Section 1.5

Multiplication and Division of Real Numbers

1.5 Lecture Guide: Multiplication and Division of Real Numbers

The following box contains the common notations used to indicate multiplication of the factors x and y. The times sign “×” is not included because it is not frequently used in algebra.

Notations for the Product of the Factors x and y

xy x y x y x y x y x y

Objective 1: Use the commutative and associative properties of multiplication.

Verbally Algebraically Numerical Example

Commutative Property

Associative Property

The product of two _______________ in either order is the same.

Factors can be ______________ without changing the product.

ab ba

ab c a bc

5 6 6 5

2 3 4

2 3 4

Properties of Multiplication

Identify the property used to justify the equality of the two expressions in each equation below. Select from the following list:

I. Commutative Property II. Commutative Property of Addition of Multiplication

III. Associative Property IV. Associative Property of Addition of Multiplication

2._________

3._________

4._________

5._________

6._________

7._________

3 4( 2) 3 4 2x x 2 5 2 5x x

2 3 2 3x x x x 3 5 5 3x x x x

3 2 3 2x x 5 4 4 5x x

8. Think about the difference in the meanings of the terms “regroup” and “reorder”.

(a) Which term applies to the commutative property?

(b) Which term applies to the associative property?

Objective 2: Multiply positive and negative real numbers.

Multiplication of Two Real Numbers

Algebraically Numerical Example

Like Signs:

Unlike Signs:

Zero Factor:

Multiply the absolute values of the two factors and use a ____________ sign for the product.

Multiply the absolute values of the two factors and use a ____________ sign for the product.

The product of 0 and any other factor is ______.

3 6 ______

3 6 ______

3 6 ______

3 6 ______

3 0 ______ 0 6 ______

9. The Signs of a Sum vs. the Signs of a Product: Fill in the correct sign of the sum or product below or indicate that not enough information is given to determine the sign.

Sum Sign Product Sign

(positive)+(positive)= (positive)●(positive)=

(positive)+(negative)= (positive)●(negative)=

(negative)+(positive) = (negative)●(positive) =

(negative)+(negative)= (negative)●(negative)=

(0)+(positive)= (0)●(positive)=

(0)+(negative)= (0)●(negative)=

10. 11. 12. 4 5 3 6 2 7

Calculate each product using only pencil and paper.

13. 14. 15. 3 4 10 2 5 11 2 8 0

Calculate each product using only pencil and paper.

16. 17. 18. 100 13.51 0.1 23.5 114

7

Calculate each product using only pencil and paper.

Product of Negative Factors

The product is __________________ if the number of negative factors is even.

The product is __________________ if the number of negative factors is odd.

The number one is called the multiplicative identity because 1 is the only real number with the property that and for every real number a.

1 a a 1a a

A Factor of 1 or −1Algebraically Verbally Numerical

ExampleFor any real number a:

The product of one and any real number is that same number.

The product of negative one and any real number is the ____________ of that real number.

and

and

1 a a

1 a a

1 5

1 5

1 3

1 4

Algebraically Verbally Numerical Example

For any real number :

is undefined

For any real number a other than zero, the product of the number a and its multiplicative

inverse is 1.

Zero has no multiplicative inverse.

and

but

is undefined

Reciprocals or Multiplicative Inverses

0a 1

1aa

10

1a

14 1

4

4 31

3 4

00

1 1

0

Give the multiplicative inverse of each of the following real numbers and then multiply the number by its multiplicative inverse.

Number Multiplicative

Inverse Product

19.

20.

21.

22.

5

4

4

74

23. The multiplicative inverse of a positive number.

24. The multiplicative inverse of a negative number.

25. The additive inverse of a positive number.

26. The additive inverse of a negative number.

Determine the sign of each number.

27. Does every real number have a multiplicative inverse?

28. Does every real number have an additive inverse?

Objective 3: Divide positive and negative real numbers.

Notations for the Quotient of x Divided by y for y

x y xy

x y :x y

Algebraically Verbally Numerical Example

For any real numbers x and y with ,

Dividing two real numbers is the same as multiplying the first number by the multiplicative ____________ of the second number.

Relationship Between Division and Multiplication

0y 1 x

x y xy y

5 2

Division of Two Real NumbersLike signs: Divide the absolute values of the two numbers and use a ______________ sign for the quotient. 

Unlike signs: Divide the absolute values of the two numbers and use a ______________ sign for the quotient.

Zero dividend: for .

Zero divisor: is ______________ for every real number x.

00

x 0x

0x

Although memorization is generally not the best way to learn mathematical concepts, it is very helpful to have the following key facts memorized when performing division.

36. The Sign of a Product vs. the Sign of a Quotient: Where possible, fill in the correct sign of each product and quotient below.

Product Sign Quotient Sign

(positive)●(positive)= (positive)÷(positive)=

(positive)●(negative)= (positive)÷(negative)=

(negative)●(positive) = (negative)÷(positive) =

(negative)●(negative)= (negative)÷(negative)=

(negative or positive)●(0)= (negative or positive)÷(0)=

(0)●(negative or positive)= (0)÷(negative or positive)=

Mentally evaluate each quotient.

37. 38.12 4 12 4

Mentally evaluate each quotient.

39. 40.12 4 12 4

Mentally evaluate each quotient.

41. 42.12 0 0 12

Mentally evaluate each quotient.

43. 44.1

243

24 3

Mentally evaluate each quotient.

45. 46.234 100 234 0.001

Mentally evaluate each quotient.

47. 48. 312

4

5 46 3

Three Signs of a Fraction:

Algebraically Verbally Numerical Example

For all real numbers a and b with ,

and

Each fraction has three signs associated with it. Any two of these signs can be changed and the value of the fraction will stay the same.

0ba a a ab b b b

a a a ab b b b

34

34

Sign: ______ Sign: ______

Sign: ______ Sign: ______

7 8 7

8

78

7 8

49. Signs of a fraction: Mentally determine the sign of each expression.

(a) (b)

(c) (d)

Objective 4: Express ratios in lowest terms.

Any ratio can be written in fraction form. To express a ratio in lowest terms, simply reduce the fraction.

Ratio

Verbally Algebraically Numerical Example

The ratio of a to b is the quotient of a divided by b.

The ratio a to b can be denoted by either a : b or .

The ratio 5 to 8 can be denoted by either __________ or ____________.

ab

12 : 20 60 : 24

Write each ratio in lowest terms.

52. Twelve of 52 cards in a deck of cards are face cards. What is the ratio of face cards to all cards in the deck?

50. 51.

The mean of a set of numerical scores is an average calculated by dividing the ____________ of scores by the number of scores, and the range of a set of scores is calculated by subtracting the ____________ score minus the ____________ score.

Mean and Range

A student earned the following scores on their Beginning Algebra Exams: 77, 59, 94, 62, 71, 61. Give the range and the mean for this set of scores. Round the mean to the nearest hundredth.

53.

Range =

Mean =

54. The price of an item was decreased from $200 to $170.

(a) What is the amount of the price decrease?

(b) What is the percent of the price decrease – that is, what percent of the original price is the decrease?

Phrases Used To Indicate Multiplication:

Key Phrase Verbal Example Algebraic Example

Times "x times y"

Product "The product of 5 and 7"

Multiplied by "The rate r is multiplied by the time t"

Twenty percent of "Twenty percent of x"

Twice "Twice y"

Double “Double the price P”

Triple “Triple the coupon value V”

0.20x

rt

5 7

xy

2y

2P

3V

Phrases Used To Indicate Division:

Key Phrase Verbal Example Algebraic Example

Divided by "x divided by y"

Quotient "The quotient of 5 and 3"

Ratio "The ratio of x to 2"

xy

5 3

: 2x

55. a times six is equal to twelve.

56. The product of p and q is equal to the product of q and p.

Translate each verbal statement into algebraic form.

57. Twice x is greater than six.

58. The ratio of three to x is equal to ten.

Translate each verbal statement into algebraic form.

59. z divided by two is equal to three less than z.

60. The quotient of seven and nine is equal to the multiplicative inverse of x.

Translate each verbal statement into algebraic form.