A Primer on Balanced Binary Representations Jeffrey Shallit * Department of Computer Science University of Waterloo Waterloo, Ontario, Canada N2L 3G1 shallit@graceland.uwaterloo.ca July 1992 - Revised 1993 Abstract We discuss balanced binary representations. 1 Introduction and Definitions Every non-negative integer n can be represented essentially uniquely in base 2, as follows: n =  0≤i≤j e i 2 i where e i ∈{0, 1} and e j = 0 for n = 0. We consider the consequences of enlarging the digit set to {−1, 0, 1}. We call such an expansion a signed-digit expansion. One immediate consequence is that every integer, positive, negative, or zero, can be represented using the digits {−1, 0, 1}. In fact, Theorem 1.1 Every nonzero integer has an infinite number of signed-digit expansions. Proof. We prove this for positive integers n, the proof for negative integers being essentially identical. Write the ordinary base-2 representation of n −1 as (n −1) 2 = e j e j -1 ··· e 0 . Choose any k>j , and consider the representation of 1 as 1 k    −1 − 1 ··· − 1. Now add these two representations, term by term. The result is a representation of n using only the digits 1, 0, and −1. * Research supported in part by a grant from NSERC. 1