The Weirdest Numbers in Mathematics

There's a number, called Chaitin's constant, that is perfectly well-defined. It's a specific real number between 0 and 1. Yet it's been proven that no computer program can ever calculate all of its digits. Only finitely many of them can ever be known.

Mathematics is full of numbers like this: numbers too large to write down, numbers built from strange recipes, and numbers that exist but can't be pinned down. Here's a tour of the weirdest.

Most Numbers Can Never Be Described

You might think every number can at least be described, by a formula, a decimal pattern or a computer program. But there are only countably many possible descriptions (finite strings of symbols), and uncountably many real numbers.

So almost all real numbers are indescribable: no finite sentence, formula or program specifies any one of them. The numbers we know, like π, e and √2, are rare exceptions.

1. i: The Square Root of −1

No real number squares to −1, so for centuries it was dismissed as "imaginary." Today i is indispensable in physics and engineering, and it behaves like a quarter-turn on a plane. See The Strange World of Imaginary Numbers.

2. Liouville's Constant: The First Proven Transcendental Number

A transcendental number isn't the root of any polynomial with integer coefficients. In 1844, Joseph Liouville proved such numbers exist, and in 1851 he gave an explicit example:

L = 0.110001000000000000000001000…

It has a 1 in positions 1!, 2!, 3!, 4!, … (positions 1, 2, 6, 24, 120, …) and 0 everywhere else. The ones become so sparse that the number is approximated "too well" by fractions to be algebraic. π and e were proven transcendental only later, in 1882 and 1873.

3. Champernowne's Constant: Every Number in a Row

Write all the positive integers in order after a decimal point:

C = 0.123456789101112131415161718192021…

In 1933, David Champernowne, then a Cambridge undergraduate, proved it's normal in base 10: every digit string of a given length appears equally often in the long run. It contains your phone number, your birthday, and every other finite digit sequence. We suspect π is normal too, but nobody has proved it.

4. Chaitin's Constant Ω: Knowable Only in Principle

Computer scientist Gregory Chaitin defined Ω in the 1970s as the probability that a randomly generated program (in a specific kind of programming language) eventually halts.

Ω is a definite number. But knowing its first n digits would let you solve the halting problem for all programs up to about n bits long, which Alan Turing proved in 1936 is impossible in general. So Ω is uncomputable: no algorithm can list its digits. It's also algorithmically random: its digits have no pattern shorter than the digits themselves.

5. Graham's Number: Too Big for the Universe

Graham's number arose as an upper bound in a problem in Ramsey theory studied by Ronald Graham and Bruce Rothschild. Martin Gardner made it famous in his Scientific American column in 1977.

It's built with Knuth's up-arrow notation: 3↑3 = 3³ = 27, 3↑↑3 = 3^3^3 ≈ 7.6 trillion, and each extra arrow means iterating the previous operation. Graham's number takes 64 layers of this, each layer determining the number of arrows in the next.

Even writing the number of digits of its number of digits is impossible in the observable universe. Yet we know its last digits: it ends in …95387.

6. TREE(3): Making Graham's Number Look Tiny

TREE(n) comes from a game of building sequences of labeled trees under certain rules, related to Kruskal's tree theorem. TREE(1) = 1 and TREE(2) = 3. TREE(3) is so much larger than Graham's number that the gap between them can't be described with up-arrows. Logician Harvey Friedman showed that proving TREE(3) is finite requires surprisingly strong mathematical axioms.

7. Skewes' Number: A Bound on a Prime Surprise

The number of primes up to x is usually less than the estimate li(x). In 1914, J. E. Littlewood proved that eventually this reverses, infinitely often. In 1933, Stanley Skewes showed, assuming the Riemann hypothesis, that the first reversal happens below about 10^(10^(10^34)). That bound has since been lowered enormously, to below about 10³¹⁷, but no example has been found. It was once famous as the largest number used in a serious proof.

8. The Feigenbaum Constant: Chaos Has a Number

In 1975, physicist Mitchell Feigenbaum studied the logistic map, xₙ₊₁ = r·xₙ(1 − xₙ), used to model populations. As r increases, stable behavior splits into 2 cycles, then 4, then 8. The intervals between splits shrink by a ratio approaching:

δ ≈ 4.669201609…

Remarkably, the same constant appears in a huge variety of unrelated systems on the road to chaos, from fluids to electronic circuits.

9. −1/12 and the "Sum" of All Positive Integers

You may have seen the claim that 1 + 2 + 3 + 4 + … = −1/12. As an ordinary sum, that's false: the series grows without limit. But using a technique called zeta function regularization, a finite value, −1/12, can be consistently assigned to it. The same value shows up in physics calculations, such as the Casimir effect and string theory. It's a reminder that "=" can mean different things in different mathematical frameworks.

10. ℵ₀: The Smallest Infinity

Aleph-null is the size of the set of whole numbers. It's the smallest infinite cardinal, and there are strictly bigger ones. See Can Infinity Be Bigger Than Infinity?

An Insider Reference: Knuth's Arrows

To describe unimaginably large numbers, Donald Knuth introduced up-arrow notation in 1976:

a↑b   = aᵇ              (exponentiation)
a↑↑b  = a↑a↑…↑a         (b copies, a tower of powers)
a↑↑↑b = a↑↑a↑↑…↑↑a      (iterated towers)

Each additional arrow represents a new, vastly faster-growing operation. It turned "unimaginably large" into something you can write on a single line.

Two Concepts Worth Knowing

Transcendental Number

A transcendental number isn't a root of any non-zero polynomial with integer coefficients. π, e and Liouville's constant are transcendental; √2 is not.

Computable Number

A computable number can have its digits produced by some algorithm to any precision. π and e are computable. Chaitin's Ω isn't.

Quick Answer: What Are the Weirdest Numbers in Mathematics?

Some of the weirdest numbers include i (the square root of −1), Liouville's constant (the first explicit transcendental number), Champernowne's constant (all integers written in a row), Chaitin's constant (well-defined but uncomputable), Graham's number and TREE(3) (unimaginably large), the Feigenbaum constant of chaos, and ℵ₀, the smallest infinity.

Try Them Yourself

• Scientific Calculator: compute 3↑↑3 = 3^27
• Numbers to Words: hear how big large numbers really are
• First 1,000 Digits of Pi: the number we suspect is normal
• Logarithm Calculator: count digits of huge powers
• How Much Is a Trillion?: start with "small" big numbers
• What Is the Largest Number You Can Name?: the contest behind giant numbers

Search the first 1,000 digits of π for your birth date written as DDMM. Then remember that Champernowne's constant is guaranteed to contain it.