Chapter 1 Real Numbers and Their Operations

Definitions

A setAny collection of objects. is a collection of objects, typically grouped within braces { }, where each object is called an elementAn object within a set.. For example, {red, green, blue} is a set of colors. A subsetA set consisting of elements that belong to a given set. is a set consisting of elements that belong to a given set. For example, {green, blue} is a subset of the color set above. A set with no elements is called the empty setA subset with no elements, denoted $\varnothing$ or { }. and has its own special notation, { } or ∅.

When studying mathematics, we focus on special sets of numbers. The set of natural (or counting) numbersThe set of counting numbers {1, 2, 3, 4, 5, …}., denoted N, is

The three periods (…) is called an ellipsis and indicates that the numbers continue without bound. The set of whole numbersThe set of natural numbers combined with zero {0, 1, 2, 3, 4, 5, …}., denoted $W$, is the set of natural numbers combined with zero.

The set of integersThe set of positive and negative whole numbers combined with zero {…, −3, −2, −1, 0, 1, 2, 3, …}., denoted Z, consists of both positive and negative whole numbers, as well as zero.

Notice that the sets of natural and whole numbers are both subsets of the set of integers.

Rational numbersNumbers of the form $\frac{a}{b}$, where a and b are integers and b is nonzero., denoted Q, are defined as any number of the form $\frac{a}{b}$, where a and b are integers and b is nonzero. Decimals that repeat or terminate are rational. For example,

The set of integers is a subset of the set of rational numbers because every integer can be expressed as a ratio of the integer and 1. In other words, any integer can be written over 1 and can be considered a rational number. For example,

Irrational numbersNumbers that cannot be written as a ratio of two integers. are defined as any number that cannot be written as a ratio of two integers. Nonterminating decimals that do not repeat are irrational. For example,

The set of real numbersThe set of all rational and irrational numbers., denoted R, is defined as the set of all rational numbers combined with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set of real numbers. In summary,

Number Line

A real number lineA line that allows us to visually represent real numbers by associating them with points on the line., or simply number line, allows us to visually display real numbers by associating them with unique points on a line. The real number associated with a point is called a coordinateThe real number associated with a point on a number line.. A point on the real number line that is associated with a coordinate is called its graphA point on the number line associated with a coordinate..

To construct a number line, draw a horizontal line with arrows on both ends to indicate that it continues without bound. Next, choose any point to represent the number zero; this point is called the originThe point on the number line that represtents zero..

Mark off consistent lengths on both sides of the origin and label each tick mark to define the scale. Positive real numbers lie to the right of the origin and negative real numbers lie to the left. The number zero (0) is neither positive nor negative. Typically, each tick represents one unit.

As illustrated below, the scale need not always be one unit. In the first number line, each tick mark represents two units. In the second, each tick mark represents $\frac{1}{7}$.

The graph of each real number is shown as a dot at the appropriate point on the number line. A partial graph of the set of integers Z follows:

Example 1: Graph the following set of real numbers: $\{-1,-\frac{1}{3},0,\frac{5}{3}\}$.

Solution: Graph the numbers on a number line with a scale where each tick mark represents $\frac{1}{3}$ unit.

Ordering Real Numbers

When comparing real numbers on a number line, the larger number will always lie to the right of the smaller one. It is clear that 15 is greater than 5, but it may not be so clear to see that −1 is greater than −5 until we graph each number on a number line.

We use symbols to help us efficiently communicate relationships between numbers on the number line. The symbols used to describe an equality relationshipExpress equality with the symbol =. If two quantities are not equal, use the symbol $\ne$. between numbers follow:

These symbols are used and interpreted in the following manner:

We next define symbols that denote an order relationship between real numbers.

These symbols allow us to compare two numbers. For example,

Since the graph of −120 is to the left of the graph of –10 on the number line, that number is less than −10. We could write an equivalent statement as follows:

Similarly, since the graph of zero is to the right of the graph of any negative number on the number line, zero is greater than any negative number.

The symbols < and > are used to denote strict inequalitiesExpress ordering relationships using the symbol < for “less than” and > for “greater than.”, and the symbols $\le$ and $\ge$ are used to denote inclusive inequalitiesUse the symbol $\le$ to express quantities that are “less than or equal to” and $\ge$ for quantities that are “greater than or equal to” each other.. In some situations, more than one symbol can be correctly applied. For example, the following two statements are both true:

In addition, the “or equal to” component of an inclusive inequality allows us to correctly write the following:

The logical use of the word “or” requires that only one of the conditions need be true: the “less than” or the “equal to.”

Example 2: Fill in the blank with <, =, or >: −2 ____ −12.

Solution: Use > because the graph of −2 is to the right of the graph of −12 on a number line. Therefore, −2 > −12, which reads “negative two is greater than negative twelve.”

Answer: −2 > −12

In this text, we will often point out the equivalent notation used to express mathematical quantities electronically using the standard symbols available on a keyboard. We begin with the equivalent textual notation for inequalities:

Many calculators, computer algebra systems, and programming languages use this notation.

Opposites

The oppositeReal numbers whose graphs are on opposite sides of the origin with the same distance to the origin. of any real number a is −a. Opposite real numbers are the same distance from the origin on a number line, but their graphs lie on opposite sides of the origin and the numbers have opposite signs.

For example, we say that the opposite of 10 is −10.

Next, consider the opposite of a negative number. Given the integer −7, the integer the same distance from the origin and with the opposite sign is +7, or just 7.

Therefore, we say that the opposite of −7 is −(−7) = 7. This idea leads to what is often referred to as the double-negative propertyThe opposite of a negative number is positive: −(−a) = a.. For any real number a,

Example 3: What is the opposite of $-\frac{3}{4}$?

Solution: Here we apply the double-negative property.

Answer: $\frac{3}{4}$

Example 4: Simplify: $-\left(-\left(4\right)\right)$.

Solution: Start with the innermost parentheses by finding the opposite of +4.

Answer: 4

Example 5: Simplify: $-(-(-2))$.

Solution: Apply the double-negative property starting with the innermost parentheses.

Answer: −2

Try this! Simplify: $-\left(-\left(-\left(5\right)\right)\right)$.

Answer: −5

Absolute Value

The absolute valueThe absolute value of a number is the distance from the graph of the number to zero on a number line. of a real number a, denoted $\left|a\right|$, is defined as the distance between zero (the origin) and the graph of that real number on the number line. Since it is a distance, it is always positive. For example,

Both 4 and −4 are four units from the origin, as illustrated below:

Example 6: Simplify:

a. $\left|-12\right|$

b. $\left|12\right|$

Solution: Both −12 and 12 are twelve units from the origin on a number line. Therefore,

Answers: a. 12; b. 12

Also, it is worth noting that

The absolute value can be expressed textually using the notation abs(a). We often encounter negative absolute values, such as $-\left|3\right|$ or −abs(3). Notice that the negative sign is in front of the absolute value symbol. In this case, work the absolute value first and then find the opposite of the result.

Try not to confuse this with the double-negative property, which states that $-(-7)=+7$.

Example 7: Simplify: $-\left|-\left(-7\right)\right|$.

Solution: First, find the opposite of −7 inside the absolute value. Then find the opposite of the result.

Answer: −7

At this point, we can determine what real numbers have a particular absolute value. For example,

Think of a real number whose distance to the origin is 5 units. There are two solutions: the distance to the right of the origin and the distance to the left of the origin, namely, $\left\{\pm 5\right\}$. The symbol (±) is read “plus or minus” and indicates that there are two answers, one positive and one negative.

Now consider the following:

Here we wish to find a value for which the distance to the origin is negative. Since negative distance is not defined, this equation has no solution. If an equation has no solution, we say the solution is the empty set: Ø.

Answers

1: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

3: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

5: {2, 4, 6, 8, 10}

7: Rational

9: Rational

11: Rational

13: Irrational

15: Irrational

17: Integer, Rational

19: Rational

21: Integer, Rational

23: True

25: False

27: True

29:

31:

33:

35:

37:

39:

41: <

43: >

45: >

47: <

49: =

51: True

53: False

55: True

57: True

59: True

61: −10, −7, −6 (answers may vary)

63: −1, −2/3, −1/3 (answers may vary)

65: −15, −10, −7 (answers may vary)

67: Ten is less than twenty.

69: Negative four is not equal to zero.

71: Zero is equal to zero.

73: $-7<0$

75: $0\ge -1$

77: $-2=-2$

79: 9

81: −10

83: −5

85: 1

87: 1

89: 11

91: $-\pi$

93: Larger

95: <

97: >

99: <

101: 20

103: 33

105: 2/5

107: 0

109: −12

111: −20

113: −7

115: 8/9

117: −3

119: 45

121: 6

123: −5

125: 9

127: $\pm 9$

129: 0

131: Ø, No solution

133: $\pm 8$

135: >

137: <

139: <

Addition and Subtraction (+, −)

Visualize adding 3 + 2 on the number line by moving from zero three units to the right then another two units to the right, as illustrated below:

The illustration shows that 3 + 2 = 5. Similarly, visualize adding two negative numbers (−3) + (−2) by first moving from the origin three units to the left and then moving another two units to the left.

In this example, the illustration shows (−3) + (−2) = −5, which leads to the following two properties of real numbers.

Next, we will explore addition of numbers with unlike signs. To add 3 + (−7), first move from the origin three units to the right, then move seven units to the left as shown:

In this case, we can see that adding a negative number is equivalent to subtraction:

It is tempting to say that a positive number plus a negative number is negative, but that is not always true: $7+\left(-3\right)=7-3=4$. The result of adding numbers with unlike signs may be positive or negative. The sign of the result is the same as the sign of the number with the greatest distance from the origin. For example, the following results depend on the sign of the number 12 because it is farther from zero than 5:

Example 1: Simplify: $14+\left(-25\right)$.

Solution: Here −25 is the greater distance from the origin. Therefore, the result is negative.

Answer: −11

Given any real numbers a, b, and c, we have the following properties of addition:

Example 2: Simplify:

a. $5+0$

b. $10+\left(-10\right)$

Solution:

a. Adding zero to any real number results in the same real number.

b. Adding opposites results in zero.

Answers: a. 5; b. 0

Example 3: Simplify:

a. $\left(3+7\right)+4$

b. $3+\left(7+4\right)$

Solution: Parentheses group the operations that are to be performed first.

a.

b.

These two examples both result in 14: changing the grouping of the numbers does not change the result.

Answers: a. 14; b. 14

At this point, we highlight the fact that addition is commutative: the order in which we add does not matter and yields the same result.

On the other hand, subtraction is not commutative.

We will use these properties, along with the double-negative property for real numbers, to perform more involved sequential operations. To simplify things, we will make it a general rule to first replace all sequential operations with either addition or subtraction and then perform each operation in order from left to right.

Example 4: Simplify: $4-\left(-10\right)+\left(-5\right)$.

Solution: Replace the sequential operations and then perform them from left to right.

Answer: 9

Example 5: Simplify: $-3+\left(-8\right)-\left(-7\right)$.

Solution:

Answer: −4

Try this! Simplify: $12-(-9)+(-6)$.

Answer: 15

Often we find the need to translate English sentences involving addition and subtraction to mathematical statements. Listed below are some key words that translate to the given operation.

Example 6: What is the difference of 7 and −3?

Solution: The key word “difference” implies that we should subtract the numbers.

Answer: The difference of 7 and −3 is 10.

Example 7: What is the sum of the first five positive integers?

Solution: The initial key word to focus on is “sum”; this means that we will be adding the five numbers. The first five positive integers are {1, 2, 3, 4, 5}. Recall that 0 is neither positive nor negative.

Answer: The sum of the first five positive integers is 15.

Example 8: What is 10 subtracted from the sum of 8 and 6?

Solution: We know that subtraction is not commutative; therefore, we must take care to subtract in the correct order. First, add 8 and 6 and then subtract 10 as follows:

It is important to notice that the phrase “10 subtracted from” does not translate to a mathematical statement in the order it appears. In other words, $10-\left(8+6\right)$ would be an incorrect translation and leads to an incorrect answer. After translating the sentence, perform the operations.

Answer: Ten subtracted from the sum of 8 and 6 is 4.

Distance on a Number Line

One application of the absolute value is to find the distance between any two points on a number line. For real numbers a and b, the distance formula for a number lineThe distance between any two real numbers a and b on a number line can be calculated using the formula $d=\left|b-a\right|$. is given as,

Example 9: Determine the distance between 2 and 7 on a number line.

Solution: On the graph we see that the distance between the two given integers is 5 units.

Using the distance formula we obtain the same result.

Answer: 5 units

Example 10: Determine the distance between −4 and 7 on a number line.

Solution: Use the distance formula for a number line $d=\left|b-a\right|$, where $a=-4$ and $b=7$.

Answer: 11 units

It turns out that it does not matter which points are used for a and b; the absolute value always ensures a positive result.

Try this! Determine the distance between −12 and −9 on the number line.

Answer: 3

Multiplication and Division

We begin with a review of what it means to multiply and divide signed numbers. The result of multiplying real numbers is called the productThe result of multiplying. and the result of dividing is called the quotientThe result after dividing.. Recall that multiplication is equivalent to adding:

Clearly, the product of two positive numbers is positive. Similarly, the product of a positive number and negative number can be written as shown:

We see that the product of a positive number and a negative number is negative. Next, explore the results of multiplying two negative numbers. Consider the products in the following illustration and try to identify the pattern:

This shows that the product of two negative numbers is positive. To summarize,

The rules for division are the same because division can always be rewritten as multiplication:

The rules for multiplication and division should not be confused with the fact that the sum of two negative numbers is negative.

Example 1: Simplify:

a. $\left(-3\right)+\left(-5\right)$

b. $\left(-3\right)\left(-5\right)$

Solution: Here we add and multiply the same two negative numbers.

a. The result of adding two negative numbers is negative.

b. The result of multiplying two negative numbers is positive.

Answers: a. −8; b. 15

Given any real numbers a, b, and c, we have the following properties of multiplication:

Example 2: Simplify:

a. $5\cdot 0$

b. $10\cdot 1$

Solution:

a. Multiplying by zero results in zero.

b. Multiplying any real number by one results in the same real number.

Answers: a. 0; b. 10

Example 3: Simplify:

a. $\left(3\cdot 7\right)\cdot 2$

b. $3\cdot \left(7\cdot 2\right)$

Solution:

a.

b.

The value of each expression is 42. Changing the grouping of the numbers does not change the result.

Answers: a. 42; b. 42

At this point, we highlight that multiplication is commutative: the order in which we multiply does not matter and yields the same result.

On the other hand, division is not commutative.

Use these properties to perform sequential operations involving multiplication and division. When doing so, it is important to perform these operations in order from left to right.

Example 4: Simplify: $3\left(-2\right)\left(-5\right)\left(-1\right)$.

Solution: Multiply two numbers at a time as follows:

Answer: −30

Because multiplication is commutative, the order in which we multiply does not affect the final answer. When sequential operations involve multiplication and division, order does matter; hence we must work the operations from left to right to obtain a correct result.

Example 5: Simplify: $10÷\left(-2\right)\left(-5\right)$.

Solution: Perform the division first; otherwise, the result will be incorrect.

Answer: 25

Notice that the order in which we multiply and divide does affect the final result. Therefore, it is important to perform the operations of multiplication and division as they appear from left to right.

Example 6: Simplify: $-6\left(3\right)÷\left(-2\right)\left(-3\right)$.

Solution: Work the operations one at a time from left to right.

Try this! Simplify: $-5÷5\cdot 2\left(-3\right)$.

Answer: 6

Within text-based applications, the symbol used for multiplication is the asteriskThe symbol (*) that indicates multiplication within text-based applications. (*) and the symbol used for division is the forward slash (/).

The set of even integersIntegers that are divisible by two or are multiples of two. is the set of all integers that are evenly divisible by 2. We can also obtain the set of even integers by multiplying each integer by 2.

The set of odd integersIntegers that are not divisible by 2. is the set of all integers that are not evenly divisible by 2.

A prime numberIntegers greater than 1 that are divisible only by 1 and itself. is an integer greater than 1 that is divisible only by 1 and itself. The smallest prime number is 2 and the rest are necessarily odd.

Any integer greater than 1 that is not prime is called a composite numberIntegers greater than 1 that are not prime. and can be written as a product of primes. When a composite number, such as 30, is written as a product, $30=2\cdot 15$, we say that $2\cdot 15$ is a factorizationAny combination of factors, multiplied together, resulting in the product. of 30 and that 2 and 15 are factorsAny of the numbers or expressions that form a product.. Note that factors divide the number evenly. We can continue to write composite factors as products until only a product of primes remains.

The prime factorizationThe unique factorization of a natural number written as a product of primes. of 30 is $2\cdot 3\cdot 5$.

Example 7: Determine the prime factorization of 70.

Solution: Begin by writing 70 as a product with 2 as a factor. Then express any composite factor as a product of prime numbers.

Since the prime factorization is unique, it does not matter how we choose to initially factor the number because the end result is the same.

Answer: The prime factorization of 70 is $2\cdot 5\cdot 7$.

Some tests (called divisibility tests) useful for finding prime factors of composite numbers follow:

1. If the integer is even, then 2 is a factor.
2. If the sum of the digits is evenly divisible by 3, then 3 is a factor.
3. If the last digit is a 5 or 0, then 5 is a factor.

Often we find the need to translate English sentences that include multiplication and division terms to mathematical statements. Listed below are some key words that translate to the given operation.

Example 8: Calculate the quotient of 20 and −10.

Solution: The key word “quotient” implies that we should divide.

Answer: The quotient of 20 and −10 is −2.

Example 9: What is the product of the first three positive even integers?

Solution: The first three positive even integers are {2, 4, 6} and the key word “product” implies that we should multiply.

Answer: The product of the first three positive even integers is 48.

Example 10: Joe is able to drive 342 miles on 18 gallons of gasoline. How many miles per gallon of gas is this?

Solution: The key word “per” indicates that we must divide the number of miles driven by the number of gallons used:

Answer: Joe gets 19 miles per gallon from his vehicle.

In everyday life, we often wish to use a single value that typifies a set of values. One way to do this is to use what is called the arithmetic meanA numerical value that typifies a set of numbers. It is calculated by adding up the numbers in the set and dividing by the number of elements in the set. or averageUsed in reference to the arithmetic mean.. To calculate an average, divide the sum of the values in the set by the number of values in that set.

Example 11: A student earns 75, 86, and 94 on his first three exams. What is the student’s test average?

Solution: Add the scores and divide the sum by 3.

Answer: The student’s test average is 85.

Zero and Division

Recall the relationship between multiplication and division:

In this case, the dividendThe numerator of a quotient. 12 is evenly divided by the divisorThe denominator of a quotient. 6 to obtain the quotient, 2. It is true in general that if we multiply the divisor by the quotient we obtain the dividend. Now consider the case where the dividend is zero and the divisor is nonzero:

This demonstrates that zero divided by any nonzero real number must be zero. Now consider a nonzero number divided by zero:

The zero-factor property of multiplication states that any real number times 0 is 0. We conclude that there is no real number such that $0\cdot ?=12$ and thus, the quotient is left undefinedA quotient such as $\frac{5}{0}$, which is left without meaning and is not assigned an interpretation.. Try $12÷0$ on a calculator. What does it say? For our purposes, we will simply write “undefined.”

To summarize, given any real number $a\ne 0$, then

We are left to consider the case where the dividend and divisor are both zero.

Here any real number seems to work. For example, $0\cdot 5=0$ and $0\cdot 3=0$. Therefore, the quotient is uncertain or indeterminateA quotient such as $\frac{0}{0}$, which is a quantity that is uncertain or ambiguous..

In this course, we state that $0÷0$ is undefined.

Reducing

A fractionA rational number written as a quotient of two integers: $\frac{a}{b}$, where b is nonzero. is a real number written as a quotient, or ratioRelationship between two numbers or quantities usually expressed as a quotient., of two integers a and b, where $b\ne 0$.

The integer above the fraction bar is called the numeratorThe number above the fraction bar. and the integer below is called the denominatorThe number below the fraction bar.. The numerator is often called the “part” and the denominator is often called the “whole.” Equivalent fractionsTwo equal fractions expressed using different numerators and denominators. are two equal ratios expressed using different numerators and denominators. For example,

Fifty parts out of 100 is the same ratio as 1 part out of 2 and represents the same real number. Consider the following factorizations of 50 and 100:

The numbers 50 and 100 share the factor 25. A shared factor is called a common factorA factor that is shared by more than one real number.. We can rewrite the ratio $\frac{50}{100}$ as follows:

Making use of the multiplicative identity property and the fact that $\frac{25}{25}=1$, we have

Dividing $\frac{25}{25}$ and replacing this factor with a 1 is called cancelingThe process of dividing out common factors in the numerator and the denominator.. Together, these basic steps for finding equivalent fractions define the process of reducingThe process of finding equivalent fractions by dividing the numerator and the denominator by common factors.. Since factors divide their product evenly, we achieve the same result by dividing both the numerator and denominator by 25 as follows:

Finding equivalent fractions where the numerator and denominator have no common factor other than 1 is called reducing to lowest termsFinding equivalent fractions where the numerator and the denominator share no common integer factor other than 1.. When learning how to reduce to lowest terms, it is helpful to first rewrite the numerator and denominator as a product of primes and then cancel. For example,

We achieve the same result by dividing the numerator and denominator by the greatest common factor (GCF)The largest shared factor of any number of integers.. The GCF is the largest number that divides both the numerator and denominator evenly. One way to find the GCF of 50 and 100 is to list all the factors of each and identify the largest number that appears in both lists. Remember, each number is also a factor of itself.

Common factors are listed in bold, and we see that the greatest common factor is 50. We use the following notation to indicate the GCF of two numbers: GCF(50, 100) = 50. After determining the GCF, reduce by dividing both the numerator and the denominator as follows:

Example 1: Reduce to lowest terms: $\frac{105}{300}$.

Solution: Rewrite the numerator and denominator as a product of primes and then cancel.

Alternatively, we achieve the same result if we divide both the numerator and denominator by the GCF(105, 300). A quick way to find the GCF of the two numbers requires us to first write each as a product of primes. The GCF is the product of all the common prime factors.

In this case, the common prime factors are 3 and 5 and the greatest common factor of 105 and 300 is 15.

Answer: $\frac{7}{20}$

Try this! Reduce to lowest terms: $\frac{32}{96}$.

Answer: $\frac{1}{3}$

An improper fractionA fraction where the numerator is larger than the denominator. is one where the numerator is larger than the denominator. A mixed numberA number that represents the sum of a whole number and a fraction. is a number that represents the sum of a whole number and a fraction. For example, $5\frac{1}{2}$ is a mixed number that represents the sum $5+\frac{1}{2}$. Use long division to convert an improper fraction to a mixed number; the remainder is the numerator of the fractional part.

Example 2: Write $\frac{23}{5}$ as a mixed number.

Solution: Notice that 5 divides into 23 four times with a remainder of 3.

We then can write

Note that the denominator of the fractional part of the mixed number remains the same as the denominator of the original fraction.

Answer: $4\frac{3}{5}$

To convert mixed numbers to improper fractions, multiply the whole number by the denominator and then add the numerator; write this result over the original denominator.

Example 3: Write $3\frac{5}{7}$ as an improper fraction.

Solution: Obtain the numerator by multiplying 7 times 3 and then add 5.

Answer: $\frac{26}{7}$

It is important to note that converting to a mixed number is not part of the reducing process. We consider improper fractions, such as $\frac{26}{7}$, to be reduced to lowest terms. In algebra it is often preferable to work with improper fractions, although in some applications, mixed numbers are more appropriate.

Try this! Convert $10\frac{1}{2}$ to an improper fraction.

Answer: $\frac{21}{2}$

Multiplying and Dividing Fractions

In this section, assume that a, b, c, and d are all nonzero integers. The product of two fractions is the fraction formed by the product of the numerators and the product of the denominators. In other words, to multiply fractions, multiply the numerators and multiply the denominators:

Example 4: Multiply: $\frac{2}{3}\cdot \frac{5}{7}$.

Solution: Multiply the numerators and multiply the denominators.

Answer: $\frac{10}{21}$

Example 5: Multiply: $\frac{5}{9}\left(-\frac{1}{4}\right)$.

Solution: Recall that the product of a positive number and a negative number is negative.

Answer: $-\frac{5}{36}$

Example 6: Multiply: $\frac{2}{3}\cdot 5\frac{3}{4}$.

Solution: Begin by converting $5\frac{3}{4}$ to an improper fraction.

In this example, we noticed that we could reduce before we multiplied the numerators and the denominators. Reducing in this way is called cross cancelingCancelling common factors in the numerator and the denominator of fractions before multiplying., and can save time when multiplying fractions.

Answer: $3\frac{5}{6}$

Two real numbers whose product is 1 are called reciprocalsThe reciprocal of a nonzero number n is 1/n.. Therefore, $\frac{a}{b}$ and $\frac{b}{a}$ are reciprocals because $\frac{a}{b}\cdot \frac{b}{a}=\frac{ab}{ab}=1$. For example,

Because their product is 1, $\frac{2}{3}$ and $\frac{3}{2}$ are reciprocals. Some other reciprocals are listed below:

This definition is important because dividing fractions requires that you multiply the dividend by the reciprocal of the divisor.

Example 7: Divide: $\frac{2}{3}÷\frac{5}{7}$.

Solution: Multiply $\frac{2}{3}$ by the reciprocal of $\frac{5}{7}$.

Answer: $\frac{14}{15}$

You also need to be aware of other forms of notation that indicate division: / and —. For example,

Or

The latter is an example of a complex fractionA fraction where the numerator or denominator consists of one or more fractions., which is a fraction whose numerator, denominator, or both are fractions.

Note

Students often ask why dividing is equivalent to multiplying by the reciprocal of the divisor. A mathematical explanation comes from the fact that the product of reciprocals is 1. If we apply the multiplicative identity property and multiply numerator and denominator by the reciprocal of the denominator, then we obtain the following:

Before multiplying, look for common factors to cancel; this eliminates the need to reduce the end result.

Example 8: Divide: $\frac{\frac{5}{2}}{\frac{7}{4}}$.

Solution:

Answer: $\frac{10}{7}$

When dividing by an integer, it is helpful to rewrite it as a fraction over 1.

Example 9: Divide: $\frac{2}{3}÷6$.

Solution: Rewrite 6 as $\frac{6}{1}$ and multiply by its reciprocal.

Answer: $\frac{1}{9}$

Also, note that we only cancel when working with multiplication. Rewrite any division problem as a product before canceling.

Try this! Divide: $5÷2\frac{3}{5}$.

Answer: $1\frac{12}{13}$

Adding and Subtracting Fractions

Negative fractions are indicated with the negative sign in front of the fraction bar, in the numerator, or in the denominator. All such forms are equivalent and interchangeable.

Adding or subtracting fractions requires a common denominatorA denominator that is shared by more than one fraction.. In this section, assume the common denominator c is a nonzero integer.

It is good practice to use positive common denominators by expressing negative fractions with negative numerators. In short, avoid negative denominators.

Example 10: Subtract: $\frac{12}{15}-\frac{3}{15}$.

Solution: The two fractions have a common denominator 15. Therefore, subtract the numerators and write the result over the common denominator:

Answer: $\frac{3}{5}$

Most problems that you are likely to encounter will have unlike denominatorsDenominators of fractions that are not the same.. In this case, first find equivalent fractions with a common denominator before adding or subtracting the numerators. One way to obtain equivalent fractions is to divide the numerator and the denominator by the same number. We now review a technique for finding equivalent fractions by multiplying the numerator and the denominator by the same number. It should be clear that 5/5 is equal to 1 and that 1 multiplied times any number is that number:

We have equivalent fractions $\frac{1}{2}=\frac{5}{10}$. Use this idea to find equivalent fractions with a common denominator to add or subtract fractions. The steps are outlined in the following example.

Example 11: Subtract: $\frac{7}{15}-\frac{3}{10}$.

Solution:

Step 1: Determine a common denominator. To do this, use the least common multiple (LCM)The smallest number that is evenly divisible by a set of numbers. of the given denominators. The LCM of 15 and 10 is indicated by LCM(15, 10). Try to think of the smallest number that both denominators divide into evenly. List the multiples of each number:

Common multiples are listed in bold, and the least common multiple is 30.

Step 2: Multiply the numerator and the denominator of each fraction by values that result in equivalent fractions with the determined common denominator.

Step 3: Add or subtract the numerators, write the result over the common denominator and then reduce if possible.

Answer: $\frac{1}{6}$

The least common multiple of the denominators is called the least common denominator (LCD)The least common multiple of a set of denominators.. Finding the LCD is often the difficult step. It is worth finding because if any common multiple other than the least is used, then there will be more steps involved when reducing.

Example 12: Add: $\frac{5}{10}+\frac{1}{18}$.

Solution: First, determine that the LCM(10, 18) is 90 and then find equivalent fractions with 90 as the denominator.

Answer: $\frac{5}{9}$

Try this! Add: $\frac{2}{30}+\frac{5}{21}$.

Answer: $\frac{32}{105}$

Example 13: Simplify: $2\frac{1}{3}+\frac{3}{5}-\frac{1}{2}$.

Solution: Begin by converting $2\frac{1}{3}$ to an improper fraction.

Answer: $2\frac{13}{30}$

In general, it is preferable to work with improper fractions. However, when the original problem involves mixed numbers, if appropriate, present your answers as mixed numbers. Also, mixed numbers are often preferred when working with numbers on a number line and with real-world applications.

Try this! Subtract: $\frac{5}{7}-2\frac{1}{7}$.

Answer: $-1\frac{3}{7}$

Example 14: How many $\frac{1}{2}$ inch thick paperback books can be stacked to fit on a shelf that is $1\frac{1}{2}$ feet in height?

Solution: First, determine the height of the shelf in inches. To do this, use the fact that there are 12 inches in 1 foot and multiply as follows:

Next, determine how many notebooks will fit by dividing the height of the shelf by the thickness of each book.

Answer: 36 books can be stacked on the shelf.

Decimals

In this section, we provide a brief review of the decimal system. A real number in decimal form, a decimalA real number expressed using the decimal system. consists of a decimal point, digits (0 through 9) to the left of the decimal point representing the whole number part, and digits to the right of the decimal point representing the fractional part. The digits represent powers of 10 as shown in the set {…, 1,000, 100, 10, 1, 1/10, 1/100, 1/1,000, …} according to the following diagram:

For example, the decimal 538.3 can be written in the following expanded form:

After simplifying, we obtain the mixed number $538\frac{3}{10}$. Use this process to convert decimals to mixed numbers.

Example 1: Write as a mixed number: 32.15.

Solution: In this example, 32 is the whole part and the decimal ends in the hundredths place. Therefore, the fractional part will be 15/100, and we can write

Answer: $32.15=32\frac{3}{20}$

To convert fractions to decimal equivalents, divide.

Example 2: Write as a decimal: $\frac{3}{4}$.

Solution: Use long division to convert to a decimal.

Answer: $\frac{3}{4}=0.75$

If the division never ends, then use a bar over the repeating digit (or block of digits) to indicate a repeating decimal.

Example 3: Write as a decimal: $2\frac{5}{6}$.

Solution: Use long division to convert the fractional part to a decimal and then add the whole part.

At this point, we can see that the long division will continue to repeat. When this is the case, use a bar over the repeating digit to indicate that it continues forever:

Then write

Answer: $2\frac{5}{6}=2.8\overline{3}$

To add or subtract decimals, align them vertically with the decimal point and add corresponding place values. Recall that sometimes you need to borrow from or carry over to the adjoining column (regrouping).

Example 4: Subtract: $54.328-23.25$.

Solution: Note that trailing zeros to the right of the decimal point do not change the value of the decimal, 23.25 = 23.250. In this case, you need to borrow from the tenths place (regroup) to subtract the digits in the hundredths place.

Answer: 31.078

Multiply decimals the same way you multiply whole numbers. The number of decimal places in the product will be the sum of the decimal places found in each of the factors.

Example 5: Multiply: $5.36\times 7.4$.

Solution: The total number of decimal places of the two factors is 2 + 1 = 3. Therefore, the result has 3 decimal places.

Answer: 39.664

When dividing decimals, move the decimal points of both the dividend and the divisor so that the divisor is a whole number. Remember to move the decimal the same number of places for both the dividend and divisor.

Example 6: Divide: $33.3216÷6.24$.

Solution: Move the decimal point to turn the divisor into a whole number: 624. Move the decimal points of both the divisor and dividend two places to the right.

Next, divide.

Answer: 5.34

It is often necessary to round offA means of approximating decimals with a specified number of significant digits. decimals to a specified number of decimal places. Rounding off allows us to approximate decimals with fewer significant digits. To do this, look at the digit to the right of the specified place value.

1. If the digit to the right of the specified place is 4 or less, then leave the specified digit unchanged and drop all subsequent digits.
2. If the digit to the right of the specified place is 5 or greater, then increase the value of the digit in the specified place by 1 and drop all subsequent digits.

Recall that decimals with trailing zeros to the right of the decimal point can be dropped. For example, round 5.635457 to the nearest thousandth:

Round the same number 5.635457 to the nearest hundredth:

After rounding off, be sure to use the appropriate notation ( $\approx$ ) to indicate that the number is now an approximation. When working with US currency, we typically round off to two decimal places, or the nearest hundredth.

Example 7: Calculate and round off to the nearest hundredth.

a. 1/3 of $10.25.

b. 1/4 of $10.25.

Solution: In this context, the key word “of” indicates that we should multiply.

a. Multiplying by $\frac{1}{3}$ is equivalent to dividing by 3.

b. Multiplying by $\frac{1}{4}$ is equivalent to dividing by 4.

Answers: a. $3.42; b. $2.56

Definition of Percent

A percentA representation of a number as part of 100: $N\%=\frac{N}{100}$. is a representation of a number as a part of one hundred. The word “percent” can be written “per cent” which means “per 100” or “/100.” We use the symbol ( % ) to denote a percentage:

For example,

Percents are an important part of our everyday life and show up often in our study of algebra. Percents can be visualized using a pie chartA circular graph divided into sectors whose area is proportional to the relative size of the ratio of the part to the total. (or circle graph), where each sector gives a visual representation of a percentage of the whole. For example, the following pie chart shows the percentage of students in certain age categories of all US community colleges.

Source: American Association of Community Colleges.

Each sector is proportional to the size of the part out of the whole. The sum of the percentages presented in a pie chart must be 100%. To work with percentages effectively, you have to know how to convert percents to decimals or fractions and back again.

Percents to Decimals

Applying the definition of percent, you see that $58\%=\frac{58}{100}=0.58$. The same result can be obtained by moving the decimal two places to the left. To convert percents to decimals, either apply the definition or move the decimal two places to the left.

Example 8: Convert to a decimal: 152%.

Solution: Treat $152\text{\% as }152.0\text{\%}$ and move the decimal two places to the left.

Answer: 1.52

Example 9: Convert to a decimal: $2\frac{3}{4}\%$.

Solution: First, write the decimal percent equivalent,

Next, move the decimal two places to the left,

At this point, fill in the tenths place with a zero.

Answer: 0.0275

Try this! Convert to a decimal: 215%.

Answer: 2.15

Decimals and Fractions to Percents

To convert a decimal to a percent, convert the decimal to a fraction of 100 and apply the definition of percent, or equivalently, move the decimal to the right two places and add a percent sign.

Example 10: Convert 0.23 to a percent.

Solution: First, convert the decimal to a fraction of 100 and apply the definition.

You can achieve the same result by moving the decimal two places to the right and adding a percent sign.

Answer: 23%.

Alternatively, you can multiply by 1 in the form of 100%.

Example 11: Convert 2.35 to a percent.

Solution: Recall that 1 = 100%.

You can achieve the same result by moving the decimal two places to the right and adding a percent sign.

Answer: 235%

Example 12: Convert $5\frac{1}{5}$ to a percent.

Solution:

Answer: 520%

Sometimes we can use the definition of percent and find an equivalent fraction with a denominator of 100.

Example 13: Convert $\frac{13}{25}$ to a percent.

Solution: Notice that the denominator 25 is a factor of 100. Use the definition of percent by finding an equivalent fraction with 100 in the denominator.

Answer: 52%

This is a very specialized technique because 100 may not be a multiple of the denominator.

Example 14: Convert $\frac{1}{3}$ to a percent.

Solution: Notice that the denominator 3 is not a factor of 100. In this case, it is best to multiply by 1 in the form of 100%.

Answer: $33\frac{1}{3}\%$

Try this! Convert to a percent: $\frac{2}{3}$.

Answer: $66\frac{2}{3}\%$

Percents to Fractions

When converting percents to fractions, apply the definition of percent and then reduce.

Example 15: Convert to a fraction: $28\%$.

Solution:

Answer: $\frac{7}{25}$

Applying the definition of percent is equivalent to removing the percent sign and multiplying by $\frac{1}{100}$.

Example 16: Convert to a fraction: $66\frac{2}{3}\%$.

Solution: First, convert to an improper fraction and then apply the definition of percent.

Answer: $\frac{2}{3}$

Try this! Convert to a fraction: $3\frac{7}{31}\%$.

Answer: $\frac{1}{31}$

Example 17: Using the given pie chart, calculate the total number of students that were 21 years old or younger if the total US community college enrollment in 2009 was 11.7 million.

Solution: From the pie chart we can determine that 47% of the total 11.7 million students were 21 years old or younger.

Source: American Association of Community Colleges.

Convert 47% to a decimal and multiply as indicated by the key word “of.”

Answer: In 2009, approximately 5.5 million students enrolled in US community colleges were 21 years old or younger.