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How long is the diagonal of a square?

(or.. Rootin' Two-tin')

If we draw a square with sides of length 1 metre, how long is the diagonal? The answer is "the square-root of 2". The square-root of a number is a number, which, when multiplied by itself (squared) gives the original number.

So the square-root of 9 (written as 9) is 3 since 3 squared = 3 x 3 = 9. Since 2 is not a square number, its square-root is not a whole number:
2 = 1.41421 35623 73095 04880 16887 ...

Can we write this as a fraction such as 99/70? 99/70 is quite close to the square root of 2 since 99/70 = 1.4142 85...

The ancient Greek mathematicians thought that the only numbers were whole numbers and fractions, but then found evidence that the square-root of 2 could not be written as a fraction, no matter how big the numbers were in the fraction!

Such numbers - that are neither whole number nor fractions - are called irrational numbers. This doesn't mean that they aren't logical, but means they are not the ratio of two numbers (rational numbers), the ir- acting like the "ir" in the word "irrelevant" (meaning not relevant).

This sequence of fractions gives better and better approximations to 2:

Can you ...
• ... spot the pattern in the numbers on the top and in the numbers on the bottom and so find the next fraction in the series? [Hint: use the previous two numbers on top to get the next number on top. Does you pattern work on the bottom numbers also?]
• .. use your calculator to make a table of the decimal fractions for these fractions. How close is each to the "true" value given above?