# Decimal without a zero

**Decimal without a zero** is a base ten positional numeral system which does not use a digit to represent zero; it instead has a digit to represent ten, such as "X."

As with conventional decimal, each digit position represents a power of ten, so for example 123 is "one hundred, plus two tens, plus three units." All positive integers which are represented solely with non-zero digits in conventional decimal (such as 123) have the same representation in decimal without a zero. Those which use a zero need to be rewritten, so for example 10 becomes X, 20 1X, 100 9X, 101 X1, 302 2X2, 1000 99X, 1110 XXX, 2010 19XX, and so on.

Addition and multiplication in decimal without a zero are essentially the same as with conventional decimal, except that carries occur when a position exceeds ten, rather than when it exceeds nine. So to calculate 643 + 759, there are twelve units (write 2 at the right and carry 1 to the tens), ten tens (write X with no need to carry to the hundreds), thirteen hundreds (write 3 and carry 1 to the thousands), and one thousand (write 1), to give the result 13X2 rather than the conventional 1402.

It is a simple system for counting and basic arithmetic for positive integers, but clearly decimal without a zero cannot represent the number zero. It can also be confusing when presenting decimal fractions: the conventional 41/20 = 2.05 comes out as 41/1X = 1.X5 (one unit plus ten tenths plus five hundredths), looking as if it could be less than two; the system has problems presenting fractions less than a tenth, except as ratios.