Multiplying and Dividing Real Numbers;
1.4 Properties of Real Numbers
1. Multiply integers.
2. Multiply more than two numbers.
3. Multiply rational numbers.
4. Find the multiplicative inverse of a number.
5. Divide rational numbers.
Multiply integers.
2 3 = 6
Ways to Show Multiplication
Factors Product 2∙3
2(3)
(2)(3)
2 3
2x
Properties of Multiplication
Property Example Word Form Symbolic Form
Multiplicative 0(4) = 0 The product of a
Property of 0 number and 0 is 0. 0∙a=0
5(0) = 0
Multiplicative The product of a
Identity 4(1) = 4 number and 1 is that 1 ∙a = a
number.
Commutative Changing the order
Property of 3(5) = 5(3) of factors does not ab = ba
Multiplication affect the product.
Associative Changing the
Property of (2 ∙ 3) ∙ 4 = 24 grouping of three or a(bc) = (ab)c
Multiplication 2 ∙ (3 ∙ 4) = 24 more factors does not
affect the product.
Distributive A sum multiplied by
Property of 2(3 + 4) = 2(3) + 2(4) a factor is equal to
Multiplication the sum of that factor a(b + c) = ab + ac
over Addition 3(x + 2) = 3x + 3(2) multiplied by each
addend.
Multiplying Numbers with Different Signs
3(4) = 12
3(3) = 9 When multiplying two numbers
3(2) = 6 that have different signs, the
3(1) = 3 product is negative.
3(0) = 0
3(–1) = –3
3(–2) = –6
3(–3) = –9
Multiplying Numbers with the Same Sign
–3(4) = –12
–3(3) = –9 When multiplying two numbers
–3(2) = –6 that have the same sign, the
–3(1) = –3 product is positive.
–3(0) = 0
–3(–1) = 3
–3(–2) = 6
–3(–3) = 9
Multiplying Integers
–2(7) Different = answer negative = –14
3(–5) Different = answer negative
= –15
–6(–2) Same = answer positive
= 12
5(–2) Different = answer negative = –10
3(6) Same = answer positive = 18
– 3(– 8) Same = answer positive = 24
Multiplying Integers
–2(–2) = +4
–2(–2) (–2) = –8
–2(–2) (–2) (–2) = +16
–2(–2) (–2) (–2) (–2) = –32
The product of an even number of negative factors
is positive, whereas the product of an odd number
of negative factors is negative.
Multiplying Integers
–2(3)(5) = –30
3(–5)(–2) = 30
–6(–2)(–1)(–3) = 36
5(–2)(–1) = 10
2(6)(1)(2) = 24
– 7(–3)(–2) = –42
Multiply (–6)(–3)(7).
a) 126
b) 126
c) –63
d) 63
1.4 Copyright © 2011 Pearson Education, Inc. Slide 1- 9
Multiply (–6)(–3)(7).
a) 126
b) 126
c) –63
d) 63
1.4 Copyright © 2011 Pearson Education, Inc. Slide 1- 10
Divide integers. Dividend 8
4 Quotient
8 2 = 4 Divisor 2
Quotient
Dividend Quotient
Divisor Dividend
Divisor
Division is defined in terms of multiplication.
8 2 = 4 because 4 ∙ 2 = 8.
Same sign rules as multiplication!
Dividing Signed Numbers
When dividing two numbers that have the same
sign, the quotient is positive.
When dividing two numbers that have different
signs, the quotient is negative.
Dividing Integers
–21 ÷ 7 Different = answer negative = –3
15 ÷ (–3) Different = answer negative
= –5
–16 ÷ (–2) Same = answer positive
=8
10 Different = answer negative = –5
2
18
Same = answer positive =3
6
36
9
Same = answer positive =4
Multiplying Rational Numbers
6 2 62 12 4
11 3 11 3 33 11
2
6 2 4
11 3 1
11
Multiplying Rational Numbers
Reduce common factors.
Multiply numerators.
Multiply denominators.
Multiplying Rational Numbers
1 4 2
7 12
2
9 14 2
3
1 3
1
5 3 3
8 10 16
2
Multiplying Rational Numbers
1 1
2 3
3
6 6 12 3
15 8 16 15 25
5 5
2
1
Dividing Rational Numbers
4 2 24 3 6
11 3 11 2 1 11
Dividing Rational Numbers
Change divide to times and write the reciprocal
or multiplicative inverse of the divisor
(second fraction)
Reduce common factors.
Multiply numerators.
Multiply denominators.
Dividing Rational Numbers
1 3 1 2 2
3 2 3 3 9
2
1
3 9 3 10 2
5 10 5 9 3
1 3
Multiplying Decimal Numbers
Multiply (–7.6)(0.24).
7.6 1 place
0.24 + 2 places
304
+ 1520
1824 3 places
1.824
–1.824 Different signs = negative.
Multiplying Decimal Numbers
Multiply (–6.6)(–0.5).
6.6 1 place
0.5 + 1 places
330
3.30 2 places
Same signs = positive.
3.30
Dividing Decimal Numbers 3.5 .07
5
50..
.07 3.5 7 350.
35
0
00
00
Move the decimal point in the divisor so it is an integer.
Move the decimal point in the dividend the same number of
places.
Place the decimal in the quotient directly above the new
decimal point in the dividend.
Dividing Decimal Numbers
44.64 ÷ (3.6)
. 12.4
3.6 44.64 36 446.4 36 446.4 Same signs = positive.
36 12.4
86
72
144
144
0
Divide14.7 ÷ (0.03)
a) 49
b) 49
c) 490
d) 490
1.4 Copyright © 2011 Pearson Education, Inc. Slide 1- 23
Divide14.7 ÷ (0.03)
a) 49
b) 49
c) 490
d) 490
1.4 Copyright © 2011 Pearson Education, Inc. Slide 1- 24
Division Involving 0
20
5 because 5 4 20
4
0
0 because 0 4 0
4
3 What times 0 equals 3? Undefined.
0
0 Indeterminate.
What times 0 equals 0?
0
Division Involving 0
7
Undefined
0
0
0
9
3
Undefined
0
0
Indeterminate
0
