Pick up any sheet of A4 paper. It is 210 millimetres wide and 297 long, and neither number looks like anything a person would choose.
They are both consequences of a single decision. The system begins with a sheet called A0 whose area is exactly one square metre. A1 is A0 folded in half; A2 is A1 folded in half; and so on. A4 is the fourth fold, so its area is a sixteenth of a square metre — about 625 square centimetres.
The clever part is the ratio. The designers wanted a sheet that, halved again and again, always keeps the same proportions — so that a page designed for one size scales without distortion to any other.
Only one shape can do that, and it is the shape whose long side is the square root of two times its short side. Check the sheet in your hand: 297 divided by 210 is 1.414, and the square root of two is 1.414.
Two everyday facts fall straight out of this. A photocopier's «A4 to A3» button enlarges by 141 percent, not 200, because doubling the area means multiplying each side by only the square root of two. And an A4 sheet is thinner and longer than a sheet of American Letter paper, which was chosen by custom rather than by arithmetic.
A whole industry, in other words, is standing on one irrational number.