The Gaussian Integers

In 1832, Gauss extended the integers into the complex plane, considering numbers of the form a + bi where a and b are whole numbers. Within this system — the Gaussian integers — new 'primes' emerge. The ordinary prime 5 is no longer prime: it factors as (2 + i)(2 − i). But 3 remains a Gaussian prime. Gauss used this system to prove results in number theory that had resisted ordinary integers for decades. It showed that choosing the right number system can make previously hard problems obvious.