Today while arbitrarily browsing through some notes posted by some of the mathematics enthusiasts, I came across the following question:"What is 0/0?"
Well, division by zero is region of great obscurity in mathematics, especially for Mathematics students. Many physics books declare that division by zero yields infinity, while many mathematics books write that something divided by zero is undefined. Students just stow their brains with these terms and notions, without experiencing their true flavors. Well, in this note, I will try to explain the meaning of division by zero through a very simple discussion.
Well, what is division? What to you mean by dividing a number 'a' by a number 'b'? It means simply subtracting 'b' from 'a' repeatedly until you get either a 0, or a positive number which is smaller than 'b'. The number of times you perform this process is your quotient, and the resulting non-negative number is your remainder. Let us divide 15 by 5 using this definition. We see that 15-5=10, and 10-5=5, and 5-5=0,and we stop the process. Clearly, the number of subtractions performed here is 3, which is our quotient here and 0 is our remainder here. Similarly let us divide, say, 16 by 3. I am writing directly: 16-3-3-3-3-3=1, where 0<1<3. Thus here quotient is 5 and remainder is 1. These results exactly tally with the results which we can arrive at, using the well known long division process. It has to be so, as long division is essentially the repeated subtraction in disguise. This can be proved easily using the Euclid's division Lemma (well, I am discussion the same Lemma in a very casual fashion, without going into the mathematical rigor). Now let us try to divide a non zero number, say, 6, by 0. We see, 6-0=6, 6-0=6, 6-0=6..ad infinitum. Clearly, this process will never end and so we will never get a quotient, neither will we get a remainder. This is because, after every subtraction what remains is 6 itself. So, such a non-zero number divided by zero will simply give us no value, and hence is meaningless, or in mathematical terminology, "undefined". However, in case of 0/0, we see, in the first step only,0-0 gives 0. Thus we have to perform subtraction only once to get a zero as our remainder. But does that mean quotient is 1? We see that, if subtract 0 from the 0 obtained as a remainder in the very first step once again, we will again get a 0. So that would mean that the quotient is 2. Using the same logic, one can argue that the quotient is 3, or, 4, or 5,or.....
Well, that's the fundamental difference between non-zero/zero and zero/zero. While the first one gives no value, the second one has no FIXED value. That is why we call the first one UNDEFINED (i.e., meaningless), while the second one is called INDETERMINATE (i.e.,lacking a fixed value). And oh, yes, one more thing, why do physicists declare non-zero/zero as infinity? Simply because what they call 0 in the denominator is actually a "vanishingly small quantity of infinitesimal magnitude". Suppose we divide 6 by such a small quantity. Then at every step, 6 will be decreased by an infinitely small quantity, and after infinitely many such subtractions, 6 will finally reduce to 0. Hence the number of subtractions performed being infinity, non-zero divided by infinitesimal will give us an infinity. Now see how easy everything becomes after an explanation.
"Omne Ignotum Pro Magnifico!"
Word of Caution: Do not write 6/0=undefined, or 0/0=indeterminate. We use the "is equal to" symbol only when both the right and the left hand sides have real values. But "undefined" is not a real value. It is only another way of saying "I don't know what that means". So don't use "=" sign here.