Thursday, September 23, 2010

Why anything raise to the power 0 is 1??

Everyone is surrounded by numbers or to be specific MATHS!! Pragmatically everybody has to apply mathematics concepts in one way or other. Even a small child starts his cognition by starting counting. It is so important in human world that without it one cannot think of being able to live.
But wait a minute, if it is indispensable and so natural, is it really natural? Confused?? Mathematics is not at all natural in the sense that it is purely man made and cognitive or to be more specific thought process provoked ; not at all science!! I mean what is the proof that 1+1 always equals to 2 and why not 3 or any other number.

Mathematics is the way we define it. It must have started with addition and then its inverse subtraction, multiplication and then division; in the same terms we can talk of more sophisticated stuffs like differentiation and integration. During the discussion in the class I realized the importance of cognition and thought process while developing something that should be proved by Abstraction and Generalization; there has to be a balance between the two otherwise this will result in either too many Definitions or less of distinction between the entities. The example of triangle turning into a circle was a thought provoking and effective enough to prove the concept of generalization and abstraction.

From the childhood I was astonished by some of the facts and rules of the mathematics language like infinity concept and multiplying some number by itself infinite times. The same thing applies to number zero 0. When you multiply this with anything then it gives you nothing; when you divide something by this no you get an unknown number. There are many instances where this zero doesn't follow the basic mathematics rules for instance:-

0 x 1 = 0 ….A
0 x 3 = 0 ….B

Equating A and B we get,
0 x 1 = 0 x 3 ….C

Now, dividing C on both sides by zero we get,
0/0 x 1 = 0/0 x 3

On simplifying,
1 = 3 ….D ( How can that be possible)

Yes D is not possible but there are no flaws in our basic mathematics…. Till the time we consider zero as a special no and say that "We cannot divide the equations by a special no called zero". So is it that the no zero is somewhat special? Or we are abstracting the things right so that we can work upon with the basic mathematics….??
One may give division problem by zero an excuse but what about raising anything to the power zero?? In my sense, if you call it a function then the curve is a straight line parallel to y-axis passing through x=1; I mean why do we need such a type of function-"anything raise to power zero." I started off from the concept of multiplication i.e. where the concept of powers started.

Let us assume any arbitrary number y then, y x y = y2. This is in accordance to the definition of symbol where any number of times that number is multiplied to itself is represented as power(raised entity; in our case 2). Now if we say y0 at this very moment then it is ridiculous because one cannot define any quantity being multiplied to itself by zero number of times. So let us wait and see. Now if you multiply it again by itself then it becomes y3 i.e. the powers are additive when one multiplies a number by itself.

Now if this is true then the inverse should also hold true i.e. if you divide the no by itself then the powers should be subtracted. Let's do it if you divide y3 by y then y x y x y / y gives you y x y which is y2 which is true so the rule now becomes if you multiply a number by itself then the powers are added whereas if you divide the no by itself then the powers are subtracted. This looks good, but can we construct y0? Yes indeed we can but what does it mean now?

If one is able to subtract the same power then surely 0 will come as a power; this means that y0 can be defined as y x y/ y x y ie y2-2 = y0. So inversely, y0 is nothing but the number divided by itself.

This turns out to be one so the rule anything raise to zero makes one holds true for any arbitrary number y. This is not a fact but in order to keep the previous rule (about the addition and subtraction of powers) intact we created another rule that anything raise to power zero is one. So the rule is not necessity but a side product of basic mathematics.

Patrick Suppes 1957 (1999 Dover edition), Introduction to Logic, Dover Publications,

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