Download 1.4 Multiplying and Dividing Real Numbers; Properties of Real Numbers

1.4
1.
2.
3.
4.
5.
Multiplying and Dividing Real Numbers;
Properties of Real Numbers
Multiply integers.
Multiply more than two numbers.
Multiply rational numbers.
Find the multiplicative inverse of a number.
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
Multiplicative
Property of 0
Multiplicative
Identity
Example
Word Form
0(4) = 0
5(0) = 0
The product of a
number and 0 is 0.
0∙a=0
4(1) = 4
The product of a
number and 1 is that
number.
1 ∙a = a
Changing the order
3(5) = 5(3)
of factors does not
affect the product.
Changing the
(2 ∙ 3) ∙ 4 = 24
grouping of three or
2 ∙ (3 ∙ 4) = 24
more factors does not
affect the product.
A sum multiplied by
Distributive
2(3
+
4)
=
2(3)
+
2(4)
a factor is equal to the
Property of
sum of that factor
Multiplication
multiplied by each
over Addition 3(x + 2) = 3x + 3(2)
addend.
Commutative
Property of
Multiplication
Associative
Property of
Multiplication
Symbolic Form
ab = ba
a(bc) = (ab)c
a(b + c) = ab + ac
Multiplying Numbers with Different Signs
3(4) = 12
3(3) = 9
3(2) = 6
3(1) = 3
3(0) = 0
3(–1) = –3
3(–2) = –6
3(–3) = –9
When multiplying two numbers
that have different signs, the
product is negative.
Multiplying Numbers with the Same Sign
–3(4) = –12
–3(3) = –9
–3(2) = –6
–3(1) = –3
–3(0) = 0
–3(–1) = 3
–3(–2) = 6
–3(–3) = 9
When multiplying two numbers
that have the same sign, the
product is positive.
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.
8  2 =
Dividend
4
Dividend
Divisor
8
 4 Quotient
2
Quotient
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
Different = answer negative
= –5
10
2
18
6
 36
9
Same = answer positive
=3
Same = answer positive
=4
Multiplying Rational Numbers
6 2
62
4
12




11 3
11  3
11
33
2
6 2

11 3
1
4

11
Multiplying Rational Numbers
Reduce common factors.
Multiply numerators.
Multiply denominators.
Multiplying Rational Numbers
1
3
4 2
7 12

9 14 2
1
2

3
1
5  3
  
8  10 
2
3

16
Multiplying Rational Numbers
1
2
1
3
3
  6   6  12 


 
 15  8 16  15 
5
2
1
5
3

25
Dividing Rational Numbers
4 2 24 3
6
  

11 2 1 11
11 3
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
  
3
2
3 3
3
9
 
5
10
2

9
2
1
3 10
  
5
93
1
2

3
Multiplying Decimal Numbers
Multiply (–7.6)(0.24).
7.6
0.24
304
+ 1520
1824
1 place
+ 2 places
3 places
1.824
–1.824
Different signs = negative.
Multiplying Decimal Numbers
Multiply (–6.6)(–0.5).
6.6
0.5
330
3.30
3.30
1 place
+ 1 places
2 places
Same signs = positive.
Dividing Decimal Numbers
.07 3.5
3.5  .07
5
50..
7 350.
35
00
0
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
3.6 44.64
.
36 446.4
44.64 ÷ (3.6)
12.4
36 446.4
36
Same signs = positive.
12.4
86
72
144
144
0
Divide 14.7 ÷ (0.03)
1.4
a)
49
b)
49
c)
490
d)
490
Copyright © 2011 Pearson Education, Inc.
Slide 1- 23
Divide 14.7 ÷ (0.03)
1.4
a)
49
b)
49
c)
490
d)
490
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

0
0

0
What times 0 equals 3?
What times 0 equals 0?
Undefined.
Indeterminate.
Division Involving 0
7

0
0

9
Undefined
0
3

0
Undefined
0

0
Indeterminate