Dividing by zero can’t be that bad, can it? Let’s just try… *KABOOM!*
Naw, just kidding. It’s supposed to be impossible, but’s lets see what happens if you suppose one simple rule:
1÷0 = ∞
Okay, steady as she goes. How about numbers other than 1?
x÷0 = x * (1÷0) = x * ∞
Still works, although a little strangely (more than one kind of infinity!?).
Now, here comes the weird part:
1÷0 = ∞ ; 1÷0*0 = ∞*0 ; ∞*0 = 1
0∞ is one? Okay, now that’s flat out weird. Zero times anything should be zero, right? Actually, not necessarily. Think about this:
Imagine you have a perfectly flat plane (physically impossible, but whatever). This plane, by definition, is of zero depth. This means that if you stacked another one on top of it, you’d still have zero depth. No matter how many you stack (how many multiples of zero you have), the depth is still zero. In order to achieve any depth whatsoever, you’d need an infinite number! If you agree with this idea (which you might not), then you agree that ∞ * 0 does have a value greater than zero. Actually, it has a value of exactly one.
If you’ve somehow managed to follow my insane mathematical ramblings so far, then it won’t come as a surprise to you that:
1÷0 = ∞ ; 1 = 0*∞ ; 1÷∞ = 0
In fact, that is the most sensible part of this whole post. You’re splitting one up into an infinite number of parts, so each part would be of no size at all. Makes sense.
Now to address the problems that normally go along with division by zero. Here’s the classic:
0 * 1 = 0 * 2
(0 * 1) ÷ 0 = (0 * 2) ÷ 0
1 = 2
Obviously one doesn’t equal two, so something’s wrong here.
My solution to this problem, and any more complex ones that follow the same principle, is this: In order for zero and infinity to be valid where all other numbers would be in an equation (even in division), they must be treated similarly to the imaginary unit i or -1 - they cannot be simplified with other numbers while multiplied.
Although most numbers can be multiplied together and simplified, like so:
2 * 2 = 4 (you don’t say?)
There are a choice few who must remain how they are:
97 * i
In this expression, i and 97 cannot be reconciled. i * 97 is just i * 97, and nothing more. I must stand on its own like a variable, even though it is a constant.
I am proposing that to be able to give zero the inalienable rights that all numbers deserve (are you seeing a possible movement here: the “Numerical Rights Movement?”), you must treat it similarly. x * 0 is no longer 0, it is just x * 0. Therefore:
0 * 1 ≠ 0 * 2
problem solved! This prevents the potential for information loss that always bothered me in algebra. This rule extends to division by infinity:
x ÷ ∞ = x * (1 ÷ ∞) = x * 0
so you can’t say
1÷∞ = 2÷∞ ; (x÷∞) * ∞ = (2÷∞) * ∞ ; 1 = 2
Because the first statement is equivalent to:
1 * 0 = 2 * 0
Which is not okay.
So far, division by zero and infinity both don’t seem that hard. You can you use them just like any other numbers, and everything seems to work fine. There is, however, one issue. Consider simplifying these fractions:
(1 ÷ 0) * (1 ÷ 0)
It seems easy enough, but there are in fact two ways to do it, both of which seem equally valid, but which give you different results. The first way is to treat both fractions as infinity:
(1 ÷ 0) * (1 ÷ 0) = ∞ * ∞ = ∞^2
Fine, but you can also multiply the fractions, then divide!
(1 ÷ 0) * (1 ÷ 0) = 1 ÷ 0 = ∞
∞ = ∞^2? That doesn’t seem right… Either exponentation of infinity does nothing, or there are restrictions placed on infinity.
Don’t worry - this isn’t necessarily a death-blow to the idea. Imaginary numbers have similar problems with expressions like √-1 * √-1:
√-1 * √-1 = √(-1 * -1) = √1 = 1
√-1 * √-1 = i * i = i^2 = -1
-1 most certainly does not equal 1, but that doesn’t mean imaginary numbers aren’t incredibly useful. Using them just imposes certain restrictions - it shows that, under extreme conditions, proceses that seem equivalent actually have subtle differences.
There’s one last issue, then you can begin reading sane posts again. What is (-1)^∞? Is it positive? negative? imaginary? complex? Is ∞ even, or odd, or even an integer? (-1)^∞ is like to trying to predict what number a gameshow wheel will land on if you spin it for eternity then stop it.
For now, let’s just say that it’s 3, because that seems harmless enough. :)