Rational Numbers
A rational number is a number that can be written as a simple fraction (ie as a ratio).
Example 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 |
| 0.111... |
1/9 |
Yes |
√2
(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.
Formal Definition of Rational Number
More formally we would say:
A rational number is a number that is expressed in the form p/q
where p and q are integers and q is not equal to zero.
So, a rational number is:
p / q
Where q is not zero
Examples:
| p |
q |
Rational Number |
| 1 |
1 |
1 |
| 1 |
2 |
0.5 |
| 55 |
100 |
0.55 |
| 1 |
1000 |
0.001 |
| 253 |
10 |
2.53 |
| 7 |
0 |
No! "q" can't be zero! |
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 it was therefore 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!
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