Concept

Squeeze

An argument that traps a quantity strictly between two bounds where nothing of its kind can lie. The classical cases put a number that would have to be whole strictly between nought and one, which is how both the irrationality of e and that of pi are established.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

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.

number · irrationality

The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number.

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.

number · irrationality

Named alongside it

The objects these essays reach for when they reach for this one.

Factorial Integrality Irrationality Proof by contradiction Convergence rate Derivative Exact arithmetic Exponential constant Geometric series Integration by parts Pi Polynomial

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