# Why 0/0 not = 1

Ahmed Jubair
if( 2/2=1,5/5=1) then it must be that 0/0=1.but Again,it couldn't be (-1) also i think because if its-5/5=5,-2/2=-2.but 0 have no value and its the low valuenumber.so no need a( - )before it.so(- 0/0 is not =-1)
then why its undefined???why not 1

ProfuselyQuarky
Gold Member
Dividing by 0 is undefined.

You can't divide anything by 0 .... including 0!

Ahmed Jubair
blue_leaf77
Homework Helper
2/2=1 because 2=2x1, likewise 8/8=1 because 8=8x1 is satisfied. But now, which number when multiplied by zero yields zero? It's anything, 0 = 0x3 = 0x100 = 0x1000. Then how will you define 0/0?

Ahmed Jubair
phyzguy
Because there are many ways in which the limit 0/0 could be reached. Your comment (2/2=1, 5/5=1, ...) is implicitly defining:

$$\frac{0}{0}=\lim_{x\to 0} \frac{x}{x} = 1$$

But this is not the only possibility. Why couldn't I define:

$\frac{0}{0}=\lim_{x\to 0} \frac{2x}{x} = 2$ or : $\frac{0}{0}=\lim_{x\to 0} \frac{x}{2x} = 1/2$

or in an infinite number of other ways? That is why it is undefined.

Mark44
Mentor
if( 2/2=1,5/5=1) then it must be that 0/0=1.but Again,it couldn't be (-1) also i think because if its-5/5=5,-2/2=-2.
I don't understand what you're doing here. -5/5 = -1, not 5, and -2/2 = -1, not 2
Ahmed Jubair said:
but 0 have no value
Certainly 0 has a value.
Ahmed Jubair said:
and its the low valuenumber
???
It's the smallest number that isn't negative.
Ahmed Jubair said:
.so no need a( - )before it.so(- 0/0 is not =-1)
then why its undefined???why not 1

jim mcnamara
fresh_42
Mentor
I prefer to say: ##0## is no element of the multiplicative group. Therefore the question whether there is an inverse or not simply doesn't exist.
One could now object: But ##1## as the neutral element of multiplication is part of the additive group, it even generates it.
My answer then would be: ##1## has a natural usage for addition, ##0## hasn't for multiplication. The definition ##0 \cdot 1 = 0## simply is a necessity for the distributive law which is the only connection between both operations.

ProfuselyQuarky
ProfuselyQuarky
Gold Member
I prefer to say: ##0## is no element of the multiplicative group. Therefore the question whether there is an inverse or not simply doesn't exist.
One could now object: But ##1## as the neutral element of multiplication is part of the additive group, it even generates it.
My answer then would be: ##1## has a natural usage for addition, ##0## hasn't for multiplication. The definition ##0 \cdot 1 = 0## simply is a necessity for the distributive law which is the only connection between both operations.
That's a nice explanation. I really love how such a simple question can have such a variety of legitimate answers.

DrewD
Division is a repeated subtraction and you keep doing it until you reach zero or a dead end ( reminder)

So for example:
20 -5 -5 -5 -5 = 0
So the result of 20/5 = 4 (how many times did you repeat the 5?)

Now for the zero:

0 - 0 = 0 ... Well I reached zero (so 0/0 = 1)
0 - 0 - 0 - 0 = 0 I reached zero too ( 0/0 = 3 )
So you can see that you can make infinite answers. So when you divide 0/0, you can't just choose one of the answers because Why not the others too?

That is how I see it which is similar to Blue_Leaf way

One could also use a proof by contradiction here.
Let a=b, if we multiply both sides by a..
a^2=ab ,now subtract b^2 from both sides.
a^2-b^2=ab-b^2
Now simply factorise:
(a-b)(a+b)=b(a-b) , now divide both sides by (a-b).
So we're left with, (a+b)=b
Using our original definition of a=b, we can simplfy this to 2b=b which implies that 2=1, which is mathematically incorrect. The mathematics of my steps were valid until the point where I divided both sides by (a-b), [a-b=0]. So as you can already tell, dividing anything by zero is not possible. Many other good reasons have been explained here. Try a graphical approach if you're really interested, and plot as many graphs as you can that pass through (0,0), the trend you will notice is that there are an infinite amount of ways to approach 0 and thus we cannot give it's division a value.

Biker
It's an axiom that division by zero is undefined, conceptually it makes sense because it makes no sense to ask ' how much nothing goes into something '.
It's also interesting to note that you cant divide any number by another and obtain a non-approximate zero.
Pure mathematics makes my head hurt.

fresh_42
Mentor
It's an axiom that division by zero is undefined, ...
It is not. 0 has nothing to do with multiplication. There is no need for an inverse!

It is not. 0 has nothing to do with multiplication. There is no need for an inverse!
Hmm I was using 'axiom' in it's broadest sense though your point is well taken, thanks.

Mark44
Mentor
One could also use a proof by contradiction here.
Let a=b, if we multiply both sides by a..
a^2=ab ,now subtract b^2 from both sides.
a^2-b^2=ab-b^2
Now simply factorise:
(a-b)(a+b)=b(a-b) , now divide both sides by (a-b).
No, this isn't valid. Since a = b, by assumption, then a - b = 0, so you're dividing by zero.
If you do that, all bets are off, which you explain below.
whit3r0se- said:
So we're left with, (a+b)=b
Using our original definition of a=b, we can simplfy this to 2b=b which implies that 2=1, which is mathematically incorrect. The mathematics of my steps were valid until the point where I divided both sides by (a-b), [a-b=0]. So as you can already tell, dividing anything by zero is not possible. Many other good reasons have been explained here. Try a graphical approach if you're really interested, and plot as many graphs as you can that pass through (0,0), the trend you will notice is that there are an infinite amount of ways to approach 0 and thus we cannot give it's division a value.

Mark44
Mentor
It's an axiom that division by zero is undefined, conceptually it makes sense because it makes no sense to ask ' how much nothing goes into something '.
It's also interesting to note that you cant divide any number by another and obtain a non-approximate zero.
"non-approximate zero"? What is that?
If you divide any nonzero number by itself, you get 1.
Marcus-H said:
Pure mathematics makes my head hurt.

Mark44
Mentor
Time to put this thread to bed. The question has been asked and answered. Division by zero is undefined, and that's all you need to say.

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