The Importance of Zero

Many years ago a Roman citizen was walking down the street when he came across a friend.


"Good day, Marius. Say, I missed the game last night. What was the final score?"


"Well, Cassius, it was a low scoring affair. The score was Patricians IV, Plebians , uh, uh... . I don't know. We don't have a number for nothing!"


That story might be fictional, but the fact is that the Roman numeral system had no symbol for zero. Numbers first came into being when people began to use them for counting. The phrase "counting numbers" refers to the numbers 1, 2, 3, and so on. When we count we start with one, not zero. Zero wasn't needed for counting.


Not only did everyday Roman citizens fail to see a need for zero, mathematicians of the time failed to see it as well. Today, the situation is very different. Zero is part of our number system and we are very comfortable using it. In fact, the system we use to write numbers, called positional notation (or place-value notation), is only possible because of zero.


In the positional notation system, the number of days in a year – three hundred sixty five - is written as "365".  "365" stands for three hundreds, six tens, and five ones. If we used those same three digits, 3, 6, and 5, and put them in a different order, the number would be different. "635" stands for six hundreds, three tens, and 5 ones. The point is that in the positional notation system the value of each digit in a number depends on the place it is in.


Now think about the number two hundred. It has two hundreds, no tens, and no ones. But we couldn't write that as "2", because 2 written that way is in the ones position. We have to put the 2 in the hundreds position. And we can only move it over to the hundreds position by showing the tens and ones positions as well. Since the number two hundred has no tens and no ones we put zero in each of those positions. So the number is written as 200. This is only possible because of the digit zero.


Aside from the fact that zero is absolutely essential when writing numbers using positional notation, let's not forget that it is also needed to represent a quantity - the quantity of nothing. When a team scores no points, we use "0" to express that quantity.


But how could it be that civilizations such as the Egyptians and Greeks were able to make important discoveries in mathematics without seeing the need for the number zero? Constance Reid, in her book "From Zero to Infinity", offers an explanation.


The Greeks were fascinated by numbers and their properties, and made great advances in number theory, much of which does not need the number zero. When it came to doing calculations, in commerce and elsewhere, they used counting boards. Counting boards used columns of beads, and positional notation, much like the abacus of today. To represent the number 203, you placed two beads in the leftmost column, no beads in the next column, and three beads in the last column.


If you are representing a number on a counting board, a zero is shown by placing no beads in its column. If you did a calculation and wanted to write down the result, as represented by the several columns of beads, you would use the symbols available to you, which, for the Greeks, were the digits 1, 2, etc. But what symbol did the Greeks use for a column with no beads? We believe they may have used a dot, dash or circle. But there was no symbol for zero. It would be many years before the need for a symbol for zero was recognized.


There is no doubt that zero is an important and essential part of our number system. Students also learn in their studies that zero has some special properties that distinguish it from the other digits. For example:


- Adding zero to a number does not change the number. 5 + 0 = 5
- Multiplying a number by zero results in zero. 5 · 0 = 0
- You cannot divide a number by zero. 5/0 is not allowed.


In today's world of computers zero plays a very important role. When you view computers on a micro level you see that they are made of circuits that are either turned "on" or turned "off". The "on" state is represented by "1" and the "off" state by "0". And the programming code that is written to control computers, when reduced to its simplest form, is composed of 1s and 0s. It is hard to imagine how computers could have been created without the concept of zero and the digit we use to represent it.


So let's hear it for zero. Its importance can hardly be overstated.