Articles
γ, the harmonic remainder
Take the first n reciprocals, subtract ln n; what survives is γ — and no one can prove it irrational.
Reader signal: 0 up, 0 down. Pure Maths 9 min 12.6 ℵ1202Apéry’s theorem
ζ(3) is irrational. Apéry’s 1978 proof is an integer race that the logarithm loses.
Reader signal: 1 up, 0 down. Pure Maths 14 min 25.9 ℵ0078Sixty-four bits of Ω
The halting probability is a definable real whose digits outrun every formal system. Here are 64 of them.
Reader signal: 0 up, 0 down. CS Theory 10 min 21.4 ℵ2314Gelfond’s constant
e^π is transcendental by a two-line identity; π^e remains unclassified. Includes a famous near-integer.
Reader signal: 0 up, 0 down. Applied Maths 8 min 19.6 ℵ1131Random Fibonacci
Flip coins in the Fibonacci recurrence and it still grows exponentially — at Viswanath’s rate.
Reader signal: 0 up, 0 down. Programming 9 min 17.3 ℵ1902Brun’s constant and a division bug
Twin-prime reciprocals converge to Brun’s constant; computing it exposed the Pentium’s divider.
Reader signal: 0 up, 0 down. Programming 8 min 14.8 ℵ0693Nats, bits, and ln 2
Cross-entropy in nats, code length in bits: ln 2 is the exchange rate, and it is a series worth knowing.
Reader signal: 0 up, 0 down. AI 7 min 8.4 ℵ1303The audioactive constant
Look-and-say sequences grow geometrically at λ, an eigenvalue of Conway’s 92-element chemistry.
Reader signal: 0 up, 0 down. Programming 8 min 13.7 ℵ0567The omega constant
ω is the one real number with ω·e^ω = 1 — the value of Lambert’s W at 1, and a fixed point you can iterate to.
Reader signal: 0 up, 0 down. Applied Maths 6 min 7.2 ℵ0123Normal by construction
Champernowne’s constant is normal because it was built to be. No natural constant is known to be.
Reader signal: 0 up, 0 down. CS Theory 7 min 9.8 ℵ1618Fibonacci hashing
Knuth’s multiplicative hash uses 1/φ because the golden ratio is the hardest number to approximate.
Reader signal: 0 up, 0 down. Programming 8 min 10.5 ℵ0739The Dottie number
Press cos on a calculator long enough and it stops at 0.739085. Banach says why.
Reader signal: 0 up, 0 down. Applied Maths 5 min 5.6 ℵ2718Derangements and 1/e
Count the permutations that fix nothing and 1/e appears, good to the nearest integer.
Reader signal: 0 up, 0 down. Pure Maths 6 min 6.9 ℵ1414√2, by descent
The oldest irrationality, run as an infinite descent, plus the continued fraction that pins it.
Reader signal: 0 up, 0 down. Pure Maths 5 min 3.1 ℵ314122/7, an overestimate
A one-line integral proves the schoolroom approximation errs high, and says by exactly how much.
Reader signal: 0 up, 0 down. Pure Maths 6 min 11.2