where , where , and where every remainder is strictly smaller than its predecessor: . The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers. The quotients are generally found by rounding the real and complex parts of the exact ratio (such as the complex number ) to the nearest integers. The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function is defined, which converts every Gaussian integer into an ordinary integer. After each step of the Euclidean algorithm, the norm of the remainder is smaller than the norm of the preceding remainder, . Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. The final nonzero remainder is , the Gaussian integer of largest norm that divides both and ; it is unique up to multiplication by a unit, or .