Irrationality — the series

1. 

   The square that cannot shrink

   The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

   part 1 · number
2. 

   Which roots refuse to be fractions

   The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.

   part 2 · number
3. 

   A tail too small to be a whole number

   If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

   part 3 · number
4. 

   An integral that cannot be a whole number

   Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.

   part 4 · number
5. 

   Approached too fast to be algebraic

   An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

   part 5 · number
6. 

   The fraction Lambert built for the tangent

   The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

   part 6 · number
7. 

   The pattern in e's continued fraction

   Written as a continued fraction, e is 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8 — two ones, then the next even number, for ever. Euler found the pattern and proved it with a differential equation. A proof from 2006 needs only three integrals, each of which turns out to be exactly the error of one of e's own convergents.

   part 7 · number
8. 

   The race that makes ζ(3) irrational

   Roger Apéry's 1978 proof that the sum of the reciprocal cubes is not a fraction comes down to a race between two numbers. A whole-number multiplier grows by a factor of ten every 0.79 steps; the gap it multiplies shrinks by a factor of ten every 0.65. The gap wins, by a margin of about seventeen per cent, and that margin is the whole proof.

   part 8 · number
9. 

   How close two sums of square roots come

   Two different sums of square roots of whole numbers are never equal: square roots of different squarefree numbers are independent, so no cancellation can make them agree. But they can come very close. Among sums of two square roots of numbers up to 320, √88 + √289 and √171 + √177 agree to eight decimal places. How many digits must be computed, in the worst case, to tell two such sums apart is not known — the best proof allows a number of digits doubly exponential in the length of the sums, the examples suggest only a modest polynomial, and the gap between them is why nobody knows whether the shortest route through points in the plane can be checked quickly.

   part 9 · number

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