# Algorithms, Numbers, and Machines

## Abstract

This chapter begins with a strange algorithm, Russian Peasant Multiplication, that may or may not work correctly. (Verifying correctness is a constant theme throughout this book, and the techniques used to decide such issues are explained thoroughly later.)

The concept of an algorithm is illustrated by more examples: testing primes, factoring an integer into primes, and finding the greatest common divisor of two integers. Representation of numbers in computers is discussed, and truncation error is shown to be an unavoidable consequence of using machines to do computation.

This chapter ends with an algorithm to solve equations, that is, to find a numerical solution that is accurate to any degree one might desire. This illustrates the formal idea of the limit of a sequence in the context of how many iterations must be done by the algorithm to guarantee a specified accuracy.