355/113 Is Good Because the Next Term Is 292

The golden ratio is the hardest number to approximate

If large terms mean good approximations, the number with no large terms at all must be the worst case — and that number is the golden ratio, whose expansion is [1; 1, 1, 1, 1, …] forever. Its convergents are ratios of consecutive Fibonacci numbers, 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, and each one improves on the last by the smallest possible margin. That is not a curiosity: a sunflower packing seeds at an angle related to the golden ratio never lets successive seeds line up, precisely because no simple fraction approximates the angle well. Being badly approximable is the useful property there.

A repeating pattern means an irrational of a particular kind

The square root of two expands as [1; 2, 2, 2, 2, …], repeating a single term forever, and every square root of a non-square whole number behaves this way — the pattern eventually cycles. That is a theorem, and it runs both ways: a continued fraction whose terms are eventually periodic is always the root of a quadratic equation with whole-number coefficients, and one that never settles into a pattern is something else. π's expansion has no pattern anybody has found, which is consistent with what is known about it. The finite case is simpler still: an expansion that stops is a rational number, and 415/93 stopping after four terms is the whole content of that statement.

Frequently asked questions

Why is 22/7 so famous if 355/113 is far better?

Because it is short enough to remember and accurate enough for most of what people did with it. 22/7 is wrong by four hundredths of a per cent, which is invisible in any measurement made with a ruler and adequate for most workshop arithmetic; 355/113 is wrong by less than a ten-millionth, which nobody needs unless the calculation is being carried to many digits. The convergents form a ladder of accuracy against memorability, and 22/7 sits at the rung where the trade is most favourable for mental arithmetic. Where more precision is needed there is usually also a calculator.

Where are continued fractions used in practice?

Wherever a ratio has to be built out of whole things. Gear trains approximate an arbitrary speed ratio with integer tooth counts, and the convergents give the smallest gears that get close enough; calendar rules approximate the length of a year with whole days and leap rules; musical temperaments approximate an octave with a whole number of steps. The same machinery also appears inside algorithms for reducing fractions and for reconstructing a rational number from a decimal, which is what lets a computer algebra system recognise that 0.142857142857 is a seventh rather than a long decimal.