Why division by zero is undefined

Suppose b and c are both non-zero. Then, in a sense that can be made precise. A similar statement applies to the numerator of a ratio except that it may be zero. There are many ways in which we can choose a and b and let them become smaller. For example, we might pick. But we could just as well pick. Then the ratio is always 2! But we just said it should equal 1! In fact, by letting a be r times as large as b we could get any ratio r we please!

It's a common strategy in teaching to simplify concepts when they are first encountered. The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code.

The reason for this is clearly explained in IEEE and quite thoroughly in this Stackoverflow post :. Division by zero an operation on finite operands gives an exact infinite result, e.

Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division. As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.

Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios. But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza. How can it be proved thoroughly, not just with math, but with an example explanation understandable by children that division by zero is truly undefined?

That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports? So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.

Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times. The answer is clearly no because any number times zero always gives you zero.

We know that any number times zero is zero. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Here's another very simple example for good measure. How much pizza will each person get? Well, you have no people to give the pizzas to. Practically speaking, this is unanswerable. In math-speak, we would say that this is undefined. My understanding of division by zero goes back to the definition of rings.

You can repeat this with other numbers as well, so the children can see that the result is arbitrary. This should make it pretty clear that if we allow division by zero, other laws cannot hold. Learn about vaccinations on campus. You can also find a link to the Nighthawks Together site under our Quicklinks menu. But first, what we need to do is familiarize ourselves with the definition of division. The definition of division states that if "a" divided by "b" equals "c" and "c" is unique, then "b" times "c" equals "a.

So let's say that 6 divided by 2 equals 3. We can all agree with that. Notice, we can say that "c" is unique. We can also figure out what the second part means. If we multiply our "b" times "c" then we should get "a". So our "b" is 2 times "c" which is 3 equals "a" which is our 6. Both of these are satisfied. So that means that 6 divided by 2 does equal 3. And we can also say that this is "defined" because it satisfies the whole definition of division.

Likewise, if it only satisfies one part of the definition, it would mean that it is "undefined. So let me clear this, and let's start with zero divided by 1. I am going to say that this equals zero because 1 times zero equals zero.