Math Basics Division by Zero

Let’s pause Geometry for a little time and start thinking of some basic rules of the game of Math. Have you ever asked “why is division by zero not allowed in Math?” Try to do 1/2 in a calculator and see what you get!!

This was also a question an immortal Indian math genius, Srinivasa Ramanujan had asked his school teacher when he was a tiny tot. Note the following two arguments against the dangers of division by zero:

(a) Suppose there are 4 apples and two persons want to divide them equally. So, it is 4/2 apples per person, that is, 2 apples per person. But, now consider a scenario in which there are 4 apples and 0 persons. So, how can you divide 4 apples amongst (or by) 0 persons? You can think of any crazy answer and keep on arguing endlessly about it!!!!

(b) The cancellation law ac = bc implies a = b does not work when c = 0. For instance, the identity 1 x 0 = 2 x 0 is true, but if you carelessly divide both sides of the equality by 0 then you will obtain 1 = 2, which is nonsense. In this case, it was obvious that you are dividing by zero; but, in other cases it can be more “hidden”.

Let me show you an example of where it can be “hidden”.

Now, multiply both sides of this equation by c. Then, you get ac=bc. Then, add ab - b^{2} to both the sides to get ac + ab - b^{2}=bc+ab-b^{2}. Hence, you get ab-b^{2}=bc+ab-b^{2}-ac. Factorizing, you obtain


b(a-b)= (b-c)(a-b)


Cancelling the factor (a-b) on both sides of above equation yields


This forces c=0 always but at the start itself we had assumed that c is an arbitrary quantity. Thus, the conclusion b=b-c must hold for any b and any c, zero or non-zero. Taking b=5, c=1, we get the absurd answer 5=4!! Again, we must have gone wrong somewhere. Where?

The nonsensical stuff can be detected in manipulating equation 1 to get equation 2. It is true that the factor

(a-b) is common to both sides. But, while striking it off, you apparently forgot to observe that it is zero! Recall that we started with the assumption a = b so that a - b = 0.

You failed to observe the eleventh Commandment of Moses: Thou shall not divide by zero!!

More later…


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