Multiplying & Dividing Real Numbers

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 1.6 in the textbook– Multiplying Real Numbers– Finding the Reciprocal of a Real Number– Dividing Real Numbers– More with Order of Operations– Expressions & Equations Using Multiplying &

Dividing

Multiplying Real Numbers

4

Multiplying Real Numbers

• Multiply as normal– How would we multiply (513)(28) without a

calculator?

• Consider the sign of the product– Multiplying two Integers of the SAME sign

• Positive

– Multiplying two Integers of DIFFERENT signs• Negative

• What happens if we multiply by 0?

5

Multiplying More than Two Real Numbers

• Multiply the numbers as usual• Consider the sign of the product

(-)(-)(-) = (+)(-) = -

(-)(-)(-)(-) = (+)(-)(-) = (-)(-) = +

(-)(-)(-)(-)(-) = (+)(-)(-)(-) = (-)(-)(-) = (+)(-) = -

:

• Notice the pattern?

Multiplying Real Numbers (Example)

Ex 1: Multiply (Remember that there are no calculators on Quiz #1!!!):

a) (-21)(54)

b) (-3.24)(-0.161)

6

7

Multiplying Fractions

• To multiply fractions:– Multiply their numerators– Multiply their denominators– Simplify

• Doing it this way leaves the possibility of simplifying large numbers– e.g. (21⁄25) * (35⁄14)

• Easier to simplify between numerators and denominators before multiplying across

• What happens if we multiply a fraction and an Integer?

• What happens if we have a mixed number involved in the multiplication?

Multiplying Fractions (Example)

Ex 2: Multiply:

a) b)

c)

8

21

16

8

3

9

10

5

33

47

9

3

2

Dividing Real Numbers

10

Dividing Real Numbers

• Divide as normal– How would we divide 399 ÷ 20 by long division?

• Consider the sign of the product– Dividing two Integers of the SAME sign

• Positive

– Dividing two Integers of DIFFERENT signs• Negative

• What happens when we evaluate 0/x?

• What happens when we evaluate x/0?

Dividing Real Numbers (Example)

Ex 4: Divide:

a) -219 ÷ -21

b) 7.38 ÷ -1.8

11

12

Reciprocals

• Reciprocal – two fractions whose product is 1– How do we find the reciprocal of a fraction?

• Exchange the numerator & denominator and keep the sign

Are 3⁄2 and 2⁄3 reciprocals? Why?

What are the reciprocals of -5⁄9, 4, and 1⁄3?

13

Dividing Fractions

• To divide fractions:– Turn the operation into multiplication by taking

the reciprocal of the SECOND fraction– Multiply the fractions as normal

• DO NOT attempt to simplify the fractions while the operation is still division!

– This is a common mistake!

Dividing Fractions (Example)

Ex 5: Divide:

a)

b)

14

3

20

9

4

15

1

3

13

More with Order of Operations

16

Exponents and Sign

• Recall that exponent notation can be written as multiplication

• Be careful when identifying the base• There is a significant difference between (-3)2 and -

32 (-3)2 = (-3 * -3) = 9

Read as negative 3 squared (base is -3)-32 = -(3 * 3) = -9

Read as the opposite of 3 squared (base is 3)• Mixing these up is a common error!• What happens when using a calculator?

More with Order of Operations (Example)

Ex 7: Simplify:

a) b)

c)

17

20

7

10

21

4

5

62

2122

22

437212 3

Expressions & Equations Using Multiplying & Dividing

Expressions & Equations Using Multiplying & Dividing

Ex 8: Evaluate the expression for the given values of the variables:

x2 – y3; x = -2, y = -3

19

Identifying Equations & Verifying Solutions (Example)

Ex 9: Tell whether the proposed solution satisfies the equation:

a) 3x + 26 = 4; x = -10

b) -10 – y = -6; y = -4

20

21

Summary• After studying these slides, you should know how to do the

following:– Multiply real numbers including integers, fractions, and decimals– Divide real numbers including integers, fractions, and decimals– Solve more complicated order of operations problems– Evaluate expressions involving multiplication or division of real

numbers– Identify an equation and verify whether the values given for the

variables satisfy the equation• Additional Practice

– See the list of suggested problems for 1.6• Next lesson

– Properties of Real Numbers (Section 1.7)