Section 1.5 Multiplication and Division of Real Numbers

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


16. 17. 18. 100 13.51 0.1 23.5 114

7


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


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


39. 40.12 4 12 4


41. 42.12 0 0 12


43. 44.1

243

24 3


45. 46.234 100 234 0.001


47. 48. 312

4

5 46 3

Three Signs of a Fraction:


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.


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.


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Section 1.5 Multiplication and Division of Real Numbers