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Question on decimal number system

Let me check how many of our mathematics enthusiasts here have given a thought to this question. We know that our decimal number system consists of the ten digits, viz., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We start our counting with ZERO(0), then we count as following:

ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE..

Now the next number is TEN(10). The very peculiar thing about this number is that in our decimal system (i.e., common man's number system) we write this number as a 1 followed by a 0. Then comes ELEVEN(11), written as a 1 followed by a 1, then TWELVE(12) written as a 1 followed by a 2, and so on. Now my question is,why is this so? Instead of writing TEN as 10, we could have written it as 1 followed by two 0's, i.e., as '100', or 0 followed by 1, i.e., as '01' or in any other way. Same argument applies in the case of ELEVEN, TWELVE, and so on. My question may seem to be an idiotic one or a meaningless one, but if you have a mathematical insight, then you will realize that it is indeed a very intriguing one. Try to account for this peculiar way in which our ancestors had designed a number system which has framed the structure of the whole of modern science and technology.

Note by Kuldeep Guha Mazumder
2 years, 5 months ago

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There are quite a few different number systems, but the most common ones used in mathematics are integer base positional number systems. This is jargon for saying that all numbers are expressed as follows, given integer base \(n\) and digits \({ a }_{ i }\)

\(\displaystyle \sum _{ i=1 }^{ \infty }{ { a }_{ i }{ n }^{ i-1 } }\)

One useful property this has, for integer bases, is that for each number, there is an unique representation written in this form. This isn't necessarily the case for non-integer bases or other positional numbering systems.

Having said that, our decimal number system is base \(10\), and uses digits \(0,1,2,3,4,5,6,7,8,9\). Note that for any integer base, the number of the digits is equal to the integer base, and thus the base number itself cannot be represented by any of them. If \("[10]"\) is a digit itself, then we are talking about base \(11\), and we end up with the sequence

\(0,1,2,3,4,5,6,7,8,9,[10],10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1[10], 20, 21,...\)

as awkward as that is, which is why nobody uses that.

Michael Mendrin - 2 years, 5 months ago

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I have to teach a few of my students now. After I become free, I will post my explanation.

Kuldeep Guha Mazumder - 2 years, 5 months ago

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Yes yes..well, that is why the decimal number system find its place above the non-positional number systems like the Roman one, in the list of convenience of use. But I have a very nice way of explaining the concept to young minds..

Kuldeep Guha Mazumder - 2 years, 5 months ago

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