Generator

irrational

A generator in the number library, called 11 times across 4 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

irrational is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

`show: "tail"`

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.

`show: "niven"`

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.

`show: "liouville"`

How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.

`show: "roots"`

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the derivative of order 7 at zero is a whole number ×12
and below 1/2 ×11
at q = 2 the tail is positive ×11
q! times the first 3 terms is a whole number ×11
the search over divisors of 2 agrees with whether 2 is a perfect square power ×8
the degree-1 integral is positive, because the integrand is ×7
the largest value of the degree-1 polynomial is (π²/4)ᵏ/k! ×7
the search over divisors of 2 agrees with whether 2 is a perfect cube power ×7
the fraction 1/1 is no closer to √2 than 1/(3q²) allows ×6
and 4 has one such root, not several ×4
the 1th truncation beats the exponent 2 ×4
and it does so with room to spare: the error is below q^(−2) ×3
some truncation beats the barrier of degree 2 ×3
a fraction's denominator is not zero ×1
and is below twice the peak, since sin integrates to two over the interval ×1
and it is closer than 1/q², which is what makes the exponent exactly two ×1
at least one of the numbers drawn has an irrational root, or the figure argues nothing ×1
between one and three increasing barrier degrees, none above the number of truncations drawn ×1
between two and five increasing degrees, each between 1 and 12 ×1
between two and ten whole numbers, each between 2 and 64 ×1
its denominator is a whole number between 1 and 200 ×1
so it is strictly between 0 and 1, where no whole number is ×1
the fraction is at least close enough to π for the picture to be about π ×1
the largest denominator tried is a whole number between 3 and 14 ×1
the number of convergents drawn is a whole number between 4 and 9 ×1
the number of tail terms summed exactly is a whole number between 12 and 60 ×1
the number of truncations of the constructed number is a whole number between 3 and 5 ×1
the numerator of the fraction π is supposed to be is a whole number between 2 and 400 ×1
the range drawn reaches a degree whose bound is below one, which is the whole argument ×1
the root taken is a whole number between 2 and 4 ×1
the tail falls as the denominator grows ×1
the view is one the family draws ×1
there are derivatives to show ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Number

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

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

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.

Number

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.

The whole library · What the figures prove