√2 is not a fraction

We have √2 trapped between 1.414 and 1.415, and could keep going. Will the squeezing ever stop? Will some decimal square to exactly 2?

No, and here's the argument. Suppose some terminating decimal (one that stops), starting 1.414… and stopping somewhere, squares to exactly 2.

A terminating decimal must have a non-zero last digit (otherwise you'd just drop the zero and it would stop one place earlier).

But when you square a decimal whose last digit isn't zero, the square also ends in a non-zero digit after the decimal point. (Multiplying the last digits gives the last digit of the answer, and no digit from 1 to 9, squared, ends in 0.)

Now 2 is 2.000…: all zeros after the point. So no terminating decimal can square to 2, and the decimal of √2 must go on for ever.

That's one kind of failure. There's a second, deeper one: √2 can't be written as a fraction m/n of counting numbers either.

Suppose it could. Then √2 = m/n, so squaring both sides gives 2 = m²/n², that is 2n² = m². Now recall from Part 1: in the prime factorisation of any square, every prime appears an even number of times (36 = 2 × 2 × 3 × 3; 49 = 7 × 7; 81 = 3 × 3 × 3 × 3).

Look at the prime 2 in 2n² = m². On the right, m² is a square, so 2 appears an even number of times. On the left, n² has an even number of 2s, and the extra 2 in front makes the count odd. An even count can't equal an odd count, so no such m and n exist. This beautiful proof is Euclid's, from the Elements of about 300 BCE.

So √2 is neither a terminating decimal nor a fraction: only the endless, never-repeating decimal 1.41421356… (Numbers like this are called irrational.)