The first reading I undertook for this project was Roger Cooke’s The History of Mathematics: a brief course (Cooke, 2011). I was particularly interested in Chapters 19 & 20, which provide an overview of the development of mathematics in India over the last few millennia. The Indians have a rich history of mathematising, but there are two particular innovations I want to consider. In this post I will look at how the Indians developed our modern number system, its advantages over other ancient systems, as well some thoughts about the relevance of this history for the modern Australian classroom. In the next post I will discuss a related concept, the invention (or discovery) of the number zero.
One of the major contributions India made to modern mathematics was the development of a number system that relied on digits whose value depended on their position within a numeral. In modern parlance, this was the origins of our system of place value. Any number can be formed by adding a series of multiples of powers of ten. The power itself is indicated by its position in the numeral, and the multiple is indicated by the value of the digit in that respective place.
I’m finding that describing our modern place value system is difficult, precisely because I’m so used to it and my mind has a habit of confusing the written representation with the number itself. It seems such an obvious way of writing numbers that I struggle to think of how else it can be done. Some ancient cultures, such as the Babylonians, used a place value system, but it was complicated by the fact that different places would indicate multiples of different numbers (Cooke, 2011, p. 56). They would group their ones in a similar way to the way we do, but groups of ten ones would be groups together in sixes rather than tens. The direct descendant of this system lives with us in our time-keeping system, in which sixty seconds make a minute, sixty minutes make an hour, but only 24 hours are needed to make a day. Again, we are so used to this system that it’s difficult to consider other ways of doing it. Yet trying to calculate, for example, how many minutes a distant time and date might be, is far more difficult than adding, say, two distances measured in the metric system.
We are also aware of other number systems. The Romans used what we now call Roman numerals (I assume they just called them numerals!) which doesn’t use place value at all. Rather, it consists of a strange combination of base-five and base-ten digits which are simply added together to make a number (with some subtraction to make things easier). I am going to consider the Roman system in some detail in another post, so I won’t describe it in any more detail, except to say that it is rather unwieldy, especially when working with large numbers, and completely incapable of dealing with fractions.
As we have seen with the Babylonian system, the position of a symbol in a number could affect its value. However, the wonderful thing about the Indian system was that there was no limit to the size of numbers that could be described in it. Indeed, the Indians had a word for 1055 which appears to be far larger than any number needed by their mathematics, but was perfectly at home in it (Cook, 2011, p. 251). The ability to write and calculate arbitrarily large (or small) numbers is crucially important to our current practice of mathematics. And the ability to do this isn’t always obvious, but every primary school teacher knows the look on a student’s face when they realise that the process they use to add two-digit numbers can just as easily be used to add twenty-digit numbers.
This system of perceiving numbers came along with another important feature: the digits the ancient Indians used are the direct ancestors of the digits we use today, as can be seen in this chart (reproduced from Cook, 2011, p. 245). Some of these systems used a zero, which made them even more powerful.
This particular line of research is fascinating, although one wonders what the relevance for the classroom might be. However, on reflection, comparing number systems forces me to think about the assumptions that are built into the way I understand and teach mathematics. A base-10 system written in a strict place value format is immensely powerful, yet it is just one possible way to consider numbers. Plenty of successful civilisations have come and gone without ever having a system remotely like ours. I need to be aware that any system for writing numbers is an abstraction that simultaneously illuminates and obfuscates the reality being described. We use our system because it’s efficient and it works, not because it directly corresponds to maths in a way that, say, Roman numerals don’t.
With that said, reflecting on the differences between these systems can provide insight into how the system we use works. Seeing how adding large number would be difficult in a mixed-base system like the Babylonians use can help us (teachers and students alike) to appreciate why our system works the way it does. Seeing how numbers larger than several thousand are difficult to notate in the Roman system (without the addition of extra systems of notation) can help someone appreciate the simplicity and extensibility of the modern system.
In the next post, I will spend some time looking at the innovation that really let this Indian system of numbers: the invention of the number zero.
Cooke, R. L. (2011). The history of mathematics: A brief course. John Wiley & Sons.