Probably one of the coolest things about teaching is when a student asks a really good question that lets you detour from your original plan to talk about something really super-awesome.
That happened in my class today.
We were discussing slope and going through a few example problems with the slope formula. I decided to show them one example that resulted in an undefined slope. I gave them the points (7, 3) and (7, 10), then we worked through the problem. We got to a point where we had 7/0 on the board and I asked the students what that meant. The consensus was that the slope was undefined because "we can't divide by zero."
Then, one of my students asked: "Mr. Brenneman, why can't we divide by zero?"
I stopped. I looked at him. I said, "I love that question! Let's put aside what we're doing and talk about this!"
I then launched into a brief explanation of proof by contradiction and asked them to put aside the laws of mathematics for one second. "Let's suppose that you can divide by zero," I said. "Let's consider what 0/0 would be equal to. What do you think?"
Many students chimed in with "0." Others chimed in with "1." I asked each side to back up their reasoning.
"Well, it would be zero because you're dividing zero by another number," one student said.
"I think it would be one, because 2/2 is 1, 4/4 is 1, so 0/0 would be 1," said another.
A few minds were blown when I told them they were both right.
Assuming we can divide by zero, the quotient of 0/0 yields two distinct yet equally valid results.
Suppose we choose a number a from all of the numbers in existence. We say that 0/a = 0 (the zero property of division) and a/a = 1 (a form of the multiplicative inverse property).
In this scenario, division by zero is allowable. (This is an important distinction, because normally the two properties I mentioned above specify that a must be nonzero.) So, 0/0 = 0 by the zero property. But, 0/0 = 1 by the multiplicative inverse property.
Thus, it is reasonable to conclude that 0/0 = 0 and 0/0 = 1.
In other words, 0 = 1.
The discussion can certainly stop here, because we have arrived at a conclusion that is mathematically absurd. Furthermore, this absurdity stems from the initial assumption that we can divide by zero; hence, we must conclude that we cannot divide by zero.
But I knew that ending our discussion at 0 = 1 wouldn't have been nearly quite as fun as proceeding with even more absurdity.
So, I asked the students, "what would 1 + 1 be equal to?"
Many said 2. Some said 1. They were all correct. I showed them why.
1 + 1 certainly equals 2. But, we've already established that 1 = 0, so we can also say that 1 + 1 = 1 + 0 = 1. Or, 1 + 1 = 0 + 0 = 0.
In other words, 0 = 1 = 2.
I extended it one more time by asking the students what 1 + 1 + 1 would equal. Some said 3, some said 2, some said 1. Again, they were all correct. Using similar reasoning as the "1 + 1" case, we concluded that 0 = 1 = 2 = 3.
At that point, the students came to realize that if we kept going, eventually we would conclude that all numbers would be equal to each other.
I told the students one of my favorite mathematically absurd things to say: "If Congress legalized division by zero, we could solve all of our economic problems. We wouldn't have a $15 trillion debt, because if we can divide by zero then 15 trillion would be equal to zero. We wouldn't owe anyone $15 trillion. Problem solved!"
My students seemed to love it. Sometimes it's fun to drop what we're doing and discuss something far more interesting when the opportunity arises.