I asked three people a question that has baffled mathematicians throughout the ages. What is Zero divided by Zero? Sam said zero. No, you are wrong, said Bill, it is 1. Amanda said “both of you are dead wrong. You are weak in math. The correct answer is infinity.” Who is correct? They all seem (on the surface) to be mathematically sound. Teachers pose this question to your math students. A lively discussion should follow.
I allowed each one to defend the answer they gave. It was interesting, entertaining, and most of all informative. This discussion clearly indicated that dividing zero by zero was still a problem.
Sam: Isn’t zero a number like all the other numbers? If that is the case then 0/0 = 0, because 0/4 = 0, 0/3 = 0, 0/2 = 0, 0/1 = 0, then 0/0 = 0. QED. (Note that some mathematicians claim that zero is not a natural number and the rules of natural numbers should not be forced upon it.)
He felt quite smug as he was so sure that he had enlightened his colleagues. But Bill would have none of it.
Bill: I agree that zero is a number like all the others. For this reason I claim that 0/0 = 1. Since 4/4 = 1, 3/3 = 1, 2/2 = 1, 1/1 = 1, then 0/0 = 1. QED.
To Bill’s mind this was quite logical. Amanda was unconvinced.
Amanda: Neither of you is correct. I say that 0/0 = ∞ and I will prove it. I too agree that zero is number like the others. You are both aware that we are not allowed to divide a non-zero number by zero, but we could divide it by smaller and smaller numbers, both positive and negative, to see what happens as they approach zero. (Is this what Bishop George Berkeley (1734) called “Ghosts of Departed Quantities”?)
For example, 2/1 = 1, 2/0.1 = 20, 2/0.01 = 200, 2/0.001 =2,000, 2/0.0001 = 20,000, ……., 2/0 = ∞. QED.
The problem confronting the three colleagues is not new. Brahmagupta had this problem in 628, he said that 0/0 = 0. Bhaskara II, 500 years later, claimed that 0/0 = ∞.
Who is correct? Each one seem to be valid. Further, It can also be proven that 0/0 could equal any number. As a result 0/0 is said to be indeterminate.
By now you must have realized that Sam, Bill, and Amanda are really me, myself, and I. I do hope that you will forgive my ramblings because,