November 02, 2012

De-mystifying Mathematics (Part 2) – India’s contribution of zero to the number system

We discussed in the earlier post that mathematics can be thought as a collection of many logical/ numerical concepts and processes that make the base for applied sciences and arts. We will touch upon the various concepts in this series – may not be in the linear flow of typical text books.

One of the first uses of mathematics that we learn and encounter is numbers – for counting and basic arithmetic. We usually start associating a written representation with a number as soon as we start learning them. In my opinion, this association itself causes limitations in later stages of learning and restricts innovation and creativity!

Let us look into this in more detail. If I say the number ‘Twelve’ – we immediately think of the representation ‘12’. What if I had a symbol for twelve – say ‘µ’? I could continue to create symbols for each number. We have all played code games in our childhood. Real counting is also like that – a bit more evolved and widely accepted. Can we continue to create a symbol for each number we know? Where should I stop creating new symbols and try to reuse – at 9, at 16, at 20 or 60?

What about other ways of writing the number - Tally marks or Roman numerals? Tally marks were the most primitive representations - they resembled counting on the five fingers on a hand. Roman numerals are representations with a combination of alphabets.

The Tally marks and Roman representations have a problem of scalability and processing. It becomes extremely hard to write down large numbers and perform arithmetic operations on them. As the system evolves, one would be forced to move to an advanced representation system with placeholders and repeating symbols (like the units, tens, hundreds, etc... in our commonly used decimal system).

The placeholders give importance or weight to the value digit. The placeholder itself can be any value – 2, 8, 10, 16, 60, etc… For example, if the placeholder is SIXTY, then the first symbol would represent the number itself, the second symbol would represent SIXTY times the symbol value and so on. Though this looked like an elegant system, the problem it posed is handling blanks!

For example, if I have to represent sixty in a placeholder system of sixty, then I would have write the symbol for ‘one’ and then leave a blank to denote the first placeholder. Sixty = ‘1 ’. It would be difficult to differentiate between ‘1 and ‘1’ – except for the context and some guesswork.

The two questions that needed answers to complete the numeral system were: How many symbols are enough and what do we do with these non-existent values or blanks?

Providing answers to these two critical questions has been India’s biggest contribution to ‘mathematics’ and hence many advanced scientific inventions and discoveries.

From the Vedic times, a lot of scriptures were already referring to large and small numbers with a placeholder value of 10.

One = eka;
Ten = dasha; Eleven = ekadasha (one plus ten)
Twenty = vimsatih; Twenty one = ekavimsatih (one plus twenty)
Hundred = shat; two hundred = dwi-shat
And so on…

The second question needed an answer to ensure written representation was non-ambiguous. This is where ‘shunya’ or zero came into existence! But why was it such a big deal? We will look into this in the next post…
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