Gaussian integer
Euclidean division

Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials. This makes the Gaussian integers a Euclidean domain, and implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma, the unique factorization theorem, and the Chinese remainder theorem, all of which can be proved using only Euclidean division.

A Euclidean division algorithm takes, in the ring of Gaussian integers, a dividend a and divisor b ≠ 0, and produces a quotient q and remainder r such that

In fact, one may make the remainder smaller:

Even with this better inequality, the quotient and the remainder are not necessarily unique, but one may refine the choice to ensure uniqueness.

To prove this, one may consider the complex number quotient x + iy = a / b. There are unique integers m and n such that – 1 / 2 < x – m ≤ 1 / 2 and – 1 / 2 < y – n ≤ 1 / 2, and thus N (x – m + i (y – n)) ≤ 1 / 2. Taking q = m + in, one has

with

and

The choice of x – m and y – n in a semi-open interval is required for uniqueness.This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number ξ to the closest Gaussian integer is at most √ 2 / 2.
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