Transcript The Rational Numbers

Thinking
Mathematically
The Rational Numbers
The Rational Numbers
The set of rational numbers is the set of all numbers
which can be expressed in the form a/b, where a and b
are integers and b is not equal to 0. The integer a is
called the numerator and the integer b is called the
denominator.
Note that every integer is a rational number. For
example 17 can be written 17/1.
The Fundamental Principle of
Rational Numbers
a
If b is a rational number and c is any number other
than 0,
a c a
bc

b
a
a c
The rational numbers b and b  c are called
equivalent fractions
Lowest Terms
A rational number can be reduced to its lowest
terms by dividing the numerator and denominator
by the greatest common divisor.
Example: Reduce 24/40 to lowest terms.
The greatest common divisor of 24 and 40 is
2 x 2 x 2 = 8.
24
40
=
(24/8)
(40/8)
=
3
5
Converting a Positive Mixed
Number to an Improper Fraction
1. Multiply the denominator of the rational
number by the integer and add the
numerator to this product.
2. Place the sum in step 1 over the
denominator in the mixed number.
Converting a Positive Improper
Fraction to a Mixed Number
1. Divide the denominator into the numerator.
Record the quotient and the remainder.
2. Write the mixed number using the following
form:
remainder
quotient
original denominator
Rational Numbers and Decimals
Any rational number can be expressed as a
decimal. The resulting decimal will either
terminate, or it will have a digit that repeats or a
block of digits that repeat.
Examples:
3
=
.6000
...
=
.60
5
The “line” over the zero indicates that it is
to be repeated infinitely often.
1
3
= .3333 ... = .3
Expressing a Repeating Decimal as a
Quotient of Integers
Step 1 Let n equal the repeating decimal.
Step 2 Multiply both sides of the equation in step 1
by 10 if one digit repeats, by 100 if two digits
repeat, by 1000 if three digits repeat, and so on.
Step 3 Subtract the equation in step 1 from the
equation in step 2.
Step 4 Divide both sides of the equation in step 3 by
an appropriate number and solve for n.
Multiplying Rational Numbers
The product of two rational numbers is the
product of their numerators divided by the
product of their denominators.
Example:
3 x 4
5
7
=
(3 x 4)
(5 x 7)
=
12
35
Dividing Rational Numbers
The quotient of two rational numbers is the product
of the first number and the reciprocal of the
second number.
If a/b and c/d are rational numbers, and c/d is not 0,
then
a c a d ad
   
b d b c bc
Adding and Subtracting Rational
Numbers with Identical
Denominators
The sum or difference of two rational numbers
with identical denominators is the sum or
difference of their numerators over the common
denominator. If a/b and c/d are rational
numbers, then
a c ac
a c ac
 
and  
b b
b
b b
b
Density of Rational Numbers
If r and t represent rational numbers, with r<t,
then there is a rational number s such that s
is between r and t.
r < s < t.
Thinking
Mathematically
The Rational Numbers