Understanding Division by Zero
Recently, a friend shared a piece of literature explaining why one must not divide by zero. I am usually wary of anyone who forbids me from doing something in the abstract realm. In the physical realm — don’t bathe in kerosene & walk into a flame or do not touch a live wire or do not try to stop a moving train — all make sense. But Mathematics being the study of patterns in the countable world makes me suspicious of anyone who tells me that I can’t subscribe to a pattern. Here is the image that was shared:
That a piece of technology (a calculator) doesn’t have the capacity, is not good enough reason to my mind. That “you risk getting answers that make no sense” is also a circuitous way of explaining something (actually, it doesn’t).
A simpler way I am able to reconcile challenges in dividing by zero is to let the student attempt it & see why it fails to provide any useful result.
I’ll take the simple example of 6 ÷ 3. BTW, this symbol, ÷, is called the obelus.
What does it mean to divide? To divide is to repeatedly reduce the dividend by the divisor till there isn’t sufficient quanta of the dividend left to perform the task (of reducing it by the amount of the divisor). The quotient is the number of repetitions & the remainder is portion of the dividend that remains when the repetitions halt.
6 ÷ 3, then, is reducing 6 by 3 repeatedly. When I reduce 6 by 3, I am left with 3. When I reduce 3 by 3, I am left with 0. Since I cannot reduce 0 by 3, I stop. The number of times I performed this task is 2 times. I am left with no portion of the dividend.
0 ÷ 3 is reducing 0 by 3 repeatedly till there isn’t sufficient quanta of the dividend left to perform the task. Since I cannot reduce 0 by 3, I stop. Since I didn’t complete performing the task even once, the quotient is 0. Since I am left with no portion of the dividend the remainder is 0.
6 ÷ 0 is reducing 6 by 0 repeatedly till there isn’t sufficient quanta of the dividend left to perform the task. I reduce 6 by 0 once. I am left with 6. I reduce it once more. I am left with 6. I can keep doing this ad infinitum and I will always remain with 6. Since I haven’t stopped reducing 6 by 0, I cannot claim to have found either quotient or remainder.
As shown above, the “rules of arithmetic” DO NOT SAY “you are not allowed to divide by zero” (quote from image above) but the rules of arithmetic show that you can do it & lets you do it, letting you find for yourself that it is an indeterminate/absurd/nonsensical operation. By leaning on first principles, one can let the student find out for her~/himself about the futility of that operation.
