Division By Zero

**eNathan** · Apr18-05, 10:11 AM

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.

**cronxeh** · Apr18-05, 10:13 AM

Try dividing 1 by 0.000000000000000000000000000001. Big number?

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

**eNathan** · Apr18-05, 10:21 AM

Originally Posted by cronxeh

Try dividing 1 by 0.000000000000000000000000000001. Big number?

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

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 $LaTeX Code: \\frac {10 m} {0 mps} = infinite$ ? (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?

**HallsofIvy** · Apr18-05, 10:29 AM

Originally Posted by 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.

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!

**eNathan** · Apr18-05, 10:35 AM

Originally Posted by 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!

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.

**matt grime** · Apr18-05, 10:41 AM

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.

**snoble** · Apr18-05, 10:51 AM

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 $LaTeX Code: 2/0 \\ne -\\infty$ (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/...first1eq2.html

(Copied directly from site)
Let a=b.
then

and

This can be written as

and cancelling

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

**eNathan** · Apr18-05, 01:28 PM

Hmn I just found a time where zero messes up an equation. We can establish that
if

then $LaTeX Code: x = \\frac {y} {z}$ and $LaTeX Code: y = \\frac {x} {z}$
But this is NOT true when x or y = 0.

**gregmead** · Apr18-05, 01:31 PM

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

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

**eNathan** · Apr18-05, 01:32 PM

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

**gregmead** · Apr18-05, 01:34 PM

Originally Posted by eNathan

Oh ye, if $LaTeX Code: \\frac {x} {0} = \\infity$ then $LaTeX Code: x != \\frac {x} {\\frac{1}{0}}$

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

**gregmead** · Apr18-05, 01:42 PM

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.

**eNathan** · Apr18-05, 01:45 PM

Originally Posted by gregmead

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

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

thanks a lot!

**gregmead** · Apr18-05, 01:47 PM

I edited my last post ;-) a bit

**eNathan** · Apr18-05, 01:49 PM

One more question. $LaTeX Code: 111\\infty = 999\\infty$

Is this correct?

**gregmead** · Apr18-05, 01:56 PM

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

$LaTeX Code: \\frac {111} {999}$ = $LaTeX Code: \\frac {\\infty} {\\infty}$

I'm learning latex ;-)

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

Edit: Like moo said below