13 April 2009

Odds of Eleven and Twelve

Pronounce the following sets of numbers: [10, 11, 12, ...], [20, 21, 22, ...], [30, 31, 32,...], [40, 41, 42, ...], and so on. Notice that the first set [10, 11, 12, ...] does not follow how others sets are pronounced. You might have probably wondered about it before. In my native language Tamil all the sets follow the same patterns of pronunciation. So, I immediately noticed the odds of Eleven and Twelve when I learned English. The reason for this odds lies behind how number theory evolved over the time.

In my early childhood we lived in a small village. Everyday, the milkman delivers milk to my home and mark a line on the house wall. On the fifth day, he crosses the last four lines and starts a new one. This is how he counts the number of days he delivered milk. This is very early form of numbering system. Imagine writing 118 in this system; it would take lot of space.

Roman number system uses the same idea, except it introduces new symbols for 5 = V, 10 = X, 100 = C, etc. For example: CXVIII is 100 + 10 + 5 + 3 = 118. Hence, Roman number system is based additive principle (Also note that instead of writing 4 as IIII, it is usually written as IV, 5-1 = 4). This system requires new symbols at every major step to minimize the size. Any arithmetic calculation like addition, subtraction, multiplication, division, etc. would be a challenging task on Roman number system. It was usually handled by few mathematical experts.

Our current number system is based on positional notation. It was originally developed in India and spread across the world. Another important mathematical milestone was also achieved in India at that time; that is to introduce a symbol for nothing or zero. Introduction of zero is noted as important as invention of wheels in the scientific history. The positional notation can be based on any base great than one (i.e. 2 and above). Our standard current number system uses base 10 – decimal system. This means that we need only 10 symbols to represent any number: 0,1,2,3,4,5,6,7,8,9. And each digit value depends on its position: 1st digit = 1; 2nd digit 10; 3rd digit 100; etc. For example: 4328 = 4*1000 + 3*100 + 2*10 + 8*1. As it is based on base 10, each digit value is increased by multiples of 10. If different base is used, say 12, then it should be increased by multiples of 12 (Note that we also need 12 symbols).

The use of base 10 is based on our 10 fingers. Before the standardization of base 10, people around the globe were tried many different bases, out of which 12 and 20 were also popular. In English and German base 12 was used initially. Though the notation is now changed to decimal system (base 10), we are still using the old words for 11 and 12; and hence the first set [10, 11, 12, ...] sounds odd compare to others.

The positional number system not only simplifies how a number is written, it also simplifies all arithmetical calculation. The ancient art of computation, once confined to a few adepts, is now taught in elementary schools. There are not many instances where scientific progress has so deeply affected and facilitated everyday life!

"Mathematics is the queen of the sciences and the theory of numbers is the queen of mathematics" – Gauss.

Reference: What’s mathematics, by Richard Courant & Herbert Robbins.


Andreas Grabner said...

hi raj
you are right - i also always wondered as a kid where that came from. i remember that i originally tried to pronounce 21 somehow like eleven, 22 like 12 and so on. it sounded odd and i soon figured it out. but it never occured to me that it comes from the times when we used 12 as our base. Is this also related to 12 being a dozen?


RajK said...

Good catch Andi!

10 is divisible by only 2 and 5; where as 12 is divisible by 2,3,4 and 6. This would make base 12 number system easier for multiplication and division. This leads to base 12 and special term like dozen.

I think, the term dozen might have been used in ancient commerce even before positional number system.

ஆ.ஞானசேகரன் said...