To multiply two numbers by hand takes a few steps but it's something we're taught in school. When dealing with big numbers, really big numbers, we need to a quicker way to do things.

We give new algorithms for the computation of square roots and reciprocals of power series in C║x║ If M(n) denotes the cost of multiplying polynomials of degree n, the square root to order n costs ...

Karatsuba’s divide-and-conquer multiplication algorithm takes advantage of this saving. Consider a multiplication algorithm that parallels the way multiplication of complex numbers works.