Irrational Number

A irrational number is a real number that cannot be written as simple fraction.

Irrational means not rational

IRRATIONAL NUMBERS

History of irrational numbers

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write 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!

Some irrational numbers

= 3.1415926535897932384626433e= 2.71828182845904523536 =1.41421356237309504 =1.732050807568877293527Golden ratio =1.61803398874989484820

Visualise √2

Imagine a floor with tiles on. The tiles are half-squares. If you look at it hard, you can see a right-angled triangle in the middle, with a square on each of its sides.

The Greeks knew Pythagoras's theorem - "the square on the hypotenuse (the longest side) is equal to the sum of the squares on the other two sides." They knew that a triangle with sides of 3, 4 and 5 had a right angle (32+42=9+16=25=52) and so did a triangle with sides of 5, 12 and 13 (52+122=25+144=169=132). But this floor tile triangle has sides 1, 1 and √2. This must be true because of Pythagoras's theorem, since the longest side squared must be equal to 12+12=2. But you can also count the squares. The squares on the shorter sides have 2 tiles each, and each tile is half a square, so that makes one square. We'd expect that, since the sides are 1long. But the big yellow square has four half tiles, so that makes two squares. So its sides must be √2. Floor tiles are definitely part of the real world, so √2 is a real number. But it's not a rational, it's an irrational number.