Mathematically why...?

My teacher says you can't divide a number by zero. Why?

Let's look at some examples of dividing other numbers.

10/2 = 5 This means that if you had ten blocks, you could
separate them into five groups of two.

9/3 = 3 This means that if you had nine blocks, you could
separate them into three groups of three.

5/1 = 5 Five blocks could be separated into five groups
of one.

5/0 = ? Into how many groups of zero could you separate
five blocks?

It doesn't matter how many groups of zero you have, because they would never add up to five since 0+0+0+0+0+0 = 0. You could even have one million groups of zero blocks, and they would still add up to zero. So, it doesn't make sense to divide by zero since there is not a good answer.

If you know a little bit about multiplication, you could look at it this way:

10/2 = 5 This means that 5 x 2 = 10

9/3 = 3 This means that 3 x 3 = 9

5/1 = 5 This means that 5 x 1 = 5

5/0 = ? This would mean that the answer x 0 = 5, but
anything times 0 is always zero.

So there isn't an answer.

- Dr. Margaret

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For one thing, when you divide one number by another, you expect the result to be another number. Look at the sequence of numbers 1/(1/2), 1/(1/3), 1/(1/4), ... . Notice that the bottoms of the fractions are 1/2, 1/3, 1/4, ..., and that they're going to zero. If there's a limit to this sequence, we would take that number and call it 1/0, so let's see if there is.

Well, the sequence turns out to be 2, 3, 4, ..., and that goes to infinity. Since infinity isn't a real number, we don't assign any value to 1/0. We just say it's undefined.

But let's say we did assign a value. Let's say that infinity is a real number, and 1/0 is infinity. Then look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ..., and notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero. So again, we would want the limit of this sequence to be 1/0. But looking at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative infinity. So which would we assign to 1/0? Negative infinity or positive infinity? Instead of just assigning one willy nilly, we say that infinity isn't a number, and that 1/0 is undefined.

- Dr. Ken

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The reason is related to the associated multiplication question. If you divide 6 by 3 the answer is 2 because 2 times 3 IS 6. If you divide 6 by zero, then you are asking the question, "What number times zero gives 6?" The answer to that one, of course, is no number, for we know that zero times any real number is zero not 6. So we say that division by zero is undefined, for it is not consistent with division by other numbers.

- Dr. Robert

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Because there's just no sensible way to define it.

For example, we could say that 1/0 = 5. But there's a rule in arithmetic that a(b/a) = b, and if 1/0 = 5, 0(1/0) = 0*5 = 0 doesn't work, so you could never use the rule. If you changed every rule to specifically say that it doesn't work for zero in the denominator, what's the point of making 1/0 = 5 in the first place? You can't use any rules on it.

But maybe you're thinking of saying that 1/0 = infinity. Well then, what's "infinity"? How does it work in all the other equations?

Does infinity - infinity = 0?
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c) will not always work:

1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads to nonsense, so to avoid all the trouble we just say that it doesn't make sense to divide by zero.

What happens if you add apples to oranges? It just doesn't make sense, so the easiest thing is just to say that it doesn't make sense, or, as a mathematician would say, "it is undefined."

Maybe that's the best way to look at it. When, in mathematics, you see a statement like "operation XYZ is undefined", you should translate it in your head to "operation XYZ doesn't make sense."

- Dr. Tom

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tl;dr: You're a fuckwit walrus fucker.