0 * 1 = 0 * 2. Divide by zero, and you’ve got 1 = 2. See where that could lead to problems?
The whole point of division is uniqueness. Fix y and z, and there’s only one number x such that xy = z. If y = 0 and z = 2, what’s x?
The reason why we don’t generally say that it’s 0 is because that violates the division theorem. For any numbers a and b, there are numbers q and r such that a = bq + r, where 0 < r < b. You’re saying that q should be 0 if b is; in that case, what would you choose for r?
There are some mathematicians who do define it that way, for technical reasons, but it’s not a popular view.
Well, take a piece of graph paper (or Gnumeric or even excel, if you must) and plot out X vs 1/X as X starts getting pretty small. You’ll quickly see that 1/0 blows off to infinity, which is undefined…
e.g. if X is one, 1/X is 1
if X is 0.5, 1/X is 2
if X is 0.0000000001, then 1/X is 10000000000
and so on…
I can’t answer for the earlier ones, but processor architecture since at least the early '70s would catch a divide by zero as an underflow error (think of ‘not enough data’), and quit the program unless told to do something else.
Also, 1/0 = infinity is just one possible answer; as others have explained, finite numbers are also possible. Infinity happens to be the easiest concept for lay people to grasp when this comes up. In electrical engineering, there is the impulse function (Dirac delta for you mathmagicians) which has a finite area, but occurs in 0 elapsed time.
If you think about it, you’ll realise that there are actually a whole lot of things that are undefined in mathematics. If it is important for us to have a solution then we make one up. If not then we leave it undefined.
For example, “naturally” (arguably) only the numbers 1, 2, 3,… exist. But we want to be able to subtract one of these numbers from another. 2 - 1 is fine but 1 - 2 is not. So we “invent” negative numbers.
Similarly 1 / 2 makes no sense so we have to “invent” fractions.
And so it goes on. The square root of negative four makes no sense unless we invent complex numbers.
So it’s not an unusual situation for the answer to a mathematical question to be undefined. It’s just that for division by zero there is no useful single mapping that we should define.
You’d never terminate, because you’d never put an item in a box. In fact, you could use an infinite number of boxes, and still never put 50 items away at 0 items per box. Hence, there is no number of boxes that can hold 50 items put away at 0 items per box.
As kabbes has correctly pointed out, there are lots of undefined expressions in mathematics. And he further pointed out the crux of why 1/0 isn’t defined:
viking has shown why 1/0 = infinity, but it can be -infinity just as easily: if we take a sequence 1/X, where X approaches zero from below (X= -1, -0.1, -0.01, etc.) then 1/X = -1, -10, -100, etc., so by a parallel logic to viking’s, 1/0 should be minus-infinity.
There are two things one can do with a mathematical expression that has multiple ‘natural’ values. One is to choose one value and declare it to be the ‘principal’ value (as we do with square roots of positive numbers, e.g. the square root of 4 is 2, but -2 is also a square root of 4) so as to create a well-defined function or expression.
The other thing we can do is let the expression be undefined, which is what we do with 1/0, or with c/0 generally, where c is a nonzero real constant.
Why do we go the principal-value route sometimes, and leave things undefined other times? I don’t know. My guess is that it has to do with which way makes more sense for mathematicians in their work. We need a well-defined nth-root function. But we don’t need 1/0 to take on a particular value, and it doesn’t do us any particular good for it to do so, except to possibly confuse students when we begin to discuss infinities.
First, I would like to strongly echo what RTFirefly, division by zero cannot be equated with either positive or negative infinity.
As to the last part of the OP, there are absolutely, postively an incredible number of mathematical operations that, in certain circumstances, do not have a solution. The log of zero, the arcsin of 2, etc. You feed such values into a computer math library and it will either throw an error or GIGO. Ditto complicated or multivariable equations such as solving for x,y: x+y =2, 2x+2y=3.
Division by zero is only special since it is the only case of the “big four” operations that is not defined for the entire set of reals or integers.
Also, in the realm of computers, bit limitations also affect all basic arithmetic operations. E.g., adding 2 billion and 2 billion doesn’t give you 4 billion when adding signed 32 bit numbers. (It usually gives a large negative number, depending on the hardware.)