When the old market square in a small town was rebuilt, the plan gave the paved area as 200 square metres and said the shape was a perfect square. Everything else the builders needed had to come out of those two facts.
The side of the square is the number that, multiplied by itself, gives 200. It works out to a little over fourteen metres — 14.14 to the nearest centimetre — and it is not a tidy number, which is why the contractor wrote it on the plan in exact form instead: ten roots of two.
The diagonal is where the arithmetic turns friendly. The diagonal of any square is its side multiplied by the square root of two — and multiplying ten roots of two by another root of two gives twenty. The path across the corner of that square comes out exactly twenty metres, with no remainder at all.
The builders used it. A diagonal string line of exactly twenty metres is far easier to set out on rough ground than a side of 14.14, and once both diagonals are equal and twenty metres long, the corners are square without anybody measuring an angle.
The paving mason put it more plainly. The side, he said, is a number you round; the diagonal is a number you can trust.