Dividing by Zero

[dougkeenan]

Youse guys aren't coming off any pie, are you?

 

[Napes Weaver]

I like this...

Actually, any finite number multiplied by zero equals zero. This does not follow in the infinity case.

So does does this mean that everything multiplied by nothing equals anything?.. or just something?

 

[agentgonzo]

So does does this mean that everything multiplied by nothing equals anything?.. or just something?

Any finite number multiplied by nothing is equal to zero.
An infinite number multiplied by zero is undefined - it is possible to argue the case that infinity multiplied by zero is equal to anything you choose.

 

[Eagle]

Another reason 10/0 and 5/0 are undefined is that neither is a consistent number. I can take 10/0 split the 10 into 5/0.5 and get 5/(0.5*0) which gets me 5/0.

 

[jedidia]

Any finite number multiplied by nothing is equal to zero.
An infinite number multiplied by zero is undefined - it is possible to argue the case that infinity multiplied by zero is equal to anything you choose.

Now THIS really doesn't make any sense in my head. I thought ANYTHING multiplied with zero equals zero, and it makes darn lot of sense too. What is the argument that infinity times zero is anything else than a finite number times zero?

 

[agentgonzo]

Now THIS really doesn't make any sense in my head. I thought ANYTHING multiplied with zero equals zero, and it makes darn lot of sense too. What is the argument that infinity times zero is anything else than a finite number times zero?

As a side argument, would you agree that anything multiplied my infinity equals infinity? If so, then surely zero multiplied by infinity equals infinity, not zero?

Let x be the unbounded sequence 1, 2, 3, ...
Let y be 1/x.

What is x * y?
Easy. x * y = x / x = 1
This is true for all x, and so as you let x increase to infinity, y decreases to zero. At the limit of this sequence, x is infinity and y is zero. Therefore 1 = x * y (by before), which also equals ∞ * 0 (as x = ∞ and y = 0).

Now, what about letting a = 2x?
a * y = 2x * y = 2x / x = 2.
But again, taking the limit as x --> ∞.
As x --> ∞, a --> ∞
∞ * 0 = a * y = 2.

So you have
∞ * 0 = 1
∞ * 0 = 2

This can be done for any value you choose. This is why it is undefined and x * 0 = 0 only for finite x.

 

[Quick_Nick]

You make a good (yet strange!) point agentgonzo.
RE: the first example only
My thoughts are that you can really only define division by 0 if you define 0 as actually in infinitely small number and not nothing. As you said, y APPROACHES 0. But when x is infinity, y would be 1/infinity. Both infinities should be the same 'number'. Also, as a random thought, if y were 0, what would x be? You'd have to divide that 1 by 0! Paradox I think. (and not only would you be dividing by 0 again, but x must be 0 to fit the definition of y)

 

[agentgonzo]

You make a good (yet strange!) point agentgonzo.

Welcome to the world of pure mathematics!

My thoughts are that you can really only define division by 0 if you define 0 as actually in infinitely small number and not nothing. As you said, y APPROACHES 0.

No, I said the limit as x --> ∞. Subtly, but very importantly different.

You can define division normally for the non-zero numbers as you state.
You can define x_n = 1/y_n (y_n > 0). Let y_n tend to zero.
As y_n decreases monotonically and is bound by zero, it is convergent (to zero by definition). You can then prove that x increases monotonically with no upper bound. y_n --> 0, x_n --> ∞. (approaches, as you mention. Not talking about absolute =0 here, that is the next step).

Now, since y is convergent, and x is monotonically divergent, you can legally take the limits.
Even though y_n > 0 (strictly greater than 0), the limit is at y = 0. Similarly, x_n is finite, but at the limit, x is infinite.
thus the limit (y_n --> 0) of x_n = 1/y_n is ∞ = 1/0 (by the continuity of our function). This is how 1/0 can be legally defined.