# When a Number System Loses Uniqueness: The Case of the Maya - Uniqueness of Representation in a Place-Value System

Author(s):
Pedro J. Freitas (Universidade de Lisboa) and Amy Shell-Gellasch (Beloit College)

We now outline the proof of uniqueness of representation in a pure place-value system because this is exactly the issue we are going to address in the Maya number system. To establish uniqueness of the coefficients of the expression $a_0+a_1\,b+a_2\,b^2+\cdots+a_k\,b^k,\quad\quad\quad\quad (2)$ in which all coefficients $$a_i$$ satisfy $$0\le{a_i}\le{b-1},$$ we can use a strong induction argument. The result is true for zero. (If you want the sum (2) to be zero, you need all coefficients to be zero.)

Now take a natural number $$n$$ and assume uniqueness holds for every positive integer smaller than $$n.$$ We wish to prove that $$n$$ has a unique expression in base $$b.$$ Suppose $n=a_0+a_1\,b+a_2\,b^2+\cdots+a_k\,b^k=a^{\prime}_0+a^{\prime}_1\,b+a^{\prime}_2\,b^2+\cdots+a^{\prime}_t\,b^t,$ with, say, $$t\ge{k}.$$ We want to prove that $$k=t$$ and $${a_i}={a^{\prime}_i}$$ for $$0\le{i}\le{k}.$$ One can rearrange the equation to get $a_0-a^{\prime}_0=(a^{\prime}_1-a_1)b+(a^{\prime}_2-a_2)b^2+\cdots$ where the left side of the equation is a number between $$-b+1$$ and $$b-1$$ and the right side is a multiple of $$b.$$ Therefore both sides must be $$0,$$ and $$a_0=a^{\prime}_0.$$ Now we take, as was done in the example on page 2, the integer $$n_1$$ such that $n=a_0+bn_1\quad{\rm{or}}\quad{n_1}=\frac{n-a_0}{b},$ which is strictly smaller than $$n,$$ and is expressed as $n_1=a_1+a_2\,b+a_3\,b^2+\cdots+a_k\,b^{k-1}=a^{\prime}_1+a^{\prime}_2\,b+a^{\prime}_3\,b^2+\cdots+a^{\prime}_t\,b^{t-1}.$ By the hypothesis of strong induction, the result is true for $$n_1$$ and we get that $$k=t$$ and $${a_i}={a^{\prime}_i}$$ for $$1\le{i}\le{k}.$$

If the encoding of a number were not well-defi ned, we would venture to say that our modern society, so reliant on technology that is driven by numbers (in particular base $$2$$), would come to a grinding halt. It should also be said that we are referring to the uniqueness of a finite integer, regardless of the base. But it also applies to terminating fractional expressions. The well known case of the non-uniqueness of decimal fractions ending in repeating $$9$$'s (for example, $$1.999\dots=2$$) does not fall under this defi nition. However, since no computer system uses an infinite expansion, the dire consequences of non-uniqueness are avoided. But there was one highly advanced civilization whose place-value system did not maintain the uniqueness requirement. Although this civilization did not come to a grinding halt, it did mysteriously decline ....

Pedro J. Freitas (Universidade de Lisboa) and Amy Shell-Gellasch (Beloit College), "When a Number System Loses Uniqueness: The Case of the Maya - Uniqueness of Representation in a Place-Value System," Loci (June 2012), DOI:10.4169/loci003883