Thursday, September 14, 2017

The Bakhshali Manuscript or Much Ado About Nothing

 Michael - Thursday

I admit that I stole the second half of the title from the article the appeared in The Guardian today (that you can read HERE). It seemed too good to miss. The article is about some recent work on an ancient Indian text called the Bakhshali manuscript which is written in a form of Sanskrit on multiple pieces of tree bark. It seems to be some sort of training manual for traders. It’s quite impressive in its scope, including problems (posed in verse) in arithmetic, algebra, and geometry. The solutions are given as well (and are correct). And many of the numbers in the text contain zeroes, which are indicated by heavy dots, showing placeholders in a number position that should be 0. So what?

Section of the Bakhshali manuscript

Number systems have been around for a very long time, of course. Most cultures developed some way of representing numbers (if they wrote things down at all). We’re all familiar with the Roman system that is used to this day, in a rather pretentious fashion, for the year. It’s a horrible system, involving a few symbols and then implied addition and subtraction from them—for example IX is one less than ten, so nine, while XI is one more than ten so eleven. Some scholars have speculated that the lack of a decent number system was a not insignificant problem for the Roman Empire. The ancient Greeks—often put forward as the originators of modern mathematics— had no concept of zero and had philosophical concerns about how nothing could be something.

In fact, the western world was in no way the leader in mathematics historically. That honor belonged to other cultures and the Indian subcontinent has a strong claim. Around 700 AD when an Indian astronomer had formalized the numeral zero, and the concept was introduced to Europe, it was dismissed. Why have a number for nothing? Why would you want to count nothing? It was like writing IIIIIV! What was the point? The point was that it now allowed the sort of number system we have today, where any number can be written in a way in which its value is obvious in powers of ten (or two if you prefer that as computers do) and on which we can do arithmetic easily. (Try the sum MMMXIV + MCLIX.)

Since this is now so obvious to us, why was it ever hard? The idea of nothing, and the understanding that if you have five sheep and sell five sheep then you have none left was always obvious. But why a symbol? It really isn’t that obvious at all. Why write down that you have 0 sheep? You simply don’t have any! Children don’t find this easy, by the way.  Zero and negative numbers are confusing things when you first meet them.

What has happened recently is that the Bakhshali manuscript has been radiocarbon dated with surprising results. Firstly, the text is not a part of one document at all, but bits of different ones hundreds of years apart. Were they part of some sort of ‘mathematics library’ that collected the material together? How did a part of that collection end up buried near the village of Bakhshali in what is now Pakistan? Dan Brown could make something of this, I’m sure. But the really interesting part, at least to people interested in the history of mathematics, is that one section of the document dates to around 300 AD. That’s about 500 years earlier than the current oldest document using a zero symbol as a digit in a decimal number representation. Looks pretty certain now that we owe essentially everything we do these days with numbers to the people of the Indian subcontinent.


  1. Popularizing zeros has undoubtedly gotten some elected.

  2. I knew there was something wrong with head, Michael. Thanks to you, I now know what. It's that, unlike most people, I like having nothing in it. As a young student, I loved negative numbers and got along quite well with them right from the start. Now, I will have to go off and figure out how such a friend of negative numbers as I turned out to be such an optimist. MMMM.