1. ## Division by 0

I was thinking about why you can't divide anything by 0 (except 0). I decided that the answer would be infinity, since no sum of 0's could equal any number. But then still, I don't think infinity would be correct because NO amount would EVER equal anything.
Take 5/0 for example. The answer is infinity. But still an infinity of 0's would never give you 5. Is there actually a real answer?

2. 'Infinity' in this context would be like this: imagine you have a really mega ultra small number. Let's say .000000000000000000000000000000000000(...)00000001 , in that situation that number would be so insanely small that it would act as zero and be recognized as 0. This is true for a lot of small numbers, providing the number is small enough it is no longer represented in it's small form, it's represented as 0, so if you were to use 0 it could be an infinate number of small numbers. From one one-millionth of a number to one-quadzillionth of a number, ad infinitum. In that context the answer would be in fact, infinity. Because you could fit so many of a number like this into 5:

.0000000(put like a bazillion more zeros here)0000001 (remember, this would be recognized as 0 so it'd be 5/0 instead of 5/...)

Instead of the divison resulting in something like 49857349587349583749538745934753948752948753948573 498573453q4534258398230492830492850398453098653096 84305968405698450698405694850649856049, it would result in infinity because the number is so insanely huge.

3. You can't divide by zero because its undefined. Meaning we haven't found an answer that meets all of the conditions assocated with division and multiplication.

Lets say you take 3 / 0 = x. That would mean that x * 0 = 3

Give me a number that you can multiple by 0 and get 3.

But still an infinity of 0's would never give you 5.
I think you need to go look up infinity.

Edit:
Take 5/0 for example. The answer is infinity.
Uh no, the answer is undefined

lim x-> 0 (5 / x)

then it'd be infinity. But all that means that as you approach very very close to 0 your answer becomes unbounded

4. But why is it infinity if you actually divide by "0"? Can't they just say it's not infinity, it's not 7, it isn't even pi to the ten trillionth power divided by the cube root of 2... it's just impossible?

5. Gonna have to correct myself:

lim x->0 (5/x) doesn't actually exist. Heres why:
if you approach from the negetive side the answer is -∞ but if you approach from the positive side it is +∞.

6. >> But why is it infinity if you actually divide by "0"?

Because it isn't. It's just impossible. 0 can never quanitify itself to divide into a number, and thus the answer is undefined.

7. I asked the question during math a few weeks ago, and they said it was infinity.
Also, why is 0!=1? Wouldn't it be 0?

8. I like the explanation:

Code:
```a / 0 = b
a = b * 0
a = 0```
Which is false clearly. If you try to quantify sorts of things like infinity and non-real numbers you start to get mathematical proofs for things like 2 = 1, because add/sub/div/mul operands are really only "built" to work with real number quantities. I guess you could say it's infinity, but really it's all just an infeasible non-real concept.

>> why is 0!=1?

Why is zero not equal to one?!

9. Quote << ">> why is 0!=1?
Why is zero not equal to one?!"

No:
Why is: 0! equal to 1?
5! = 5x4x3x2x1 = 120
3! = 3x2x1 = 3
0! = 0x0 = 0

0! should be 0, not 1.

10. lim x->0 (5/x) doesn't actually exist. Heres why:
Look at Lim x->0 5 / abs(x)

Also, what is the integral from -5 to 5 1/x ... is it undefined, or should it be zero? For every amount it goes 'up' towards 'positive infinity' doesn't it go down an equal amount 'down' towards 'negative infinity' ?

You can also look at the integral of tan over a period spanning an equal distance from PI/4 radians (45 degrees) the actual accepted form of this is, I believe, ln(secant) where secant is 1/cos, ln is natural log.

Another thing, floating point units will happily perform a divide by zero. Reason being, if you graph the numbers a floating point can represent there's actually a 'hole' for really small numbers (something ^ -126 power), so it can't tell if zero is really zero or if it fits in this 'hole'.

Another thing you might like:

1.9 with a vinculum (repeating 9 to infinite number of decimal places) is equal to 2

lim(m --> infinity) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1

Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

http://www.blizzard.com/press/040401.shtml

11. 0! is defined to be 1.

When something is defined to be a certain way it just is. Its not right or wrong, true or false. It is what we use to determine if something else is true or false.

division by zero is undefined meaning it hasn't been given a definition that would let us compute the answer of a number divided by 0.

12. Look at Lim x->0 5 / abs(x)
Um ok. It still doesn't mean the limit for 5/0 exists.

edit: And the 0.999~ = 1 thing is so old and has been proven more ways then I care to count.

13. 0! is defined to be 1.

When something is defined to be a certain way it just is. Its not right or wrong, true or false. It is what we use to determine if something else is true or false.
This has all sorts of intriguing implications. People don't seem to realize the extent to which math is horribly limited. Math is a description of the reality we perceive. The idea that any mathematical construct is somehow 'right' in the absolute sense is, almost in a definitional manner, a flawed idea. Alter the brains of humans and you will see the rules and 'definition' of math change accordingly.

1 + 1 = 17 (base 10) in my perception of reality. Anybody who argues with me is a god damn moron.

Um ok. It still doesn't mean the limit for 5/0 exists.

edit: And the 0.999~ = 1 thing is so old and has been proven more ways then I care to count.
It cannot possibly be older than the concept that lim x->0 5/0 is undefined...what is your point? Maybe we should just stop talking about math altogether because there must be somebody somewhere that's already familiar with what we're talking about, eh?

14. Do you just enjoy trolling Bob? Or do you actually have a point?

15. He means zero factorial. [Edit: with the time I took to write my reply, the previous sentence is redundant, yes.] There are a few reasons. One is that it allows us to represent "n choose k" as (n! / (k! (n - k)!)). But another is that it is consistent with the relationship n! = n * (n - 1)!:

Code:
```n! = n * (n - 1)!
1! = 1 * (1 - 1)!
1 = 1 * 0!
1 = 0!```
(-1)! is not defined, because this relationship can be extended no further. Also, 0! = 1 is the value that works with the gamma function.

We could look at n! as the number of ways to order n distinct elements, and then 0! = 1 is consistent with that definition.

One way of explaining why n / 0 is undefined is simply that if you define the real numbers as "an ordered field with the least upper bound property," this comes inherently with the definition of a field. Infinity is not a real number.