# Division By Zero

1. Apr 18, 2005

### eNathan

Can somebody please tell me any case where it is logical to NOT divide by zero? I know division by zero is illogical itself, but usually when you divide by zero the result should be zero anyway.

2. Apr 18, 2005

### cronxeh

Try dividing 1 by 0.000000000000000000000000000001. Big number?

Now try 1/(10^-99999) - is that bigger than 0 or close to infinity?

3. Apr 18, 2005

### eNathan

Excellent explanation! Hmn, I actually though of this the other day. I though "hmn, when you divide by 0 shouldnt the result be infinite?"

I just though of an equation where dividing by zero violates something in 'nature' by common sense, but the result should still be infinite.

Let's say I travel 10 meters at a rate of 0 meters per second. The only way this is posible is it I have an infinite amount of time, right? Why isnt $$\frac {10 m} {0 mps} = infinite$$? (I know there is a latex symbol for infiinite I just dont know it). But thanks again for that 'infinite' explination. So my point is, what is wrong with an infinite number?

Last edited: Apr 18, 2005
4. Apr 18, 2005

### HallsofIvy

Where did you get the idea that "usually when you divide by zero the result should be zero anyway." I can't imagine any situation (except under some condition where you have 0/0 but you don't seem to be talking about that) where dividing by 0 could reasonably be interpreted at resulting in 0.
Some people say, loosely, that dividing by 0 results in infinity- but surely not 0!

5. Apr 18, 2005

### eNathan

hmn, well that is why I started this thread because I was not quite sure about it. Like I said, the other day I was thinking about it and I though that the result should either be infinite or 0.

6. Apr 18, 2005

### matt grime

And here we go again.

At least you want to look at it logically.

So, let's do so. When we divide by x where x is any nonzero number, what we're doing is "logically" multiplying by 1/x, where 1/x is the symbol that satisfies x*(1/x)=(1/x)*x=1.

Now, it is easy to show (you should do so) that in any field, it is logical that 0*x=0 for all x.

Thus, if we were to wish for dividing by zero to be defined logically then we should have

0=0*(1/0)=1

so that'd be a problem.

If we wish to divide by zero we need to pass to a larger system that wouldnt't be a field.

7. Apr 18, 2005

### snoble

Two reasons

First off you got a unique solution to your operation because the nature of the reality (how long does it take to get from here to there) precludes negative numbers. But in general who's to say that $$2/0 \ne -\infty$$ (infinity is \infty in tex by the way). So you have an ambiguity. But you also have an ambiguity with the square root and we work around that right? In fact if you construct numbers from a projective plane (borrow something by Coxeter from your local math library) you are allowed to divide by zero but in that case
infinity is the same as negative infinitity and addition is not defined on infinity. So this is not a situation you want to be dealing with.

Also if you allow x/0 other questions arrise. Like what is x/0 -y/0. What is 0/0. etc. Then you truly do get problems if you allow those to have answers as seen here http://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

(Copied directly from site)
Let a=b.
then $$a^2 =ab$$
$$a^2 +a^2 = ab+ a^2$$
$$2a^2 = ab+ a^2$$
$$2a^2 -2ab= ab+ a^2-2ab$$
and $$2a^2 -2ab= a^2 -ab$$
This can be written as $$2(a^2-ab) = 1(a^2-ab)$$
and cancelling $$a^2-ab$$ from both sides gives 1=2.

So in the end division by 0 is bad because it may lead to greater problems and there is almost no value in allowing division by 0

8. Apr 18, 2005

### eNathan

Hmn I just found a time where zero messes up an equation. We can establish that
if $$z = xy$$ then $$x = \frac {y} {z}$$ and $$y = \frac {x} {z}$$
But this is NOT true when x or y = 0.

9. Apr 18, 2005

thats where you've got an asymptote in the graph.

eg, the graph y = 1/x is undefined when x=0

10. Apr 18, 2005

### eNathan

Oh ye, if $$\frac {x} {0} = \infty$$ then $$x \ne \frac {x} {\frac{1}{0}}$$ But that depends on how you do the math.

11. Apr 18, 2005

Unfortunately you cant just state things like if x/0= becuase that makes no sence. :-)

12. Apr 18, 2005

your statment x/0 = Infinity is not technically correct, sorry I'm not good at latex so you'll have to make do with shoddey normal writing.

Its better to say that x/0 is underfined or things start getting a bit messed up, becasue otherwise it leads onto saying things like

0/0 = 1
and
Inf/Inf = 1

which is not allways right.

Last edited: Apr 18, 2005
13. Apr 18, 2005

### eNathan

Well I was not stating it as fact. But I do understand this concept of $$\frac {x} {0}$$ being underfined.

thanks a lot! :tongue2:

Last edited: Apr 18, 2005
14. Apr 18, 2005

I edited my last post ;-) a bit

15. Apr 18, 2005

### eNathan

One more question. $$111\infty = 999\infty$$

Is this correct?

16. Apr 18, 2005

hmmm, I'd say no, becuase it implies that

$$\frac {111} {999}$$ = $$\frac {\infty} {\infty}$$

I'm learning latex ;-)

$$\frac {\infty} {\infty}$$ can equal all sorts of things, but you really need an equation defined before you can calculate it.

Edit: Like moo said below

Last edited: Apr 18, 2005
17. Apr 18, 2005

### Moo Of Doom

You don't really want to do arithmetic operations with $$\infty$$, but yeah generally it is true that $$a*\infty = b*\infty$$ so long as a and b aren't zero.

18. Apr 18, 2005

### cronxeh

I think we all agreed that division by zero is immoral..

19. Apr 18, 2005

doing algebra with infinities generally turns pretty ugly.

you can say that $$\frac {1} {\infty} = 0$$ but then things get messed up when you start rearranging that formula to stuff like:

$$\infty = \frac {1} {0}$$ then start to try doing divisions with this defanition if infinity like: $$\frac {\infty} {\infty} = \frac {\frac {1} {0}} {\frac {1} {0}}$$

becasue you end up with crazy stuff like $$\frac {\infty} {\infty} = \frac {0} {0}$$ which implies that this is the only answer for $$\frac {\infty} {\infty}$$ which is not technically correct, and also seems to show if you take $$\infty$$ to be some sort of algebraic argument then it gives $$\infty = 0$$ which is clearly wrong.

Think of infinities in this case more as some useful device for working out limits of equations rather than some kind of 'number' you can use in algeba.

20. Apr 18, 2005

If a and b = 0 then you end up with 0=0 which is correct. I would argue that this is the only time that $$a*\infty = b*\infty$$ is correct, becuase here your suggesting that the two infinities are actual numbers, and are somehow different from each other.