the additive inverse is 8, since[latex]\,\left(-8\right)+8=0.[/latex]

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[latex]\,\frac{1}{a},[/latex] that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if[latex]\,a=-\frac{2}{3},[/latex] the reciprocal, denoted[latex]\,\frac{1}{a},[/latex] is[latex]\,-\frac{3}{2}\,[/latex] 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.

[latex]\,3\cdot 6+3\cdot 4[/latex]
[latex]\,\left(5+8\right)+\left(-8\right)[/latex]
[latex]\,6-\left(15+9\right)[/latex]
[latex]\,\frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)[/latex]
[latex]\,100\cdot \left[0.75+\left(-2.38\right)\right][/latex]
[latex]\begin{array}{cccc}\hfill 3\cdot 6+3\cdot 4& =& 3\cdot \left(6+4\right)\hfill & \phantom{\rule{7em}{0ex}}\text{Distributive property}\hfill \\ & =& 3\cdot 10\hfill & \phantom{\rule{7em}{0ex}}\text{Simplify}\hfill \\ & =& 30\hfill & \phantom{\rule{7em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill \left(5+8\right)+\left(-8\right)& =& 5+\left[8+\left(-8\right)\right]\hfill & \phantom{\rule{3em}{0ex}}\text{Associative property of addition}\\ & =& 5+0\hfill & \phantom{\rule{3em}{0ex}}\text{Inverse property of addition}\hfill \\ & =& 5\hfill & \phantom{\rule{3em}{0ex}}\text{Identity property of addition}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill 6-\left(15+9\right)\hfill & =& 6+\left[\left(-15\right)+\left(-9\right)\right]\hfill & \phantom{\rule{2em}{0ex}}\text{Distributive property}\hfill \\ & =& 6+\left(-24\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \\ & =& -18\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill \frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)& =& \frac{4}{7}\cdot \left(\frac{7}{4}\cdot \frac{2}{3}\right)\hfill & \phantom{\rule{6em}{0ex}}\text{Commutative property of multiplication}\hfill \\ & =& \left(\frac{4}{7}\cdot \frac{7}{4}\right)\cdot \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Associative property of multiplication}\hfill \\ & =& 1\cdot \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Inverse property of multiplication}\hfill \\ & =& \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Identity property of multiplication}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill 100\cdot \left[0.75+\left(-2.38\right)\right]& =& 100\cdot 0.75+100\cdot \left(-2.38\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Distributive property}\\ & =& 75+\left(-238\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \\ & =& -163\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\,\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right][/latex]
[latex]\,5\cdot \left(6.2+0.4\right)[/latex]
[latex]\,18-\left(7-15\right)[/latex]
[latex]\,\frac{17}{18}+\left[\frac{4}{9}+\left(-\frac{17}{18}\right)\right][/latex]
[latex]\,6\cdot \left(-3\right)+6\cdot 3[/latex]
11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
33, distributive property;
26, distributive property;
[latex]\,\frac{4}{9},[/latex] commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
0, distributive property, inverse property of addition, identity property of addition

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as[latex]\,x+5,\frac{4}{3}\pi {r}^{3},[/latex] or[latex]\,\sqrt{2{m}^{3}{n}^{2}}.\,[/latex]In the expression[latex]\,x+5,[/latex] 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.

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

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

[latex]\frac{4}{3}\pi {r}^{3}[/latex]
[latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]
[latex]2\pi r\left(r+h\right)[/latex]
[latex]4{y}^{3}+y[/latex]

Evaluating an Algebraic Expression at Different Values

Evaluate the expression[latex]\,2x-7\,[/latex]for each value for x.

[latex]\,x=0[/latex]
[latex]\,x=1[/latex]
[latex]\,x=\frac{1}{2}[/latex]
[latex]\,x=-4[/latex]
Substitute 0 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(0\right)-7\\ & =& 0-7\hfill \\ & =& -7\hfill \end{array}[/latex]
Substitute 1 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}2x-7& =& 2\left(1\right)-7\hfill \\ & =& 2-7\hfill \\ & =& -5\hfill \end{array}[/latex]
Substitute[latex]\,\frac{1}{2}\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(\frac{1}{2}\right)-7\hfill \\ & =& 1-7\hfill \\ & =& -6\hfill \end{array}[/latex]
Substitute[latex]\,-4\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(-4\right)-7\\ & =& -8-7\hfill \\ & =& -15\hfill \end{array}[/latex]

Evaluate the expression[latex]\,11-3y\,[/latex]for each value for y.

[latex]\,y=2[/latex]
[latex]\,y=0[/latex]
[latex]\,y=\frac{2}{3}[/latex]
[latex]\,y=-5[/latex]

Evaluate each expression for the given values.

[latex]\,x+5\,[/latex]for[latex]\,x=-5[/latex]
[latex]\,\frac{t}{2t-1}\,[/latex]for[latex]\,t=10[/latex]
[latex]\,\frac{4}{3}\pi {r}^{3}\,[/latex]for[latex]\,r=5[/latex]
[latex]\,a+ab+b\,[/latex]for[latex]a=11,b=-8[/latex]
[latex]\,\sqrt{2{m}^{3}{n}^{2}}\,[/latex]for[latex]\,m=2,n=3[/latex]
Substitute[latex]\,-5\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill x+5& =& \left(-5\right)+5\hfill \\ & =& 0\hfill \end{array}[/latex]
Substitute 10 for[latex]\,t.[/latex] [latex]\begin{array}{ccc}\hfill \frac{t}{2t-1}& =& \frac{\left(10\right)}{2\left(10\right)-1}\hfill \\ & =& \frac{10}{20-1}\hfill \\ & =& \frac{10}{19}\hfill \end{array}[/latex]
Substitute 5 for[latex]\,r.[/latex] [latex]\begin{array}{ccc}\hfill \frac{4}{3}\pi {r}^{3}& =& \frac{4}{3}\pi {\left(5\right)}^{3}\\ & =& \frac{4}{3}\pi \left(125\right)\hfill \\ & =& \frac{500}{3}\pi \hfill \end{array}[/latex]
Substitute 11 for[latex]\,a\,[/latex]and –8 for[latex]\,b.[/latex] [latex]\begin{array}{ccc}\hfill a+ab+b& =& \left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right)\\ & =& 11-88-8\hfill \\ & =& -85\hfill \end{array}[/latex]
Substitute 2 for[latex]\,m\,[/latex]and 3 for[latex]\,n.[/latex] [latex]\begin{array}{ccc}\hfill \sqrt{2{m}^{3}{n}^{2}}& =& \sqrt{2{\left(2\right)}^{3}{\left(3\right)}^{2}}\hfill \\ & =& \sqrt{2\left(8\right)\left(9\right)}\hfill \\ & =& \sqrt{144}\hfill \\ & =& 12\hfill \end{array}[/latex]
[latex]\,\frac{y+3}{y-3}\,[/latex]for[latex]\,y=5[/latex]
[latex]\,7-2t\,[/latex]for[latex]\,t=-2[/latex]
[latex]\,\frac{1}{3}\pi {r}^{2}\,[/latex]for[latex]\,r=11[/latex]
[latex]\,{\left({p}^{2}q\right)}^{3}\,[/latex]for[latex]\,p=-2,q=3[/latex]
[latex]\,4\left(m-n\right)-5\left(n-m\right)\,[/latex]for[latex]\,m=\frac{2}{3},n=\frac{1}{3}[/latex]
[latex]\,\frac{121}{3}\pi[/latex];

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[latex]\,2x+1=7\,[/latex]has the unique solution of 3[latex][/latex] because when we substitute 3 for[latex]\,x\,[/latex]in the equation, we obtain the true statement[latex]\2\left(3\right)+1=7.[/latex]

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[latex]\,A\,[/latex]of a circle in terms of the radius[latex]\,r\,[/latex]of the circle:[latex]\,A=\pi {r}^{2}.\,[/latex]For any value of[latex]\,r,[/latex] the area[latex]\,A\,[/latex]can be found by evaluating the expression[latex]\,\pi {r}^{2}.[/latex]

Using a Formula

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

A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.

Evaluate the expression[latex]\,2\pi r\left(r+h\right)\,[/latex]for[latex]\,r=6\,[/latex]and[latex]\,h=9.[/latex]

The surface area is[latex]\,180\pi \,[/latex]square inches.

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm 2 ) is found to be[latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]See Figure 4. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.

Given that [latex]L=32[/latex] and [latex]W=24[/latex], plug the numbers into the formula [latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]

[latex]\,A=\left(32+16\right)\left(24+16\right)-32\cdot 16.\,[/latex]

[latex]\,A=\left(48\right)\left(40\right)-512.\,[/latex]

Simplifying Algebraic Expressions

Simplify each algebraic expression.

[latex]3x-2y+x-3y-7[/latex]
[latex]2r-5\left(3-r\right)+4[/latex]
[latex]\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)[/latex]
[latex]2mn-5m+3mn+n[/latex]
[latex]\begin{array}{cccc}\hfill 3x-2y+x-3y-7& =& 3x+x-2y-3y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 4x-5y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill 2r-5\left(3-r\right)+4& =& 2r-15+5r+4\hfill & \phantom{\rule{10em}{0ex}}\text{Distributive property}\hfill \\ & =& 2r+5r-15+4\hfill & \phantom{\rule{10em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 7r-11\hfill & \phantom{\rule{10em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}\hfill \left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)& =& 4t-\frac{5}{4}s-\frac{2}{3}t-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Distributive property}\hfill \\ & =& 4t-\frac{2}{3}t-\frac{5}{4}s-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \frac{10}{3}t-\frac{13}{4}s\hfill & \phantom{\rule{4em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\begin{array}{cccc}2mn-5m+3mn+n& =& 2mn+3mn-5m+n& \phantom{\rule{5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \text{ }5mn-5m+n\hfill & \phantom{\rule{5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
[latex]\frac{2}{3}y-2\left(\frac{4}{3}y+z\right)[/latex]
[latex]\frac{5}{t}-2-\frac{3}{t}+1[/latex]
[latex]4p\left(q-1\right)+q\left(1-p\right)[/latex]
[latex]9r-\left(s+2r\right)+\left(6-s\right)[/latex]
[latex]\,-2y-2z\text{ or }-2\left(y+z\right);[/latex]
[latex]\,\frac{2}{t}-1;[/latex]
[latex]\,3pq-4p+q;[/latex]
[latex]\,7r-2s+6[/latex]

Simplifying a Formula

A rectangle with length[latex]\,L\,[/latex]and width[latex]\,W\,[/latex]has a perimeter[latex]\,P\,[/latex]given by[latex]\,P=L+W+L+W.\,[/latex]Simplify this expression.

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

[latex]A=P\left(1+rt\right)[/latex]

Key Concepts

Rational numbers may be written as fractions or terminating or repeating decimals.
Determine whether a number is rational or irrational by writing it as a decimal.
The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.
The order of operations is used to evaluate expressions.
The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.
Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.
Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.
Is[latex]\,\sqrt{2}\,[/latex]an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

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.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

For the following exercises, simplify the given expression.

[latex]10+2\,×\,\left(5-3\right)[/latex]
[latex]6÷2-\left(81÷{3}^{2}\right)[/latex]
[latex]18+{\left(6-8\right)}^{3}[/latex]
[latex]-2\,×\,{\left[16÷{\left(8-4\right)}^{2}\right]}^{2}[/latex]

[latex]-2[/latex]

[latex]4-6+2\,×\,7[/latex]
[latex]3\left(5-8\right)[/latex]

[latex]-9[/latex]

[latex]4+6-10÷2[/latex]
[latex]12÷\left(36÷9\right)+6[/latex]
[latex]{\left(4+5\right)}^{2}÷3[/latex]
[latex]3-12\,×\,2+19[/latex]
[latex]2+8\,×\,7÷4[/latex]
[latex]5+\left(6+4\right)-11[/latex]
[latex]9-18÷{3}^{2}[/latex]
[latex]14\,×\,3÷7-6[/latex]
[latex]9-\left(3+11\right)\,×\,2[/latex]
[latex]6+2\,×\,2-1[/latex]
[latex]64÷\left(8+4\,×\,2\right)[/latex]
[latex]9+4\left({2}^{2}\right)[/latex]
[latex]{\left(12÷3\,×\,3\right)}^{2}[/latex]
[latex]25÷{5}^{2}-7[/latex]
[latex]\left(15-7\right)\,×\,\left(3-7\right)[/latex]
[latex]2\,×\,4-9\left(-1\right)[/latex]
[latex]{4}^{2}-25\,×\,\frac{1}{5}[/latex]
[latex]12\left(3-1\right)÷6[/latex]

For the following exercises, solve for the variable.

[latex]8\left(x+3\right)=64[/latex]
[latex]4y+8=2y[/latex]

[latex]-4[/latex]

[latex]\left(11a+3\right)-18a=-4[/latex]
[latex]4z-2z\left(1+4\right)=36[/latex]
[latex]4y{\left(7-2\right)}^{2}=-200[/latex]
[latex]-{\left(2x\right)}^{2}+1=-3[/latex]

[latex]±1[/latex]

[latex]8\left(2+4\right)-15b=b[/latex]
[latex]2\left(11c-4\right)=36[/latex]
[latex]4\left(3-1\right)x=4[/latex]
[latex]\frac{1}{4}\left(8w-{4}^{2}\right)=0[/latex]

For the following exercises, simplify the expression.

[latex]4x+x\left(13-7\right)[/latex]
[latex]2y-{\left(4\right)}^{2}y-11[/latex]

[latex]-14y-11[/latex]

[latex]\frac{a}{{2}^{3}}\left(64\right)-12a÷6[/latex]
[latex]8b-4b\left(3\right)+1[/latex]

[latex]-4b+1[/latex]

[latex]5l÷3l\,×\,\left(9-6\right)[/latex]
[latex]7z-3+z\,×\,{6}^{2}[/latex]

[latex]43z-3[/latex]

[latex]4\,×\,3+18x÷9-12[/latex]
[latex]9\left(y+8\right)-27[/latex]

[latex]9y+45[/latex]

[latex]\left(\frac{9}{6}t-4\right)2[/latex]
[latex]6+12b-3\,×\,6b[/latex]

[latex]-6b+6[/latex]

[latex]18y-2\left(1+7y\right)[/latex]
[latex]{\left(\frac{4}{9}\right)}^{2}\,×\,27x[/latex]

[latex]\frac{16x}{3}[/latex]

[latex]8\left(3-m\right)+1\left(-8\right)[/latex]
[latex]9x+4x\left(2+3\right)-4\left(2x+3x\right)[/latex]

[latex]9x[/latex]

[latex]{5}^{2}-4\left(3x\right)[/latex]

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighbor’s car.

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

[latex]\frac{1}{2}\left(40-10\right)+5[/latex]

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[latex]\,\pi .\,[/latex]Is the circumference of a quarter a whole number, a rational number, or an irrational number?

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[latex]\,g\,[/latex]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.

[latex]g+400-2\left(600\right)=1200[/latex]

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must spend the budget such that[latex]\,2,500,000-x=0.\,[/latex]What property of addition tells us what the value of x must be?

inverse property of addition

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

[latex]0.5{\left(12.3\right)}^{2}-48x=\frac{3}{5}[/latex]
[latex]{\left(0.25-0.75\right)}^{2}x-7.2=9.9[/latex]
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:[latex]\,\sqrt{-18-4\left(5\right)\left(-1\right)}.[/latex]
Determine whether the simplified expression is rational or irrational:[latex]\,\sqrt{-16+4\left(5\right)+5}.[/latex]
The division of two whole numbers will always result in what type of number?
What property of real numbers would simplify the following expression:[latex]\,4+7\left(x-1\right)?[/latex]

Media Attributions

1.1 Figure 1 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
1.1 Figure 2 © OpenStax Algebra and Trignometry is licensed under a CC BY (Attribution) license
1.1 Distributive Property Graphic © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
1.1 Figure 3 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
1.1 Figure 4 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license

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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

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1.5: Properties of Real Numbers

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Learning Objectives

By the end of this section, you will be able to:

Use the commutative and associative properties
Use the properties of identity, inverse, and zero
Simplify expressions using the Distributive Property

Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add $8+9$ or $9+8$, the results are the same—they both equal 17. So, $8+9=9+8$. The order in which we add does not matter!

Similarly, when multiplying two numbers, the order does not affect the result. If we multiply $9·8$ or $8·9$ the results are the same—they both equal 72. So, $9·8=8·9$. The order in which we multiply does not matter! These examples illustrate the Commutative Property .

COMMUTATIVE PROPERTY

\[\begin{array}{lll} \textbf{of Addition} & \text{If }a \text{ and }b \text{are real numbers, then} & a+b=b+a. \\ \textbf{of Multiplication} & \text{If }a \text{ and }b \text{are real numbers, then} & a·b=b·a. \end{array} \]

When adding or multiplying, changing the order gives the same result.

The Commutative Property has to do with order. We subtract $9−8$ and $8−9$, and see that $9−8\neq 8−9$. Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative .

Division is not commutative either . Since $12÷3\neq 3÷12$, changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!

Addition and multiplication are commutative.
Subtraction and division are not commutative.

When adding three numbers, changing the grouping of the numbers gives the same result. For example,$(7+8)+2=7+(8+2)$, since each side of the equation equals 17.

This is true for multiplication, too. For example, $\left(5·\frac{1}{3}\right)·3=5·\left(\frac{1}{3}·3\right)$, since each side of the equation equals 5.

These examples illustrate the Associative Property .

ASSOCIATIVE PROPERTY

\[\begin{array}{lll} \textbf{of Addition} & \text{If }a,b, \text{ and }c \text{ are real numbers, then} & (a+b)+c=a+(b+c). \\ \textbf{of Multiplication} & \text{If }a,b,\text{ and }c \text{ are real numbers, then} & (a·b)·c=a·(b·c). \end{array} \]

When adding or multiplying, changing the grouping gives the same result.

The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

\[\begin{array}{cc} (10−3)−2\neq 10−(3−2) & (24÷4)÷2\neq 24÷(4÷2) \\ 7−2\neq 10−1 & 6÷2\neq 24÷2 \\ 5\neq 9 & 3\neq 12 \end{array}\]

When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.

Example $\PageIndex{1}$

Simplify: $18p+6q+15p+5q$.

\[\begin{array}{lc} \text{} & 18p+6q+15p+5q \\ \text{Use the Commutative Property of addition to} & 18p+15p+6q+5q \\ \text{reorder so that like terms are together.} & {} \\ \text{Add like terms.} & 33p+11q \end{array}\]

Example $\PageIndex{2}$

Simplify: $23r+14s+9r+15s$.

$32r+29s$

Example $\PageIndex{3}$

Simplify: $37m+21n+4m−15n$.

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.

EXAMPLE $\PageIndex{4}$

Simplify: $(\frac{5}{13}+\frac{3}{4})+\frac{1}{4}$.

$ \begin{array}{lc} \text{} & (\frac{5}{13}+\frac{3}{4})+\frac{1}{4} \\ {\text{Notice that the last 2 terms have a common} \\ \text{denominator, so change the grouping.} } & \frac{5}{13}+(\frac{3}{4}+\frac{1}{4}) \\ \text{Add in parentheses first.} & \frac{5}{13}+(\frac{4}{4}) \\ \text{Simplify the fraction.} & \frac{5}{13}+1 \\ \text{Add.} & 1\frac{5}{13} \\ \text{Convert to an improper fraction.} & \frac{18}{13} \end{array}$

EXAMPLE $\PageIndex{5}$

Simplify: $(\frac{7}{15}+\frac{5}{8})+\frac{3}{8}.$

$1 \frac{7}{15}$

EXAMPLE $\PageIndex{6}$

Simplify: $(\frac{2}{9}+\frac{7}{12})+\frac{5}{12}$.

$1\frac{2}{9}$

Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity . The Identity Property of Addition that states that for any real number $a,a+0=a$ and $0+a=a.$

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity. The Identity Property of Multiplication that states that for any real number $a,a·1=a$ and $1⋅a=a.$

We summarize the Identity Properties here.

IDENTITY PROPERTY

\[\begin{array}{ll} \textbf{of Addition} \text{ For any real number }a:a+0=a & 0+a=a \\ \\ \\ \textbf{0} \text{ is the } \textbf{additive identity} \\ \textbf{of Multiplication} \text{ For any real number } a:a·1=a & 1·a=a \\ \\ \\ \textbf{1} \text{ is the } \textbf{multiplicative identity} \end{array}\]

What number added to 5 gives the additive identity, 0? We know

$alt$

The missing number was the opposite of the number!

We call $−a$ the additive inverse of $a$. The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number $a,a+(−a)=0.$

What number multiplied by $\frac{2}{3}$ gives the multiplicative identity, 1? In other words, $\frac{2}{3}$ times what results in 1? We know

$alt$

The missing number was the reciprocal of the number!

We call $\frac{1}{a}$ the multiplicative inverse of a . The reciprocal of a number is its multiplicative inverse. This leads to the Inverse Property of Multiplication that states that for any real number $a,a\neq 0,a·\frac{1}{a}=1.$

We’ll formally state the inverse properties here.

INVERSE PROPERTY

\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end{array}\]

The Identity Property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

What about division involving zero? What is $0÷3$? Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So, $0÷3=0.$

We can check division with the related multiplication fact. So we know $0÷3=0$ because $0·3=0$.

Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact:

$alt$

Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4. We conclude that there is no answer to $4÷0$ and so we say that division by 0 is undefined .

We summarize the properties of zero here.

PROPERTIES OF ZERO

Multiplication by Zero: For any real number a ,

\[a⋅0=0 \; \; \; 0⋅a=0 \; \; \; \; \text{The product of any number and 0 is 0.}\]

Division by Zero: For any real number a , $a\neq 0$

\[\begin{array}{cl} \dfrac{0}{a}=0 & \text{Zero divided by any real number, except itself, is zero.} \\ \dfrac{a}{0} \text{ is undefined} & \text{Division by zero is undefined.} \end{array}\]

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

EXAMPLE $\PageIndex{7}$

Simplify: $−84n+(−73n)+84n.$

$\begin{array}{lc} \text{} & −84n+(−73n)+84n \\ \text{Notice that the first and third terms are} \\ \text{opposites; use the Commutative Property of} & −84n+84n+(−73n) \\ \text{addition to re-order the terms.} \\ \text{Add left to right.} & 0+(−73n) \\ \text{Add.} & −73n \end{array}$

EXAMPLE $\PageIndex{8}$

Simplify: $−27a+(−48a)+27a$.

$−48a$

EXAMPLE $\PageIndex{9}$

Simplify: $39x+(−92x)+(−39x)$.

$−92x$

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.

EXAMPLE $\PageIndex{10}$

Simplify: $\frac{7}{15}⋅\frac{8}{23}⋅\frac{15}{7}$.

$\begin{array}{lc} \text{} & \frac{7}{15}⋅\frac{8}{23}⋅\frac{15}{7} \\ \text{Notice the first and third terms} \\ {\text{are reciprocals, so use the Commutative} \\ \text{Property of multiplication to re-order the} \\ \text{factors.}} & \frac{7}{15}·\frac{15}{7}·\frac{8}{23} \\ \text{Multiply left to right.} & 1·\frac{8}{23} \\ \text{Multiply.} & \frac{8}{23} \end{array}$

EXAMPLE $\PageIndex{11}$

Simplify: $\frac{9}{16}⋅\frac{5}{49}⋅\frac{16}{9}$.

$\frac{5}{49}$

Simplify: $\frac{6}{17}⋅\frac{11}{25}⋅\frac{17}{6}$.

$\frac{11}{25}$

The next example makes us aware of the distinction between dividing 0 by some number or some number being divided by 0.

Simplify: a. $\frac{0}{n+5}$, where $n\neq −5$ b. $\frac{10−3p}{0}$ where $10−3p\neq 0.$

$\begin{array}{lc} {} & \dfrac{0}{n+5} \\ \text{Zero divided by any real number except itself is 0.} & 0 \end{array}$

$\begin{array}{lc} {} & \dfrac{10−3p}{0} \\ \text{Division by 0 is undefined.} & \text{undefined} \end{array}$

EXAMPLE $\PageIndex{14}$

Simplify: a. $\frac{0}{m+7}$, where $m\neq −7$ b. $\frac{18−6c}{0}$, where $18−6c\neq 0$.

a. 0 b. undefined

EXAMPLE $\PageIndex{15}$

Simplify: a. $\frac{0}{d−4}$, where $d\neq 4$ b. $\frac{15−4q}{0}$, where $15−4q\neq 0$.

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times $9 so $27 and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.

DISTRIBUTIVE PROPERTY

$\begin{array}{lc} \text{If }a,b \text{,and }c \text{are real numbers, then} \; \; \; \; \; & a(b+c)=ab+ac \\ {} & (b+c)a=ba+ca \\ {} & a(b−c)=ab−ac \\{} & (b−c)a=ba−ca \end{array}$

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.

EXAMPLE $\PageIndex{16}$

Simplify: $3(x+4)$.

$\begin{array} {} & 3(x+4) \\ \text{Distribute.} \; \; \; \; \; \; \; \; & 3·x+3·4 \\ \text{Multiply.} & 3x+12 \end{array}$

Simplify: $4(x+2)$.

EXAMPLE $\PageIndex{18}$

Simplify: $6(x+7)$.

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example would look like this:

$alt$

EXAMPLE $\PageIndex{19}$

Simplify: $8(\frac{3}{8}x+\frac{1}{4})$.

EXAMPLE $\PageIndex{20}$

Simplify: $6(\frac{5}{6}y+\frac{1}{2})$.

EXAMPLE $\PageIndex{21}$

Simplify: $12(\frac{1}{3}n+\frac{3}{4})$

Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.

EXAMPLE $\PageIndex{22}$

Simplify: $100(0.3+0.25q)$.

EXAMPLE $\PageIndex{23}$

Simplify: $100(0.7+0.15p).$

EXAMPLE $\PageIndex{24}$

Simplify: $100(0.04+0.35d)$.

When we distribute a negative number, we need to be extra careful to get the signs correct!

EXAMPLE $\PageIndex{25}$

Simplify: $−11(4−3a).$

$\begin{array}{lc} {} & −11(4−3a) \\ \text{Distribute. } \; \; \; \; \; \; \; \; \; \;& −11·4−(−11)·3a \\ \text{Multiply.} & −44−(−33a) \\ \text{Simplify.} & −44+33a \end{array}$

Notice that you could also write the result as $33a−44.$ Do you know why?

Simplify: $−5(2−3a)$.

$−10+15a$

EXAMPLE $\PageIndex{27}$

Simplify: $−7(8−15y).$

$−56+105y$

In the next example, we will show how to use the Distributive Property to find the opposite of an expression.

Simplify: $−(y+5)$.

$\begin{array}{lc} {} & −(y+5) \\ \text{Multiplying by }−1 \text{ results in the opposite.}& −1(y+5) \\ \text{Distribute.} & −1·y+(−1)·5 \\ \text{Simplify.} & −y+(−5) \\ \text{Simplify.} & −y−5 \end{array} $

EXAMPLE $\PageIndex{29}$

Simplify: $−(z−11)$.

$−z+11$

EXAMPLE $\PageIndex{30}$

Simplify: $−(x−4)$.

$−x+4$

There will be times when we’ll need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the Distributive Property, which removes the parentheses. The next two examples will illustrate this.

EXAMPLE $\PageIndex{31}$

Simplify: $8−2(x+3)$

We follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

$\begin{array}{lc} {} & \text{8−2(x+3)} \\ \text{Distribute.} & 8−2·x−2·3 \\ \text{Multiply.} & 8−2x−6 \\ \text{Combine like terms.} &−2x+2 \end{array}$

EXAMPLE $\PageIndex{32}$

Simplify: $9−3(x+2)$.

$3−3x$

EXAMPLE $\PageIndex{33}$

Simplify: $7x−5(x+4)$.

$2x−20$

EXAMPLE $\PageIndex{34}$

Simplify: $4(x−8)−(x+3)$.

$\begin{array}{lc} {} & 4(x−8)−(x+3) \\ \text{Distribute.} & 4x−32−x−3 \\ \text{Combine like terms.} & 3x−35 \end{array}$

EXAMPLE $\PageIndex{35}$

Simplify: $6(x−9)−(x+12)$.

$5x−66$

EXAMPLE $\PageIndex{36}$

Simplify: $8(x−1)−(x+5)$.

$7x−13$

All the properties of real numbers we have used in this chapter are summarized here.

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Home & House Stagers in Elektrostal'

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Elektrostal', Moscow Oblast, Russia

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Featured Reviews for Home & House Stagers in Elektrostal'

Reach out to the pro(s) you want, then share your vision to get the ball rolling.
Request and compare quotes, then hire the Home Stager that perfectly fits your project and budget limits.

A home stager is a professional who prepares a house for sale, aiming to attract more buyers and potentially secure a higher selling price. They achieve this through the following techniques:

Rearranging furniture to optimize space and functionality.
Decluttering to create a clean and spacious look.
Making repairs to address visible issues.
Enhancing aesthetics with artwork, accessories, and lighting.
Introducing new furnishings to update the style.

Their goal is to present the house in the best light. Home stagers in Elektrostal' help buyers envision themselves living there, increasing the chances of a successful sale.

Decluttering
Furniture Selection
Space Planning
Art Selection
Accessory Selection

Benefits of the home staging in Elektrostal':

Attractive and inviting: Staging creates a welcoming atmosphere for potential buyers.
Faster sale: Homes sell more quickly, reducing time on the market.
Higher sale price: Staging can lead to higher offers and appeal to a wider range of buyers.
Showcasing best features: Strategic arrangement highlights positives and minimizes flaws.
Stand out online: Staged homes capture attention in online listings.
Emotional connection: Staging creates a positive impression that resonates with buyers.
Easy visualization: Buyers can easily picture themselves living in a staged home.
Competitive advantage: Staging sets your home apart from others on the market.
Affordable investment: Cost-effective way to maximize selling potential and ROI.
Professional expertise: Experienced stagers ensure optimal presentation for attracting buyers.

What does an Elektrostal' home stager do?

What should i consider before hiring an interior staging company, questions to ask potential real estate staging companies in elektrostal', moscow oblast, russia:, business services, connect with us.

IMAGES

Properties Of Real Numbers Worksheet
Real Number Properties Worksheet
Properties of real numbers graphic organizer notes study guide
Properties of Real Numbers Chart
Properties Of Real Numbers Worksheet
Real Number Properties Worksheet

VIDEO

Properties of Real Numbers
Properties of Real Numbers, Fsc(Part 1), Lec3, Chapter1
Lec-4(d)| Mathematics
"Exploring Class 9 Maths: Chapter 1 #part2 NCERT
Lecture # 3
Ex: 1.2, Q.5, Class 9th Maths, National Book Foundation, Federal Text Book Board Islamabad, APS

COMMENTS

1.1 Real Numbers: Algebra Essentials
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 ...
1.6: Properties of Real Numbers
Use the Commutative and Associative Properties. The order we add two numbers doesn't affect the result. If we add \ (8+9\) or \ (9+8\), the results are the same—they both equal 17. So, \ (8+9=9+8\). The order in which we add does not matter! Similarly, when multiplying two numbers, the order does not affect the result.
1.1: Real Numbers
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 3πr3, or √2m3n2. In the expression x + 5, 5 is called a constant because it does not vary and x is called a variable because it does.
Properties of Real Numbers
Properties of Real Numbers - MathBitsNotebook (A1) A real number is a value that represents a quantity along a continuous number line. Real numbers can be ordered. The symbol for the set of real numbers is , which is the letter R in the typeface "blackboard bold". The real numbers include: counting (natural) numbers ( ) {1, 2, 3, ...
1.6: Properties of Real Numbers
The commutative properties apply only to addition and multiplication! Addition and multiplication are commutative. Subtraction and division are not commutative. When adding three numbers, changing the grouping of the numbers gives the same result. For example, since each side of the equation equals 17.
1-4 Properties of Real Numbers
1-4 Bell Work - Properties of Real Numbers. 1-4 Exit Quiz - Properties of Real Numbers. 1-4 Guide Notes SE - Properties of Real Numbers. 1-4 Guide Notes TE - Properties of Real Numbers. 1-4 Lesson Plan - Properties of Real Numbers. 1-4 Online Activities - Properties of Real Numbers.
PDF Homework #1
Homework #1: The Properties of the Real Numbers . State a pair of inverse elements for addition: ... and then multiply the result by another number. The distributive property allows us to multiply the original two numbers individually by the third, THEN add them. Clearly, this is difficult to describe in words, so (maybe with a little research
1.1 Real Numbers: Algebra Essentials
The property states that for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a, that, when multiplied by the original number, results in the multiplicative identity, 1. a ⋅ 1 a = 1. For example, if a = − 2 3, the reciprocal, denoted 1 a, is − 3 2. because.
Real Number Properties
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
Unit 1 Real Numbers/ Properties of Real Numbers Flashcards
Study with Quizlet and memorize flashcards containing terms like Rational Numbers (Q), Natural Numbers (N), Integers (Z) and more. ... Unit 1 Real Numbers/ Properties of Real Numbers. 5.0 (1 review) Flashcards; Learn; Test; Match; Q-Chat; Get a hint. Rational Numbers (Q) Click the card to flip 👆 ...
PDF Unit 1 Packet
Unit 1 Packet - Properties & Operations with Real Numbers Answer Key . Part 1 - Properties: 1. Commutative Property of multiplication 2. Distributive Property 3. Additive Identity 4. Commutative Property of Addition . 5. Associative Property of Multiplication 6. Additive Inverse . 7. Multiplicative Identity 8.
1.10: Properties of Real Numbers
PROPERTIES OF ZERO. Multiplication by Zero: For any real number a, a ⋅ 0 = 0 0 ⋅ a = 0 The product of any number and 0 is 0 (1.10.20) (1.10.20) a ⋅ 0 = 0 0 ⋅ a = 0 The product of any number and 0 is 0. Division of Zero, Division by Zero: For any real number a, a ≠ 0 a, a ≠ 0.
1-1 Properties of Real Numbers
1-1 Bellwork - Properties of real numbers. 1-1 Exit Quiz - Properties of Real Numbers. 1-1 Guided Notes SE - Properties of Real Numbers. 1-1 Guided Notes TE - Properties of Real Numbers. 1-1 Lesson Plan - Properties of Real Numbers. 1-1 Online Activity - Properties of Real Numbers. 1-1 Slide Show - Properties of Real Numbers.
PDF 1.1 Real Numbers
1.1 Real Numbers A. Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . ... E. Properties of Real Numbers 1. Commutative: we can add or multiply in any order commutative property of addition commutative property of multiplication (note: means multiply) 2. Associative: In repeated ...
Solved Lindsen Rackley Name: Unit 1: Equations &
Question: Lindsen Rackley Name: Unit 1: Equations & Inequalities Date: $-26 2021 Homework Real Numbers & Properties Directions: Name ALL SETS to which each number belongs. 1. 2.49 3.06 Q,R Q, B N, W, 2,Q,R *Z,Q,R 36 6. 1.125 5. Q R IR 7. Place the LETTER of each value in its location in the real number system below.
ELEKTROSTAL HOTEL
Elektrostal Hotel, Elektrostal: See 25 traveler reviews, 44 candid photos, and great deals for Elektrostal Hotel, ranked #1 of 2 B&Bs / inns in Elektrostal and rated 4 of 5 at Tripadvisor.
Real estate in Elektrostal, Moscow Oblast, Russia
* calculated weighted mean of apartment cost per 1 square foot/meter in Elektrostal secondary housing market. Among prices in range from 30 to 200 thousand Rub/m² for Elektrostal. Among apartments with area in range: from 20 to 350 m², from 215 to 3767 ft².
Best 15 Real Estate Agents in Elektrostal', Moscow Oblast, Russia
Search 402 Elektrostal' real estate agents to find the best real estate agent for your project. See the top reviewed local real estate agents in Elektrostal', Moscow Oblast, Russia on Houzz. ... Вадим Оришак - Bosco Properties. Send Message. ... 1M+ Total Number of Reviews Left by Homeowners.
1.1E: Real Numbers
This page titled 1.1E: Real Numbers - Algebra Essentials (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
1.5: Properties of Real Numbers
of Addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c). of Multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c). When adding or multiplying, changing the grouping gives the same result. The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will ...
Best 15 Home & House Stagers in Elektrostal', Moscow Oblast, Russia
This pro works to prepare your Elektrostal', Moscow Oblast, Russia home for the local real estate market, with the main objective to make your house desirable to potential buyers. Home staging services in Elektrostal', Moscow Oblast, Russia can be a major factor in helping your place sell quickly and easily, so don't skip out on this crucial ...

1.1 Real Numbers: Algebra Essentials

Classifying a Real Number

Writing Integers as Rational Numbers

Identifying Rational Numbers

Irrational Numbers

Differentiating Rational and Irrational Numbers

Real Numbers

Classifying Real Numbers

Sets of Numbers as Subsets

Sets of Numbers

Differentiating the Sets of Numbers

Performing Calculations Using the Order of Operations

Order of Operations

Using the Order of Operations

Using Properties of Real Numbers

Commutative Properties

Associative Properties

Distributive Property

Identity Properties

Inverse Properties

Properties of Real Numbers

Evaluating Algebraic Expressions

Describing Algebraic Expressions

Evaluating an Algebraic Expression at Different Values

Using a Formula

Simplifying Algebraic Expressions

Simplifying a Formula

1.1 Section Exercises

Real-World Applications

1-4 Properties of Real Numbers

Properties of Real Numbers - Word Docs & PowerPoint

Properties of Real Numbers - PDFs

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1.1 Real Numbers: Algebra Essentials

Classifying a Real Number

Writing Integers as Rational Numbers

Identifying Rational Numbers

Irrational Numbers

Differentiating Rational and Irrational Numbers

Real Numbers

Classifying Real Numbers

Sets of Numbers as Subsets

Sets of Numbers

Differentiating the Sets of Numbers

Performing Calculations Using the Order of Operations

Order of Operations

Using the Order of Operations

Using Properties of Real Numbers

Commutative Properties

Associative Properties

Distributive Property

Identity Properties

Inverse Properties

Properties of Real Numbers

Evaluating Algebraic Expressions

Describing Algebraic Expressions

Evaluating an Algebraic Expression at Different Values

Using a Formula

Simplifying Algebraic Expressions

Simplifying a Formula

Key Concepts

Real-World Applications

Media Attributions

Share This Book

Real Number Properties

Example: Multiplying by zero

1-1 Properties of Real Numbers

Properties of Real Numbers - Word Docs & PowerPoints

Properties of Real Numbers - PDFs

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ELEKTROSTAL HOTEL

Analysis of real estate market in Moscow Oblast, Russia:

1.5: Properties of Real Numbers

Learning Objectives

Use the Commutative and Associative Properties

COMMUTATIVE PROPERTY

ASSOCIATIVE PROPERTY

Example \(\PageIndex{1}\)