Rational Numbers

A Rational Number is a real number that can be written as a simple fraction (i.e. as a ratio).


1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

Here are some more examples:

Number As a Fraction Rational?
5 5/1 Yes
1.75 7/4 Yes
.001 1/1000 Yes
-0.1 -1/10 Yes
0.111... 1/9 Yes
(square root of 2)
? NO !

Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are not rational they are called Irrational.

Another famous irrational number is Pi (π):

Rational Number

Formal Definition of Rational Number

More formally we would say:

A rational number is a number that can be in the form p/q
where p and q are integers and q is not equal to zero.

So, a rational number can be:


Where q is not zero


p q p / q =
1 1 1/1 1
1 2 1/2 0.5
55 100 55/100 0.55
1 1000 1/1000 0.001
253 10 253/10 25.3
7 0 7/0 No! "q" can't be zero!

Using Rational Numbers

add, subtract, multiply and divide

If a rational number is still in the form "p/q" it can be a little difficult to use, so I have a special page on how to:

Add, Subtract, Multiply and Divide Rational Numbers


Pythagoras' Student

The ancient greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction), but one of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!