Concept

Integrality

The fact that a rational root of a polynomial whose leading coefficient is one must be a whole number. It turns an infinite search over fractions into a finite search over the divisors of the constant term, and it is what decides which roots of whole numbers are irrational.

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

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.

number · irrationality

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.

Irrationality Factorial Proof by contradiction Squeeze Algebraic number Convergence rate Derivative Divisor Exact arithmetic Exponential constant Geometric series Infinite descent

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