Concept

Factorial

The product of all the whole numbers up to a given one, written with an exclamation mark. It counts the orderings of that many things, and it outgrows every fixed base raised to the same power — which is what makes it the multiplier in almost every squeeze argument.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The constant that counts what does not happen

Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.

analysis · the exponential

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.

Integrality Irrationality Proof by contradiction Squeeze Approximation Convergence rate Counting argument Derangement Derivative e, the number Exact arithmetic Exponential constant

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