2. A irrational number is a real number
that cannot be written as simple
fraction.
Irrational means not rational
3.
4. 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!
6. 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.
7. 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
