Download 2.3 Rational Numbers

Section 2.3
Rational Numbers
A rational number is a number that may be written in the form
a
for any integer a and
b
any nonzero integer b.
Why is division by zero undefined?
For example, we know that 3 × 4 =
12 and so
12
= 3 . We get one and only one
4
number for this quotient.
0
= n . But remember, n was “any
0
number”, not a unique one! This is one example as to why division by 0 is
undefined!
We also know that any number, n × 0 =
0 so
The rational numbers are ratios of whole numbers. The formal symbol for the rational
numbers is  . Integers are may also be called rational numbers since any integer n may
n
be written as = n .
1
Note:
a
−a a
=
= −
b −b
b
Reducing Fractions
To reduce a fraction, we must find its greatest common factor, gcf, and divide it out. The
resulting fraction is in simplest form (lowest terms).
If finding the gcf is too much work, start by dividing out a common factor (you may use
divisibility rules here), the fraction will become simpler. Continue in this manner until
you see that its gcf is 1.
Example 1: Reduce
240
to lowest terms.
360
Section 2.3 – Rational Numbers
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Mixed Numbers and Improper Fractions
The form a
The form
b
is called a mixed numeral.
c
a
where a > b is called an improper fraction.
b
We can convert from a mixed numeral to an improper fraction and vice versa.
Mixed numeral to an improper fraction:
b
a⋅c + b
a =
c
c
Improper fraction to a mixed numeral: simply divide
Example 2: Convert the following mixed numbers to improper fractions.
a. 3
9
10
b. − 1
5
7
Example 3: Convert the following improper fractions to mixed numbers.
a.
9
4
Section 2.3 – Rational Numbers
b.
− 135
4
2
Every rational number when expressed as a decimal number will be either a terminating
or a repeating decimal number.
Example of terminating decimal is −
Example of repeating decimal is
3
=
−0.12 since upon dividing, the division ended.
25
1
= 0.3333… = 0.3 since upon dividing, the division
3
did not end and it repeated.
Converting Decimal Numbers to Fractions
First recall that
1
1
1
= 0.001 , etc.
= 0.1 ,
= 0.01 ,
100
10
100
Example 4: Convert the following terminating decimal numbers to a quotient of integers.
a. 0.6
b. 0.0622
c. 1.02
Section 2.3 – Rational Numbers
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Converting Repeating Decimal Numbers to a Fractions
Example 5: Convert 0.35 to a quotient of integers.
Example 6: Convert 12.142 to a quotient of integers.
Section 2.3 – Rational Numbers
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Addition and Subtraction of Fractions
In order to add or subtract fractions, we first need a common denominator. Once we
write each fraction using the same denominator, simply add or subtract the numerators
and keep the same denominator. Reduce the answer if possible.
If any of the numbers are mixed numerals, first convert them to an improper fraction then
add/subtract as described above.
Example 7: Perform the indicated operation.
3
−8
a.
+2
13
13
b.
5 3
−
6 8
2
1
c. 1 − 2
4
5
Section 2.3 – Rational Numbers
5
Multiplication of Fractions
In order to multiply fractions, first make sure the fractions are in simplest form. Then
multiply the numerators together and multiply the denominators together.
If any of the numbers are mixed numerals, first convert them to an improper fraction then
multiply as described above.
Division of Fractions
The reciprocal of a fraction,
a
b
is the fraction .
b
a
In order to divide fractions, change the division to multiplication and find the reciprocal
of the second fraction (the one that is dividing). Then follow the rules for multiplying
fractions.
Example 8: Evaluate the following.
 −3   22 
a.    
 2   15 
b.
2
1
÷3
3
7
7 2 
c.  −  ÷ 22
 2 18 
Section 2.3 – Rational Numbers
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