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Rational Numbers (fractions)
A rational number is any number that can be represented by the ratio of two integers.

Examples: , , ,


The horizontal bar is called a vinculum and acts both as a grouping symbol (serves as parenthesis) and to indicate division. These numbers are also called fractions, where the number above the vinculum is called the numerator and the number below the vinculum is called the denominator. Many times we'll say '6' over '4', meaning, .

Properties:

A. Unlimited Representation
Every rational number can be represented in an unlimited number of ways.
Example: = = = ...
Example: 1 = = = = ...
(The previous example is used heavily to manipulate rational numbers, more on this later.)

B. Perfect Rationals are the Integers
A perfect rational is a fraction whose denominator is 1.
Example: , , , ...
which are integers 1, 2, 3, 4, ...

C. Changing forms
Recall that multiplying any number by 1 (the multiplicative identity) does not change that number. This result holds for rational numbers as well. In other words recall that
1 = = = = ... = = = ...
Examples:
Write so that the denominator is 16
Well, 4 * 4 = 16, and multiplying any number by 1 does not change that number, the version of 1 we need is .

So, we have: * =

Write so that the denominator is 21

3 * 7 = 21, we have * =
Write so that the numerator is -10

5 * -2 is -10, we have * =

D. Writing Rationals in their simplest form
Factor the numerator and the denominator into their prime factors, then use the previous result to simplify. This operation is usually called canceling common factors.
Examples:
= = * *
= 1 * 1 *
=

= = * *
= 1 * 1 *
=
Operations:

E. Multiplication (first since addition requires multiplication)
When multiplying two or more rational numbers together, the result is the product of the numerators over the product of the denominators.
Example:


Since a rational has an unlimited number of forms, we usually write them in their simplest form, that is, the form with the smallest possible numerator and smallest possible denominator. A systematic way to do this will follow shortly.
We would write as for the answer, although, technically, both are correct.
Example:
* * = =
Example:
* * = = =

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