Dividing Zero by Zero.

[Jim B.]

Why is dividing zero by zero an "illegal" mathematical action. It has been a while since I took mathematics in school. But I know on my calculator, dividing zero by zero invariably gives you an "E" error. Zero times zero equals zero. So why can't you just turn the equation around and say zero divided by zero equals zero?

Also, what does it "approach" as the numerator and denominator nears zero? I know for one divided by zero, it approaches infinity as the denominator approaches zero.

Thank you in advance to all who reply :)

[Manduck]

The problem is that any number could be the answer. How many times can you subtract 0 from 0? One would work. So would 0, or a billion, or pi.

[treis]

We just had a thread about this but the answer is becuase 0/0 can be any number. Say you have a function f(x,y)=a*x/y and you want to know what will be at x=0 and y=0. According to you it will be 0 but if you graph it you will see that the value should be whatever a is. If you are having trouble picturing what that graph will look like consider a plot of the function along the line x=y. You will see that the fuction becomes a*x/x and the x's cancel out leaving the value of the graph at "a" for all x.

0/0 doesn't have to be 1 either. Take for example the equation y=(x^(2)-9)/(x-3). For x=3 you get 0/0. If you look at the graph the value you would expect at x=3 is 6. In fact, I can manipulate equations to make 0/0 be any number from minus infinity to infinity.

[Cunctator]

Previous thread (http://boards.straightdope.com/sdmb/showthread.php?t=343097&highlight=indeterminate+undefined)

[sturmhauke]

I didn't truly understand the reason you can't divide by zero until I learned about limits in calculus. I can try to explain, but if you haven't taken calculus you might not get it either.

Consider the graph of y=1/x. The sign of y always matches the sign of x, so the graph is a hyperbola with one branch in the upper right quadrant (I) and one in the lower left (III), and asymtotically bounded by the x- and y-axes (that is, the hyperbola approaches but never touches the axes).

Now look at a few values of x and y and see the pattern:

x=3, y=1/3
x=2, y=1/2
x=1, y=1
x=1/2, y=2
x=1/3, y=3
x=1/10, y=10
x=1/100, y=100
x=1/1,000,000, y=1,000,000

x=-3, y=-1/3
x=-2, y=-1/2
x=-1, y=-1
x=-1/2, y=-2
x=-1/3, y=-3
x=-1/10, y=-10
x=-1/100, y=-100
x=-1/1,000,000, y=-1,000,000

As you can see, as x approaches 0 from the positive side, y approaches infinity; as x approaches 0 from the negative side, y approaches negative infinity. Therein lies the problem. If y "shot off" in the same direction at all times, it could perhaps be said that division by zero was an infinite quantity, but since y is sometimes positive and sometimes negative, you cannot say that y has any specific value when x is zero. Therefore division by zero is undefined.

[party store]

Just in case you're like me and absolutely hate math, here's the way I think about it:

I have zero apples. I give each of my zero apples to zero people. How many apples did each person get?

Well, my question is pure gibberish because there were no people to get the non-existent apples I was handing out, so hence the answer is undefined.

I think perhaps, that the mathematical function of something infinitely approaching zero can be expressed in everyday terms as a question gradually approaching stupid. Then, again, I suck at math, so I could be way off base here.

[Giles]

Just in case you're like me and absolutely hate math, here's the way I think about it:

I have zero apples. I give each of my zero apples to zero people. How many apples did each person get?
I think that's the wrong model, because it's modelling 0 times 0. A better model is:

I have zero apples. How many people can I give zero apples to, and have no apples left over?

And the answer to that is, any number of people, so there's no good way to define0 divided by zero.

[chrisk]

Zero times zero equals zero. So why can't you just turn the equation around and say zero divided by zero equals zero?


one times zero equals zero, therefore zero divided by zero equals one

two times zero equals zero, therefore zero divided by zero equals two

three times zero equals zero, therefore zero divided by zero equals three

four times zero equals zero, therefore zero divided by zero equals four

(Repeat indefinitely for all positive integers.)


I know other people have answered the question, but I just REALLY wanted to post that. ;)

[FlyingCowOfDoom]

Hijack:

If the temperature today is 0 degrees, and the weatherman says it will be twice as cold tomorrow, what will the temperature be tomorrow?

--FCOD

[chrisk]

Hijack:

If the temperature today is 0 degrees, and the weatherman says it will be twice as cold tomorrow, what will the temperature be tomorrow?

--FCOD

Umm... since the weatherman is stupid enough to use a term like that, I think it will be... 4 degrees tomorrow, not quite as cold. :D

(American degrees or celsius?)

[FlyingCowOfDoom]

(American degrees or celsius?)

Either.

--FCOD

[BobLibDem]

Hijack:

If the temperature today is 0 degrees, and the weatherman says it will be twice as cold tomorrow, what will the temperature be tomorrow?

--FCOD

If twice as cold = half as much heat, then twice as cold would be halfway from zero of the temperature scale used to absolute zero.

So twice as cold as 0F = -230F or so
And twice as cold as 0C = -137C or so

[FlyingCowOfDoom]

So twice as cold as 0F = -230F or so
And twice as cold as 0C = -137C or so
Phew...guess I better not forget my coat tomorrow.

--FCOD

[Squee]

I remember a high school teacher explaining a proof on the board... he basically pointed out a picture, showing a line. His point was saying if you cut the line in half, you have half, cut it again infinite times, you will still never get to the end.

Ironically enough, he was also my calculus teacher.

If I'm a retard, I blame the school system. = )

[LionelHutz405]

I always thought a number divided by zero results in infinity. Not ‘undefined’.
The reason the calculator displays ‘e’ is not because it is mathematically an error, but because you have exceeded its ability to display the answer. You can get this ‘e’ result without dividing by zero; you just have to exceed the physical abilities of the calculator. (Such as repeatedly squaring a number.)

[Giles]

I always thought a number divided by zero results in infinity. Not ‘undefined’.
Mathematicians use several definitions of infinity, but with none of the normal meanings of the word does it make any sense to say that a number divided by zero is infinity.

If you take infinity to be the cardinal number of the set of natural numbers (1, 2, 3, ...), then multiplying that by zero would surely give you zero. (What is the cardinal number of the set of ordered pairs (x,y), where x is a member of the empty set, and y is a natural number? The answer is zero, because you get the empty set again).

However, saying that a number n divided by zero is infinity means that infinity times zero is the number n. That's false, so it's false to say that a n divided by 0 is infinity.

[Trunk]

I always thought a number divided by zero results in infinity. Not ‘undefined’.
The reason the calculator displays ‘e’ is not because it is mathematically an error, but because you have exceeded its ability to display the answer. You can get this ‘e’ result without dividing by zero; you just have to exceed the physical abilities of the calculator. (Such as repeatedly squaring a number.)
Well, a number divided by zero can be thought of as infinity, but zero divided by zero is totally undefined.

The reason the calculator displays 'e' is because it is mathematically an error. If you use a "fancy" calculator, like some mathematical software, 1/0 will probably return "Inf" and 0/0 will return "error" or "NaN" (not a number).

If you try to do EITHER in a programming language, you'll get something crazy back. . .maybe an exception, an "overflow error" or just strange behavior.

[ultrafilter]

Well, a number divided by zero can be thought of as infinity....

No it can't.

[chrisk]

I always thought a number divided by zero results in infinity. Not ‘undefined’.


Well, 1/x as x approaches zero from above will 'limit to infinity' (in the sense that you will get higher and higher values as you get closer and closer to zero.)

On the other hand, if you approach zero from below, the limit will approach negative infinity. So that doesn't really help us much.

[LionelHutz405]

However, saying that a number n divided by zero is infinity means that infinity times zero is the number n. That's false, so it's false to say that a n divided by 0 is infinity.

Yep that makes sense. So I’ll stop thinking like that. :)

[Frylock]

"I think perhaps, that the mathematical function of something infinitely approaching zero can be expressed in everyday terms as a question gradually approaching stupid. Then, again, I suck at math..."

Party Store, just want you to know, I now quote you in my signature at another discussion board I participate in. (I change it every few weeks.) If there were a prize for this, you would have won it.

:p

-FrL-

[panamajack]

Also, what does it "approach" as the numerator and denominator nears zero? I know for one divided by zero, it approaches infinity as the denominator approaches zero.


This has been adumbrated in previous posts, but just to make it clear, the numerator and denominator don't necessarily change and approach anything.

If one or both are functions, then you can look at what the value of the function is as it approaches some point at which it is undefined. And take the limit (which is most definitely not the "value of the function" at that point, since the function's undefined). As has been made clear in examples above, simply knowing that the numerator or denominator function approaches zero is not enough to tell you what the limit is .

So "one divided by zero" is simply that, and doesn't approach infinity on its own (and even as a function, it may go positive or negative).

[jsmith]

If you use a "fancy" calculator, like some mathematical software, 1/0 will probably return "Inf" and 0/0 will return "error" or "NaN" (not a number).


I am not to sure about this. Every calculator and software that I have capable of returning infinity as a result will return undefined if you try to divide by zero.

[Mathochist]

Mathematicians use several definitions of infinity, but with none of the normal meanings of the word does it make any sense to say that a number divided by zero is infinity.

The real projective line. Thank you, come again.

[Chronos]

If the temperature today is 0 degrees, and the weatherman says it will be twice as cold tomorrow, what will the temperature be tomorrow?We say a temperature is "hot" or "cold" to the extent that it deviates from our comfortable temperature (approximately 25 Celsius or 77 Farenheit). So a temperature which is twice as cold is twice as far away from our comfortable temperature. Thus, -77 Farenheit is twice as cold as 0 Farenheit, and 103 Farenheit is twice as hot as 90 Farenheit.

[kaon]

As you can see, as x approaches 0 from the positive side, y approaches infinity; as x approaches 0 from the negative side, y approaches negative infinity. Therein lies the problem. If y "shot off" in the same direction at all times, it could perhaps be said that division by zero was an infinite quantity, but since y is sometimes positive and sometimes negative, you cannot say that y has any specific value when x is zero. Therefore division by zero is undefined.
Many years ago when I wondered about 0/0, I plotted y = 1/x and observed the same as you.
My question, why don't we say 0/0 = "plus/minus infinity"

(similar to how we say if x = y^2, then y = plus/minus square-root of x) (hmm... i'm sure some doper is gonna tell me this isn't strictly correct..)

[Giles]

The real projective line. Thank you, come again.
OK, that's a geometric object. How are you defining multiplication and division on it? Presumably for all points that map to real numbers, it's the equivalent of normal multiplication and division on the reals. Do you define infinity times any non-zero number as infinity? (I can't think of any other sensible definition). And what is infinity times zero?

For cardinal numbers, as I said earlier, it makes sense for infinity times zero to be zero, but I can't think of a sensible definition here. If you make it anything other than zero or infinity, then I think multiplication can't be associative:

If zero x infinity = n,
then:
2 x (zero x infinity) = 2n
but:
(2 x zero) x infinity = n

[Mathochist]

Many years ago when I wondered about 0/0, I plotted y = 1/x and observed the same as you.
My question, why don't we say 0/0 = "plus/minus infinity"

(similar to how we say if x = y^2, then y = plus/minus square-root of x) (hmm... i'm sure some doper is gonna tell me this isn't strictly correct..)

First of all, the two situations aren't analogous. The first has limits and the second has two actual solutions.

On top of that, why would a graph of 1/x tell you anything about 0/0? It'll tell you about 1/0 maybe, but not 0/0. Your suggestion could only remotely apply to a nonzero number over 0.

[Mathochist]

OK, that's a geometric object.

No, it's an algebraic object, as in algebraic geometry. As for the multiplication, I didn't say it was a ring. You asserted that mathematicians never use "x/0 = infinity".

In this particular case, functions are defined on RP1 and people study the polology. How do you study the behavior of f at infinity on the projective line? Look at f(1/x) as x varies in a neighborhood of zero.

[simster]

To Infinity.... and Beyond!

(I have nothing of value to add, but I had to state the above for your amusement.)

[Giles]

No, it's an algebraic object, as in algebraic geometry. As for the multiplication, I didn't say it was a ring. You asserted that mathematicians never use "x/0 = infinity".

In this particular case, functions are defined on RP1 and people study the polology. How do you study the behavior of f at infinity on the projective line? Look at f(1/x) as x varies in a neighborhood of zero.
I haven't come across the term "polology" before, and I can find a definition in my usual mathematics reference works.

And I still don't think that there is a sensible meaning for x/0 on the projective real line, because x/0 = infinity would mean that infinity times 0 = x. However, that would be true for all x except zero and infinity, so infinity times 0 would not be unique, and uniqueness is a uiseful property for algebraic operations.

[Mathochist]

I haven't come across the term "polology" before, and I can find a definition in my usual mathematics reference works.

It's the study of the nature and location of the poles of functions.

And I still don't think that there is a sensible meaning for x/0 on the projective real line, because x/0 = infinity would mean that infinity times 0 = x. However, that would be true for all x except zero and infinity, so infinity times 0 would not be unique, and uniqueness is a uiseful property for algebraic operations.

Didn't I just say that it wasn't a ring?