Every candidate for a rational square root, tried
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.
With nothing chosen
The tail that would have to be a whole number
Every candidate for a rational square root, tried
The polynomial that squeezes π
How closely a fraction can come, and the barrier that says no closer
The tangent built from a continued fraction
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.
- a(0) is the whole number given by the binomial sum ×61
- the denominator of b(0) divides 2·lcm³ ×61
- the denominator of b(0) divides lcm² ×61
- term 0 follows the pattern ×30
- past the threshold the tail at level 1 lies strictly between 0 and 1 ×14
- the search over divisors of 2 agrees with whether 2 is a perfect square power ×14
- 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 degree-1 integral is positive, because the integrand is ×9
- the fraction 1/1 is no closer to √2 than 1/(3q²) allows ×9
- the largest value of the degree-1 polynomial is (π²/4)ᵏ/k! ×9
- and 4 has one such root, not several ×7
- the search over divisors of 2 agrees with whether 2 is a perfect cube power ×7
- Aₙ = q·e − p for the convergent 1/1 ×5
- Bₙ = p − q·e for the convergent 1/0 ×5
- Cₙ = p − q·e for the convergent 2/1 ×5
- the 1th truncation beats the exponent 2 ×5
- and it does so with room to spare: the error is below q^(−2) ×4
- some truncation beats the barrier of degree 2 ×4
- the search over divisors of 2 agrees with whether 2 is a perfect fourth power ×4
- at x = 0.3 each deeper convergent is closer to tan x ×3
- the fraction converges to tanh 0.5 ×3
- √2 is 1; 2, 2, 2, … ×1
- √2: all 16 terms are certain ×1
- a fraction's denominator is not zero ×1
- and (1 + tanh ½)/(1 − tanh ½) is e ×1
- and ends below one millionth ×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
- Apéry's fractions approximate only like one over the denominator to a power a little above one ×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
- e follows its pattern ×1
- e or its square root ×1
- e: all 16 terms are certain ×1
- every convergent is within one over its denominator squared ×1
- for √2, q² times the error never falls below a third ×1
- for e the lowest values drift down, slowly ×1
- golden ratio: all 16 terms are certain ×1
- its denominator is a whole number between 1 and 200 ×1
- q times the error falls at every step ×1
- so it is strictly between 0 and 1, where no whole number is ×1
- the bounds fix at least as many terms as are drawn ×1
- the constant is ζ(2) = π²/6 ×1
- the constant is ζ(3) ×1
- the denominator is a whole number between 1 and 5 ×1
- the error shrinks like (√2 − 1)⁴ⁿ ×1
- the error shrinks like φ⁻⁵ⁿ ×1
- the fraction is at least close enough to π for the picture to be about π ×1
- the golden ratio is all ones ×1
- the largest denominator tried is a whole number between 3 and 14 ×1
- the lcm grows slowly enough for both proofs ×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 is a whole number between 1 and 5 ×1
- the numerator of the fraction π is supposed to be is a whole number between 2 and 400 ×1
- the product of the whole-number multiplier and the error falls to nought ×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
- twelve levels at x = 1 give tan 1 to double precision ×1
- ζ(2) or ζ(3) ×1
- π: all 16 terms are certain ×1
- π's fifth term is 292 ×1
- ψ(n) is the logarithm of the lcm ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
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.
NumberAn 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.
NumberApproached 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.
NumberThe 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.
NumberThe 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.
NumberThe 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.
NumberWhich 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.