Download 6.3 Multiplication and Division of Rational Numbers

Math 365 Lecture Notes © S. Nite 8/18/2012
Section 6-3
Page 1 of 5
6.3 Multiplication and Division of Rational Numbers
Multiplication of Rational Numbers
Repeated Addition Model
Example: 4 ⋅
2
3
Definition of Multiplication of Rational Numbers
If
a
c
a c a⋅c
and are any rational numbers, then ⋅ =
.
b
d
b d b⋅d
Example:
4 7
⋅
21 44
Properties of Multiplication of Rational Numbers
Theorem 6-10: Multiplicative Identity and Multiplicative Inverse of Rational
Numbers
a. The number 1 is the unique number such that for every rational number
a
.
b
a a a
1⋅   = =   ⋅1
b b b
a
b
b. For any nonzero rational number ,
b
is the unique rational number such that
a
a b
b a
⋅ =1= ⋅
b a
a b
The multiplicative inverse is also called the reciprocal.
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 6-3
Page 2 of 5
Theorem 6-11
a. Distributive Property of Multiplication Over Addition for Rational Numbers
If
e
a c
, , and are any rational numbers, then
b d
f
a c e  a c  a e 
 + = ⋅ + ⋅ 
b  d f   b d   b f 
b. Multiplication Property of Equality for Rational Numbers
a
c
a
c
e
and are any rational numbers such that = , and is any rational
b
d
b
d
f
a e c e
numbers, then ⋅ = ⋅ .
b f d f
If
c. Multiplication Property of Inequality for Rational Numbers
a
c
e
> and > 0, then
b
d
f
a
c
e
If > and < 0, then
b
d
f
If
(i)
(ii)
a
⋅
b
a
⋅
b
e c e
> ⋅
f d f
e c e
< ⋅
f d f
d. Multiplication Property of Zero for Rational Numbers
If
a
a
a
is any rational number, then ⋅ 0 = 0 = 0 ⋅ .
b
b
b
Example:
A bicycle is on sale at
2
of its original price. If the same price is $440, what was the
3
original price?
Multiplication with Mixed Numbers
1
8
Example: 3 ⋅ 2
1
3
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 6-3
Example: Solve for x:
Page 3 of 5
x x
+ =1
a b
Division of Rational Numbers
Definition of Division of Rational Numbers
a
c
a c e
e
and are any rational numbers, then ÷ = if, and only if, is the unique
b
d
b d f
f
c e a
rational number such that ⋅ = .
d f b
If
Theorem 6-12: Algorithm for division of Fractions
If
a
c
c
a c a d
and are any rational numbers and ≠ 0 , then ÷ = ⋅ .
b
d
d
b d b c
Alternate Algorithm for Division of Rational Numbers
a c ad bc
ad
÷ =
÷
= ad ÷ bc , or
b d bd bd
bc
Example: Jenn has 35
requires
1
yards of material available to make towels. Each towel
2
3
yards of material. How many towels can Jenn make? How much material
8
will be left over?
Estimation and Mental Math with Rational Numbers
1
Example:  5  ⋅ 12
 6
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 6-3
Page 4 of 5
Extending the Notion of Exponents
Definition of a to an Integer Power
a⋅
a ⋅
...
⋅ a , where m is a positive integer and a is any rational number.
1. a m = a⋅
m factors
0
2. a = 1
3. a-m =
1
am
For any nonzero rational number a and for any integers m and n,
Theorem 6-13: a m ⋅ a n = a m + n
Theorem 6-14:
am
= a m−n
n
a
Theorem 6-15: (a m ) = a mn
n
m
a
am
Theorem 6-16:   = m
b
b
Theorem 6-17: (ab )m = a m b m
Theorem 6-18
a
For any nonzero rational number and any integer m,
b
a
 
b
−m
b
= 
a
m
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 6-3
Theorem 6-19: Properties of Exponents
a. a0 = 1
b. a-m =
1
an
c. am an = am+n
am
d. n = a m−n
a
e. (am)n = amn
m
a
am
f.   = m
b
a
g.  
b
b
−m
b
= 
a
m
h. (ab)m = ambm
Example: (a −2 + b −2 )
−1
Example: 202 ÷ 24
Example: (x 4 y −3 )
−3
Page 5 of 5