# Dividing 0 by 0

by repugno
Tags: dividing
 P: 80 Greetings all, What happens if you divide 0 by itself? I had an argument with my math teacher about this; she reasons that there could be 2 answers to this question. - If you divide a number by itself it is always one - If you divide by 0 it is infinity The first explanation I believe is illogical since you cannot have nothing divided by nothing giving you 1. I can also not imagine how dividing 0 by 0 can give you an infinite number. What I can understand is, if I have nothing then I divide this into 0 parts I will end up with nothing, because there was nothing in the first place. Which is considered to be correct? Also, why if we divide a number by 0 we get infinity? I have an object like an apple, I then decide to cut it up into no parts. How could I then have an infinite number of apple pieces? Any help would be much appreciated. Thank you.
 PF Patron Sci Advisor P: 2,188 Just to show why you shouldn't take a limits of functions necessarily as real values: $$x \rightarrow \infty \ , |x^\frac{1}{x}| \rightarrow 1 \ , x \in [1,\infty)$$ If we now make a new number, infinity, taking equalities from limits as you have done and assuming we can perform basic algebra on it: $$\infty^\frac{1}{\infty} = 1 \therefore 1^\infty = \infty$$ but: $$x \rightarrow \infty \ , 1^x \rightarrow 1$$
P: 1,210

## Dividing 0 by 0

It is easy to show that:
x>=0
x*1      1
--- = x*--- = x*1 = x
1       1
But when:
x>0
y*x      x
--- = y*--- = y*1 = y
x       x

x=0
y*x      x
--- = y*--- = y*(if x/x=1 then y*1 = y , but if x/x=0 then y*0 = 0)
x       x
Becauce a unique result cannot be more than one value, we can learn that any devision by 0 has no unique result.
P: n/a
 Quote by repugno What happens if you divide 0 by itself? I had an argument with my math teacher about this; she reasons that there could be 2 answers to this question. - If you divide a number by itself it is always one - If you divide by 0 it is infinity
Just to reiterate Organic's point, division is a purely abstract process that has nothing to do with apples or anything real. Math is just a bag of tricks, and even though we can apply those tricks to solve real problems, the tricks themselves are meaningless abstractions.

The reason you can't divide any number by zero is because there is no number, real or imaginary, which could be used to represent the answer without creating paradoxes. Let's say you define X = 0/0 and leave the value of X unknown. That means X is just a replacement for 0/0. Harmless enough, right? However, you can't do anything with X. You can't multiply it, divide it, add it, subtract it. The result of 0/0 is a useless number. That's why you can't do it.

Last but not least, it's not true that a finite number divided by zero equals infinity. Strictly speaking, division by zero is undefined. It's not a knowable quantity.
 P: 1,210 I want to say that x/0 is undifined or useless if we understand it through Boolean Logic, Fuzzy Logic or any other logical system which is basd on 0 XOR 1 connective. But there can be different forms of logic, which are not 0 XOR 1 connective, for example: http://www.geocities.com/complementarytheory/BFC.pdf
 P: 112 The reason we have no zero divisors is because if there was a zero divisor then: $$\exists 0^-^1 : 0^-^1 \cdot 0 = 1$$ But of course this contradicts the fact that: $$a \cdot b=0 \Leftrightarrow a \vee b=0$$

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