Download A.2 The Algebra of Rational Numbers

E - BOOK FOR COLLEGE ALGEBRA
A2
King Fahd University of Petroleum & Minerals
The Algebra of Rational Numbers
 Algebraic Operations
 Least Common Denominator
 Properties of Rational Numbers
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E - BOOK FOR COLLEGE ALGEBRA
King Fahd University of Petroleum & Minerals
Algebraic Operations

Prime Numbers
A prime number is a natural numbers p > 1
that can not be written as product of two natural
numbers other than one or itself. Otherwise p is
called a composite number.
For example, the natural numbers 2, 3, 5, 7, 11,
13, 19
………. are prime, while the natural
numbers 4, 6, 8, 10, 12, 14, 16, 18, …..… are
composite.
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Algebraic Operations

Prime Factorization
Every non-zero integer n can be written uniquely
as a product of the form (prime factorization)
n  u  p1  pr
where u=±1 and p1…….pr are prime factors of n.

Equivalent Fraction
Two rational numbers a/b and c/d are equivalent
if there is an integer n≠0 such that
a
nc

b
nd
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E - BOOK FOR COLLEGE ALGEBRA
Example 1
Find the prime factorization of the
following numbers.
270 
a)
23 5
3
b)
.
1500   1  2  3  5
c)
.
 3600   1  2  3  5
d)
.
3
4
1470 
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3
2
2
2357
2
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 2
2

3
a)
Find four equivalents to each given
reduced rational numbers.
10

15
6

9
34
8

51
12
b)
1
2
5
6
7
.   
 
 
 
4
8
20
24
28
c)
.
5
10
50
15
20
.
  
 
 
 
3
6
30
9
12
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E - BOOK FOR COLLEGE ALGEBRA
Algebraic Operations
For any rational numbers
a
c
and
, we define
b
d
1. Multiplication
a
b
2. Division
a
c
a d
ad




b
d
b c
bc
3. Addition
a
c
ad  bc


b
d
bd
4. Subtraction
a
c
ad  bc


b
d
bd
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c
ac

d
bd
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 3
Perform the indicated operation and
write the answer in reduced form.
a)
3
4 15
4  15
23
6




11
5 22
1  11 11
5 1  22
2
b)
. 33  35  5
8 5
8 3
8
c)
. 2  3  40  210  250  5
70
d)
20
1400
1400
28
. 7  3  28  30  2   1
10 4
40
40
20
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E - BOOK FOR COLLEGE ALGEBRA
King Fahd University of Petroleum & Minerals
Least Common Denominator (LCD)

The least common denominator of two
fractions a/b and c/d is an integer N such that
both b and d divide N, and N is the least
possible such integer. That is, there are two
integers s and r such that N = bs and N = dr
and N is the least such integer. In this case we
have
a
c
as
cr
as
cr
as  cr






b
d
bs
dr
N
N
N
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 4
a)
Find the least common denominator of
each of the following sets of fractions.
7
4
 1 5
, , 
 ,
 30 12 54 85 
30
12
85
54




235
2
2 3
5  17
3
23
Factor all the
denominators
into their prime
factorizations
LCD  2  3  5  17
2
3
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1
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 4
b)
Find the least common denominator of
each of the following sets of fractions.
2
6 
 17 51
, 
,
 ,

490 343 
 14 280
14 
280 
490 
343 
27
3
2 57
2
257
73
Factor all the
denominators
into their prime
factorizations
LCD  2  5  7
3
1
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3
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Properties of Rational Numbers
Addition Properties
Multiplication Properties
Closure
x + y is a rational number
x
y is a rational number
Commutative
x+y = y+x
x
y = y
Associative
(x + y) + z = x + (y + z)
(x
Identity
y)
x
z = x
(y
z)
There exist a rational number There exist a rational number
0 such that
0+x = x
Inverse
x has additive inverse
– x such that – x + x = 0
Distributive
(x+y)
z
=
x
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z + y
1 such that 1
x = x
x has multiplicative inverse
1/x such that x (1/x) = 1
z
E - BOOK FOR COLLEGE ALGEBRA
Example 5
King Fahd University of Petroleum & Minerals
Give the fractional and decimal representation
of each of the percentages.
(a)
a)
37%
is equivalent to
(b)
b)
6.9%
is equivalent to
(c) .0.021%
c)
is equivalent to
(d) .157%
d)
is equivalent to
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37
100
69
100
21
10000
157
100
or 0.37
or 0.069
or 0.0021
or 1.57
E - BOOK FOR COLLEGE ALGEBRA
King Fahd University of Petroleum & Minerals
Give the percentage representation of
Example 6 each of the numbers.
1
(a)
a)
5
21
(b)
b)
50
is equivalent to
is equivalent to
(c) . 0.394
c)
is equivalent to
(d) .12.5
d)
is equivalent to
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20
100
42
100
39.4
100
1250
100
or 20%
or 42%
or 39.4%
or 1250%
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 9
a)
a) Find the 12% of 98.
b) What percentage is 54 of 450?
12
 98
12% of 98 is equal to
100
 0.12  98  11.76
b)
54 is
54
of 450
450

0.12 of 450
 12% of 450
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Challenge
Example: Argue that
3 is not a rational number.
Solution: If
3 was a rational number, it would have been possible
a
a
to find a completely reduced rational number such that 3 = , then
b
b
a2
3 = 2 or a 2  3b 2
b
From this we see that 3 must be a factor of a 2 and hence 3 must be a
factor of a. Then 32 must be a factor of a 2 or a 2 = 32 q. Then b2  3q.
Then 3 must be a factor of b 2 and hence 3 must be a factor of b. But 3
a
cannot be a factor of both a and b because
is completely reduced.
b
a
This contradiction proves that 3 can not be equal to
for any
b
integers a , b  0 and hence 3 is an irrational number.
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