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…
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