The zero/infinity paradox

A nice, simple paradox, which manages to completely confuse mathematical theory. Most people would agree with the following statements:

1. Anything divided by zero is infinity. This is obvious when you start dividing a number by increasingly smaller numbers. eg:

10/1=10
10/0.5=20
10/0.00000000001=1,000,000,000,000
etc.

2. Zero multiplied by anything is zero. Therefore, it follows that zero divided by anything is zero, since division by n is the same as multiplication by 1/n. i.e. 0/2=0 X 1/2 = 0

So, here's the paradox: What is zero divided by zero?

It's undefined. What did you expect? A miracle, perhaps?

There's a reason your first year Calculus prof made your life miserable with epsilons and deltas (and if she didn't, there's a reason she should have done). Some things are numbers: 0, 17, 0.1234567891011121314..., pi. But not everything is a number; some things just aren't: a banana, a dodecahedron, "1 divided by zero". When we call something a number, we expect to be able to perform various operations on it. Deep, complex, mathematical operations like addition and multiplication. (Maybe even a few more complicated ones, but these 2 will do.)

So who cares about some snotty mathematician's prejudices that "that's not what I call a number!"? Well, you should! Say 1/0 is a number; call it x. Well, 2/0 should be 2x (= x+x). Now we have that 1/x = 1/(1/0) = 0, which is cool, until you realise that then 1/(2x) = 0/2 = 0 = 1/x, so 2x=x and therefore 1/0=x=0, which is clearly wrong.

Conclusion: It's not possible meaningfully to define 1/0 to be a number. Sorry.

Sorry to be a bitch, but anything (except zero) divided by zero is undefined. Zero divided by zero, on the other hand, has its own special term.

It is not undefined, but indeterminate.

In reality, you can pull all sorts of crazy shit with infinity. Infinity divided by infinity, for example. Anything divided by itself is one, right? Wrong. Any real number divided by itself is 1. And infinity is unfortunately not a real number.

Picture this. There are an infinite number of ways this writeup could be cooled. Someone could do the honours, or the system could go haywire, or God Himself could do it, or...Also, there are a infinite number of Universal possibilities: eg, this writeup could turn into a butterfly and fly off my monitor.
Thus, the probability of this writeup being cooled is infinity on infinity. If this is equal to one, then I'm guaranteed a cool! I wish. :)

Conclusion: Zero on zero is indeterminate. Infinity is not a element of the reals.

I think that for non-mathematicians, a small explanation would be in order. Ralix states that anything divided by zero is infinity. Excluding bananas, and other non-numbers which ariels has been kind enough to point out, let me be so bold as to exclude many other things, and assume that Radix is talking about real numbers (or for the non-mathematicians - any number that YOU would consider to be a number). Two types of people will say that anything divided by zero is infinity:

1. People who don't know enough and think that it's true (which ariels showed nicely that it's not), and
2. People who use it as a short form of "when a number is divided by x, as x approaches zero, the result approaches infinity."

The latter is what people like my first Calculus professor and myself and Cantor mean when they say that y/0 = infinity.

Let's look at 0 / 0 using that definition. A number cannot be divided by zero (that's what asymptotes on graphs are there for). So, zero divided by x, as x approaches infinity, is zero. When x=0, it is undefined.

At this point my mathematics teacher would hesitantly say "There are many different types of infinity," whilst scratching his head trying to think of a way to simplify his thoughts to something we could understand.

My other mathematics teacher would have slaughtered you (or me) before we even mentioned zero, let alone dividing by it.

Ralix, you say that: "Anything divided by zero is infinity. This is obvious when you start dividing a number by increasingly smaller numbers."

However this is an oversimplification, and quite incorrect. What you actually mean to say is that the limit of 1/x (as x approaches zero) would approach infinity.

What is zero divided by zero?
Well, it could be anything you want it to be, really... Keep in mind that anything divided by zero is undefined, so what you actually mean to say is:
What is the limit of x/y as x, y approach zero?

If you think about two numbers which are getting closer and closer to zero, divided by each other, the result could be anything, depending on how quickly they each approached zero.

Take this example:

2x/x = 2    (right?)
But surely the limit of 2x as x approaches zero would be zero.
Similarly the limit of x as x approaches zero is also zero!

So you see your quotient is entirely dependant on how quickly your divisor and dividend are converging to zero.

Another more complex example would be the following:

Evaluate the limit, as x approaches zero, of:

   sin(x) - x    
sin(x) - x.cos(x)

At first look you'd probably think "sin(0) - 0 == 0 - 0 == 0, and if the numerator is 0 then the whole thing is zero, right?" Wrong. As I said, it's all to do with the rate of convergence...

sin(x) can be approximated by the following infinite series:

sin(x) ~ x - x³/3! + x⁵/5! - ... + (-1)^r ( x^2r+1 / (2r+1)! ) + ...

Hence it is reasonable to assume that for x approaching 0,

sin(x) ~ x - x³/3!

The index of x increases by two with each term, and x is approaching 0, which means the significance of each subsequent term diminishes very quickly... so we can ignore anything with a coefficient of x⁴ or less.

Similarly we can say that:

cos(x) ~ 1 - x²/2!

If we put these together, by substituting into the original we get:

      (x - x³/3!) - x      
(x - x³/3!) - x.(1 - x²/2!)

==         - x³/6        
     ( - x³/6) + (x³/2)

==      (-1/6)     
     (-1/6) + (1/2)

== (-1/6)
      (1/3)

== -1/2

Not quite the zero you were hoping for, was it?

I wouldn't call myself a mathematician, so if there's anything wrong with the above don't hesitate to /msg me.