November 19, 2012

De-mystifying Mathematics (Part 4) – 21 hundred vs. 2 thousand 1 hundred – the flexibility of the placeholder system!

We discussed the ease of representing numbers using a placeholder system and ten symbols (including ‘0’). Having understood the fundamentals, let us look at the flexibility this system offers.

As an example, if you had to find the next number in the following series –

Seven thousand seven hundred seventy seven
Eight thousand eight hundred eighty eight
Nine thousand nine hundred ninety nine

A majority of people, to whom I have asked this question, have struggled with this. If we convert the above series into a numerical representation:
The next number in the series is ‘ten thousand ten hundred ten tens and ten’. What if we leave the number as it is and write it as 10|10|10|10? It may create some confusion among readers and may take away the simplicity of the representation that we started off with. However, there is still some benefit of this multi-digit placeholder representation – it gives us a different way of representing large numbers during intermediate calculation steps before bringing it to a generally accepted nomenclature.

In the above example, it is easy to write down the next number in the series as 10|10|10|10 and then derive the answer:
10 ones         = 1 tens and 0 ones
10 tens         = 1 hundreds, 0 tens and 0 ones
10 hundreds   = 1 thousand, 0 hundreds, 0 tens and 0 ones
10 thousands = 1 ten thousand, 0 thousands, 0 hundreds, 0 tens and 0 ones

What we have ended up doing is ‘carrying over’ the additional digits into the next placeholder. So, the number in the generally accepted representation is ‘11110’eleven thousand one hundred ten!

When the concept of ‘carrying over’ is taught, it seems like a sin to leave two digits in one placeholder at any point in the calculation steps. If you understand the basics and how to deal with the ‘placeholders within placeholders’, then you do not feel afraid to take this liberty

So, feel free to explore different ways of writing a number – at least in the calculation steps. Let us see how this can help us in basic calculations.
The advantage of this being that there is no restriction on where I start from (left to right, right to left or each one individually). This is possible as the ‘carry over’ adjustment/ cleaning up are done in the second step.

We will take a look at using this flexibility for multiplication in the next post…

November 12, 2012

De-mystifying Mathematics (Part 3) – Concept of zero

We discussed in the previous post on the need to create a representation for non-existent values in the placeholder system of writing numbers. Using blank spaces may make it ambiguous as blanks are also used for separating words and numbers. We can create another symbol for this – so what is the big deal? Let us say we represented it by ‘0’ and called it ‘nothing’.

The problem arises in the definition of this symbol and its value. As we are talking about counting as the basic need, what does nothing mean? All other symbols 1 to 9 are countable. You understand what 1 apple means or 7 chairs mean. But what does nothing mean? Is ‘no apples’ the same as ‘no chairs’?

If you had just one apple on the table and took that away – you would be left with ‘no apples’. You will also be left with ‘no oranges’ or ‘no plates’. How do I deal with this symbol in basic calculations?

These are the types of questions that plagued mathematicians in Middle East and Western world even as late as 10th century. The more you think of it as a mystic symbol, the more confusing it may get. The Indian cultures had already embraced the concept of nothingness and it was an integral aspect of the spiritual traditions. ‘Shunya’ (the nothingness) and ‘Anant’ (the infinite) were easily understood and talked about in this part of the world.

The earliest documented references to this numerical symbol go back to around 5th century AD. This was the time when not only the numerical representations, but also the rules governing the usage and calculations came into existence. Rules explaining addition, subtraction, multiplication with zero were laid down and understood.

We are not sure if these existed before that time as there is no documentary evidence available so far. Even after this advancement, it took a few hundred years before the rest of the world understood and accepted this notation. The journey from ‘Shunya’ to ‘Sifr’ to ‘Zephyr’/ ‘Cipher’ to ‘Zero’ was a long one across continents and centuries…

Many great mathematicians and physicists have commented on this in the past. One of the best descriptions is by the French mathematician Pierre Simon Laplace who wrote:

“It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.”

To be able to fully appreciate the contribution of ‘Zero’ in today’s context, imagine writing numbers without the zero and placeholder system. Imagine no binary system and hence no electronics, no computer, no TV, no mobiles! On a lighter note, this is the mathematical representation of the spiritual truth – everything has come from nothingness :-)

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…