Monday, November 21, 2016 0

I expect most of you are familiar with Roman numerals. For example, V represents five, I represents one, so seven is written as VII. The full set is:

 1 I 5 V 10 X 50 L 100 C 500 D 1000 M

The key difference between the Roman system and our number system is that the position of a number makes no difference – X represents 10 wherever it appears. So, translate LXXV11 ….. the answer is 76. (The system did change over the years so that position had some relevance. For example, instead of writing VIIII for 9, it became IX, ie 1 before 10).

Roman numerals, astonishingly, are still in use today: films and television programmes often show their year of creation in Roman numerals, the Olympic Games use them, you might see them on clock faces, and the first eleven football or hockey team is often written as 1st XI. But, unlike our system which is used both for representing numbers and for doing calculations, you couldn’t possibly use Roman numerals for calculating. Imagine CLXXVI ÷ XVI.  (The answer is, of course, XI)! The Romans used the abacus, an incredibly efficient calculating device, leaving the numeral system purely for number representation. The two images below show an actual Roman abacus, made of bronze, and a scene where a slave in a wealthy family can be seen using an abacus.

The invention of zero and its use as a placeholder in numbers was not a simple linear piece of history. Many civilisations dabbled with placeholders – for example, the Babylonians and the Mayans – and others experimented with zero as a number; but it was probably in India that the two came together. The brilliant mathematician But look in the table of numerals – there is no symbol for zero. Why not? Because, to them, zero was not a number. To have a “number” of things, you must have at least one. The Romans simply used words for “none” – nullus, nihil or nil (from which many words in Latin-based languages are derived, such as null, nullify, nihilist, annihilate in English).

Brahmagupta developed the rules for zero as a number in the 7th century. For example: “The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.” He also managed to multiply by zero, but struggled with division by zero, in particular stating that “zero divided by zero equals zero.”  (Something for you
to investigate…).

And it was the brilliance of Indian mathematicians that developed 0 as a placeholder. What does this mean? Consider the numbers 2, 23, and 254. In the first, the value of the digit 2 is, of course two; in the second, it has value twenty; and in the third, two hundred. But suppose we want to write two hundred and four, in other words we need to show that there aren’t any 10’s. We can’t just leave a gap (although earlier mathematicians did use a dot). So, in 204, the zero indicates that there are no 10’s, and the 2 still stands for two hundred. We take all this for granted, but it took centuries for such a system to develop.

Traders took the new number system from India to the Islamic lands and from there, in the 12th century, Italian mathematician Fibonacci introduced it to Europe where, within a century, it had pretty well displaced Roman numerals and the abacus.

Zero as a number and as a placeholder is incredibly powerful without which the development of mathematics, and the numeracy required for our modern world, would have been very much slower.

And without zero you couldn’t have the mathematical joke: “There are 10 types of people in the world – those who understand binary numbers, and those who don’t!”