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…

October 27, 2012

De-mystifying Mathematics (Part 1) – Introduction

Mathematics is one of the subjects that evoke a wide range of emotions among students and parents alike. Being one of my favorite subjects, I wanted to understand what is it about mathematics that causes anxious moments? Is it the concepts, the way it is taught or the natural inclination of the student?



This thought stayed with me for many years. In the mean time, I came across a book on Vedic Mathematics as described by Swami Bharti Krisna Tirthaji Maharaj. It gave a new and different perspective to basic mathematical concepts and processes. I learnt these and started teaching it as an introduction to Vedic Mathematics.

However, the original question and others around India’s contribution to mathematics still remained. Over years of reading and exploring, I have understood mathematics more than what I had in my school days. As the case is with memory, I thought it might be a good idea to write down my thoughts and share it.

There is a lot of literature on the internet about the history of mathematics and its main contributors around the world across time that one can read.


Mathematics as a subject
One of the key things that I have learnt is that study of mathematics as a separate subject is not as old as the processes themselves. So, the arithmetic processes of counting, addition, multiplication are very old – but learning them under the banner of a separate subject is relatively new. Even in the Vedic classification, maths (‘Ganit’) does not appear as a veda, but is mentioned under ‘Jyotish’. This may be attributed to the astronomers who used and developed mathematical calculations extensively to determine the panchang (calendar) accurately.

The science of construction also had its basis in mathematics and developed many geometrical concepts. The hymns, music and other sciences also had a good mathematical base.

In all, mathematics was (and is) the basis of the knowledge prevalent in the ancient times across all fields – basic counting/ accounting, ritualistic hymns, music, ceremonies, construction, astronomy, astrology, etc… The question then is – what is this basis called ‘mathematics’ that we struggle with?

One usually thinks of mathematics as completely logical and unambiguous where 1+1=2. What I have now understood is that this is not the case – we will come to that in a separate post. As many great mathematicians have said in different words: Mathematics is an attempt to explain the universe (prakriti) around us. Though the universe is never completely explainable, there are certain rules and patterns that are found in here – and mathematics, as we understand, helps us understand these.

If I was told this in my school, I would have appreciated this subject much more :) I believe that if we have to continue to push the frontiers of knowledge, strong fundamentals are very necessary. Like a good chef can experiment with the ingredients to create a new dish and predict the taste and outcome, a good student of mathematics should feel confident to play around with the concepts and processes to attempt something new.

Next, we look at the basic counting and what does India’s contribution of ‘zero’ really mean – to be continued.

 

September 16, 2012

A different approach to increasing services margins


It is interesting how day to day incidents can teach you fairly advanced business concepts. Recently, we got our house painted. Painting is a fairly commodity skill set and fiercely competitive. There is a generally accepted price band and the customer would ensure that they pay within the band and not a rupee/ sq ft more than that. In this kind of an industry, as a painter, how do you make more money than others? To raise the rates that you charge for the same work, you would need to really differentiate yourself from others – and that may be quite a challenge. Not to mention the same drill you have to go with, for every client.

As the customer here, we were also offered special effects on some walls. We browsed through the catalogue of special effects and designs – and finally narrowed down on a couple of walls and the special effect that we need. The output looked great – but what struck me was that the additional effort spent to create the effect was only one hour for each wall! And the additional margin (net of extra material) was few thousand rupees!

We did not mind paying that extra amount as the output was really exceptional and the painter also made a sizeable additional margin (for the effort spent). What made this possible: New product/ offering (Royale play by Asian Paints), prepackaged combination options, standard tool kit for creating the effect and training a wider set of painters to be able to use all of these.




My primary work area being business consulting in the SCM software domain – I started thinking if we could do something similar. Software consulting services are fairly a commodity now and the competition on hourly rates is fierce. Can we then create an offering that is unique/ enhanced and we can charge a higher amount for that? Remember, the idea is to make it scalable and justify the additional effort vs. margin benefit.

This would involve creation of such an offering along with a list of prepackaged configuration options. This options list is very essential as the clients may not be able to visualize the output. In our painting example, we could not have visualized the output – nor can I explain it to anyone who has not seen these catalogs/ pictures.

The next step would be creating tools to standardize the output and training a large set of people. This will ensure repeatability of the process and scalability. Many unique services eventually die out as they as not made scalable – they remain the proprietary of their founders and/ or few highly qualified individuals.



This one incident did give me some good insights into a different perspective on competing in a already crowded marketplace. Let us hope this translates into improved margins. And hope it helps you too…