Famous Numbers and Number Systems, From Perfect Numbers to Quaternions

A perfect number equals the sum of its proper divisors, as 6 equals 1 plus 2 plus 3. Euclid proved a rule that builds an even perfect number from certain primes, and Euler later proved that every even perfect number has that form. No odd perfect number is known to exist or ruled out.

Mersenne primes, primes one less than a power of two, are the other half of Euclid's rule: every one generates an even perfect number. Marin Mersenne listed exponents in 1644, and since 1997 nearly every new one has been found by the distributed GIMPS project.

Each term is the sum of the two before it. Leonardo of Pisa introduced the sequence to Western Europe in his 1202 book Liber Abaci through a puzzle about rabbits, although Indian poets studying Sanskrit meters had met similar sequences earlier.

The ratios of consecutive Fibonacci numbers approach the golden ratio, about 1.618. Its earliest known definition is in Euclid's Elements, which speaks of cutting a line in extreme and mean ratio, and it is the positive root of phi squared equals phi plus one.

Each entry is the sum of the two above it, and each gives the number of ways to choose k items from n. Blaise Pascal's treatise of 1665 named it in the West, but Piṅgala, al-Karaji and Yang Hui described equivalent arrays centuries before.

The sequence 1, 1, 2, 5, 14, 42 counts parenthesis matchings, binary trees and triangulations of polygons. It is named for Eugene Catalan, yet Mingantu found it in the 1730s and Euler described it in 1751.

These rational numbers turn up in series for the tangent function and in formulas for sums of powers. Seki Takakazu discovered them independently of Jacob Bernoulli, whose Ars Conjectandi appeared in 1713, and Ada Lovelace used them in her 1842 algorithm note for the Analytical Engine.

A Carmichael number is composite yet passes the congruence that Fermat's Little Theorem guarantees for primes, so that theorem cannot serve as a perfect primality test. The smallest is 561, published by Robert Carmichael in 1910.

Gauss introduced complex numbers with integer real and imaginary parts in his 1832 work on quartic reciprocity. They factor uniquely into Gaussian primes, and a prime congruent to 1 modulo 4 splits into two conjugate Gaussian primes.

Kurt Hensel first described p-adic numbers in 1897. Built from digits in base p that extend infinitely to the left, they complete the rationals under a different notion of size than the real numbers do.

William Rowan Hamilton found these four-dimensional numbers on 16 October 1843 and carved the rule into Brougham Bridge. Multiplication no longer commutes, yet every nonzero quaternion has an inverse, and unit quaternions now describe rotations in graphics and robotics.