division by zero: is it really undefined?!

Why shouldn''t there be a square root of negative one?

It''s generally treated as infinity, but in some cases, it will be something else. It isn''t really undefined. It''s just undefined in computer terms because there is no way to represent infinity (yet?).

In java it just wrights the word "infinity" which sometimes has odd results...

EDIT: At least I think it does, not entirely sure.

-~-The Cow of Darkness-~-

[edited by - cowsarenotevil on April 22, 2003 11:17:57 PM]

Because you could have 0/0, which could be any real (or, i suppose, imaginary) value, depending on what function you''re trying to evaluate. Best way to define divide by zero is a limit:

f(x) = a(x)/b(x);

a(p) = b(p) = 0; for some p. So what is f(p)?
Define f(p) = lim (x->p) f(x)

This might also be unbounded in either direction.

Tom

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Original post by Cold_Steel
It''s generally treated as infinity, but in some cases, it will be something else. It isn''t really undefined. It''s just undefined in computer terms because there is no way to represent infinity (yet?).

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Just a comment here. The only math processors I''m really familiar with are the MC6888x family. They have an infinity flag that can be raised under some circumstanses, but division by zero was yet undefined/illegal.

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if you just want to work with square roots of negative numbers, do it with complex numbers, not with reals... reals are for newbies

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Original post by CWizard
Just a comment here. The only math processors I''m really familiar with are the MC6888x family. They have an infinity flag that can be raised under some circumstanses, but division by zero was yet undefined/illegal.

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That''s true, I forgot about that. I was just thinking that there was no way to represent infinity in an integer format or something like that. IEEE floats do have an overflow or something if I recall correctly, which can be treated as infinity.

Like I said, something divided by zero can have a bunch of different meanings. You guys are right in saying that it is undefined in some situations, but not in all.


Off topic- Hmmm, if we live in a closed universe, would infinity and negative infinity possibly be the same anyway? Food for thought.

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1 / 0 is greater than infinity... because no matter how many times you multiply by 0 its always 0.

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There is nothing greater than infinity by definition.

Also, is 1 / 0 negative infinity, because no matter how many negative times you multiply by 0, it''s always 0. Or is it a complex infinity...


Unfortunately for me:

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Access Denied to IP Address 203.109.249.138

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Thank you for your interest in Eric Weisstein''s World of Mathematics. Unfortunately, your client, subnet, proxy, or cache server has been identified as source of an excessive number of hits which appear to have originated from a robot or spider.

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That IP address is the IP address of the proxy my ISP has. Some loser just ruined one of my favourite sites for me

Trying is the first step towards failure.

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Original post by Cold_Steel
That''s true, I forgot about that. I was just thinking that there was no way to represent infinity in an integer format or something like that. IEEE floats do have an overflow or something if I recall correctly, which can be treated as infinity.

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IEEE floats have positive and negative infinity. They also have denormalized numbers(underflow), NaN(Not a Number - 0/0), and other fun things.

1/0 is undefined, however the limit of 1/x as it approches 0 from either side is defined. There is a difference between the limit being defined and the number being defined, as most calculus books will tell you...
Jesse

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Original post by laeuchli
1/0 is undefined

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Indeed.

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however the limit of 1/x as it approches 0 from either side is defined.

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If it''s defined, what is it??? lim (x->0) |1/x| is defined to be infinity, but 1/x doesn''t have a limit at x=0.

Cédric

Infinity isn''t a number and applying operators to it is rubbish, infinity is more of a concept. There are different ''orders'' of infinity. It''s a useful concept, but trying to compare it with anything other than itself and ways of obtaining it is a bit of a waste of time. As for 1/0, it''s rubbish that''s all. If your in a situation where you''re trying to divide by 0, you shouldn''t be. It certainly isn''t defined, but it may not necessarily be undefined as it doesn''t exist.

eg, for f(x) = 1 / (1-x)

f(1) is undefined, not really because 1/0 is undefined, but because our definition doesn''t cover f(1).

infact that last definition is wrong, it should be:

f(x) = 1 / (1-x) x is not equal to 1



All this is a matter of definitions and different viewpoints. The one this we should all agree on is that 1/0 is not defined, for reasons argued by other people.

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Original post by ragonastick
There is nothing greater than infinity by definition.

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There are different orders of infinity which are not 'equal'. I've seen it proven in some of my advanced math courses.

As for 0*0=0. Yes it is , for FINITE numbers of zeros, but once you are dealing with infinity, all bets are off.



[edited by - SpaceRogue on April 23, 2003 8:09:05 PM]

1 / 0 = x
0x = 1
0 = 1

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Original post by SpaceRogue
There are different orders of infinity which are not 'equal'. I've seen it proven in some of my advanced math courses.

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An example of this is how the set of all real numbers is larger than the set of all integers. Even though both sets have an infinite number of members, one is larger than the other. There are varying degrees of infinity. Weird, huh?

EDIT Check out the "infinite hotel" too. It sort of illustrates this sort of thing. Really cool problem. Not sure it exactly applies though.

[edited by - Cold_Steel on April 23, 2003 8:41:59 PM]

That was one of the examples.

I spent forever turning that one around and around before I could visualize it.

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however the limit of 1/x as it approches 0 from either side is defined.
If it''s defined, what is it??? lim (x->0) |1/x| is defined to be infinity, but 1/x doesn''t have a limit at x=0.

Cédric

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Ever heard of limits AT infinity? It is terminology used. Furthermore, we have to be more precise about lim x>0 (1/x). Only using a one sided limit does the limit exist. It is simply infinity. That qualifies. Such as lim x>0- (1/x) = -oo and
lim x>0+ (1/x) = oo

i think the problem, is that alot of math rulse changes if you use it, because you loose the information in x:
x*0 = 0
x/0 = inf

( x*A ) / A = x
( x*0 ) / 0 != x

//as CoffeeMug said:
1 / 0 = x
0x = 1
0 = 1

you can also prove that 1 == 2 with the use of x/0, and alot of other things taht we know is wrong

is 0/0 defined as 1 ??? (whitout a lim)
is there any math rules about x/0 ??? (exsept that its undefined)

the problem with x/0 is that its not logical in practical eksamples!!! i is sort of logic, because its defined sqrt(-1)..
shure you can call x/0 for p or f or something else... no rules will applie to it anyway... (i think in my humble opinion)
if 0 persons splitt 5 apples, how many appled does one person get? (you cant realy ansver this, and the ansver is not infinitly many)

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Original post by CoffeeMug
1 / 0 = x
0x = 1
0 = 1

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You''ve just argued that 1/0 cannot be defined to be a real number, as it would contradict the already-defined properties of a real number. This doesn''t say anything about defining 1/0 = x for some non-real x. Try again.