What happens if you divide by zero?

[Ikelepte]

Very good sir. Very good. Also ironically, you have 0 posts. This makes me happy.

Very good you too ;p True I just started.

[darkwarrior42]

(for my post, ^=power)
Yes, so 1^1 = 1
1^0 = 1
1^100=1
1^4912727393038373663 = 1
1^ -439827272990101 = 1
1^Infinity = Infinity? Just because there is no definite number for infinity? well need a better explaination

http://mathforum.org/library/drmath/view/53372.html

[rezoacken]

I'm not a huge fan of brain masturbation but I'm pretty sure 1^infinity (as in 1^x when x goes toward infinity) is equal to 1 since 1^x = 1 and then the limite of 1 is still 1.

[Deleted]

Finally, a use for my graduate education! I will explain WHY it is impossible to divide by zero. We will need the concept of identities and inverses.

First, let's talk about addition. Zero is the unique number with the property that X+0=X, no matter what number X is. Because addition with zero leaves a number 'identically' the same, we call zero the additive identity.

Now, let's think about negative numbers. For any number X, -X is formally defined as the number so that when you add it to X, you get the additive identity back. That is, X + -X = 0. For that reason, we call -X the additive inverse. The additive inverse of a number is the number you need so that when you add them, you get zero.

Whenever we subtract numbers, we are really just adding the additive inverse. Subtraction is just shorthand for that. To a formalist, 3-2 is just a shortcut for 3 + (-2).

Okay, I hope that makes sense. Let's try and do the same thing with multiplication. What number can you multiply X by and get X again, no matter what X is? Clearly, 1. 1 is the multiplicative identity. You multiply by it, and you get identically what you started with.

So what should the multiplicative inverse be? It should be the number so that when you multiply by it, you get the multiplicative identity. So the multiplicative inverse of X is 1/X, because X * (1/x) = 1.

So remember how subtraction is just a shortcut for addition by the additive inverse. Well, you guessed it, division is just a shortcut notation for multiplication by the multiplicative inverse. So to a formalist, dividing by X is just a shortcut for multiplying by 1/X.

Ok, no problems yet. Now let's think about what we are REALLY doing when we divide by zero. Dividing by zero is really multiplying by the multiplicative inverse of zero. Ok, what is the multiplicative inverse of zero? What can you multiply by zero to get the multiplicative identity? What number Y exists so that 0*Y=1?

Well... there isn't one. No matter what you multiply by zero, you get zero back. It doesn't have a multiplicative inverse, and so dividing by zero is non-sensical. IT LITERALLY HAS NO MEANING.

This man knows his shit.

[Kolmagorov]

1^infinity is what we call an indeterminate form.

Formally, because infinity is not a number, 1^infinity doesn't really mean anything under the real number system. But that's a pretty unsatisfying place to stop. Surely we can introduce the concept of infinity in a way that makes sense, right?

What I'm going to do here is a VERY rough outline of the way mathematician defined 'the extended real line'. So for now let's think of 1^infinity as being undefined, and see if it will make sense to assign a particular value to it.

Formally, infinity is the name and symbol we use to describe a sequence of numbers that are unboundedly large, either in the negative or positive direction. Thinking about it this way, let's look at 1^Y, where we allow the number Y to get bigger and bigger and bigger and bigger and bigger...

1^1 = 1
1^10 = 1
1^100 = 1
1^10000000000000 = 1

I see a pattern here. This suggests that maybe we should assign the value 1 to the expression 1^infinity.

Now, even though expressions like 5^infinity are undefined under the real numbers, consider the following sequence

5^1 = 5
5 ^10 = 9765625
5^100 = 7888609052210118054117285652827862296732064351090230047702789306640625

Wow, that's getting out of control fast! So for any X > 1, we should probably assign X^infinity = infinity. And since we can make X get arbitrarly close to 1, maybe we should assign 1^infinity = infinity.

Dammit! We have a contradiction. And until we introduce some new concepts, we can't go any further. We are stuck. 1^infinity can't be assigned a value in a way that is consistent.

Now, let's allow the concept of a limit. Let's let X be a sequence of numbers that get close to 1, and let Y be a sequence of numbers that goes off to infinity.

Then what is the behavior of the sequence X^Y? There are actually 4 possibilities depending on how fast X goes to 1 and how fast Y goes to infinity.

1) X^Y may approach 1
2) X^Y may grow without bound, that is, it goes off to infinity
3) X^Y may settle down to some number in between
4) X^Y fluctuates so much that it never settles down to any value

A famous example is the expression (1+1/X)^X as X goes to infinity. The part inside the parentheses gets close to 1, and the exponent goes to infinity. This expression actually converges to the value of e, the base of the natural logarithm. This is a case of possibility 3) happening.

So, to conclude, this is why 1^infinity is an indeterminate form. It can't be assigned a value without a contradiction, and even if we allow limits, the behavior of the sequences determines what number it settles down to or even if it settles down at all.

[Darkdruidelf]

0, -0, infinity, -infinity.

AKA. You explode the universe.

Edit: Just to clarify here, you cannot divide by 0 because 0 is not a number. 0 is a representation of nothing, I actually had this convo with my maths teacher once :P