Why is it impossible to divide by zero?

Answer 1

You can't put something in 0 groups, it is undefined arithmetically. In programming it has some applications as well as in calculus and theoretical math, but in general it has no 'logical' or well-defined meaning.

Answer 2

A mathematical reason why division by zero is undefined:

First of all, let's talk about addition and subtraction:

If you have two number a and b, and add them together, a+b=c is still a number (this is known as the "closure" of numbers under addition).

There is an additive identity element (we call it 0) such that for any number a, a+0=a.

Now what if we have a number a, and want to find a number b, such that a+b=0. Well, we have defined b as the negation of a (-a). Therefore a+(-a)=0.

Now what is subtraction?

We have define a-b as a+(-b). Therefore subtraction is addition with a negative number.

What does this have to do with dividing by zero?

Let's look at a property of 0 and any number a.

a•0 = a•(0+0) = a•0+a•0.
therefore 0=a•0+(-(a•0)) = a•0+a•0+(-(a•0)) = a•0

So for any number a, a•0=0.

Now let's look at multiplication and division:

The reason I talked about addition and subtraction is because they are related in similar ways.

First of all, for any two number a and b their product a•b=c is a number ("closure" under multiplication).

There is also a multiplicative element (we call it 1), such that for any number a a•1=a.

What if we have a number a and want to find b such that a•b=1. Well if a≠0 then we define b as 1/a. Why not do this for 0 also?

Remember that for any number b, b•0=0≠1, therefore there is no such number b, that would make that work.

Now to division:

We define a divided by b by a/b = a•(1/b) or multiplication by the inverse of b.

To the answer to your question:

a/0 would be then defined by a•(1/0), but as stated above 1/0 is not defined. Therefore a/0 cannot be defined.

This was a little long and drawn out, but hopefully it answers your question.

Answer 3

Strictly speaking, it isn't impossible to divide by zero! Here ... lets do it:

5/0 = 6.

That was easy! Here's the problem: if we do this then the whole structure of arithmetic becomes logically contradictory! For instance we could algebraically re-arrange the above equation to get:

5 = 6 * 0

and then rearrange some more to get ...

5/6 = 0.

This contradicts the fact that 5/6 is *not* zero! The point is this: you *can* divide by zero, but then arithmetic becomes unreliable and self-contradictory. That is, dividing by zero isn't *impossible*, it's just not a good idea!

Answer 4

You can divide a number by zero but then you end up having infinitely many answers. Division is defined as 'repetitative subtraction'. Example:

6 / 2 :
6-2=4
4-2=2
2-2=0

You were able to subtract 2 three times, therefore the answer is 3.

Now for 6/0:
6-0=6
6-0=6
6-0=6
....
See what happens here? Where do you stop? Because subtraction is a finite process, it is not possible to divide by zero because we do not know when to stop the subtraction process.

Answer 5

NOw we will prove this by an example.

Now consider 1/(1/2) = 2
concider 1/(1/3) = 3
consider 1/(1/4) = 4

consider 1/(1/10) = 10
consider 1/(1/100) = 100
consider 1/(1/1000) = 10000
Now you can see that as the denominator decreases the value of the fraction increases. The minimum value of denominator can be '0' and hence when the denominator becomes least the value becomes highest which is represented by infinity. This is how anything divided by zero becomes infinity, since zero is the least value.

Answer 6

ok.. let's say you buy a dozen donuts...

now.. you decide to split up that dozen donuts to as many people as possible.. giving each person zero donuts... how many people can you give zero donuts?... some people will say at this point that you can give an infinite number of people zero donuts.. ok.. let's just say you can...

but... would that make 12/0 = infinity?... if so.. then.. multiply both sides by zero... and get:

12=0... hmmm.. wait.. this is not right!!! but.. then zero times anything is zero.. right?... yes it is.. it is defined this way...

so then 12 = 0 is wrong.. then 12/0 ≠ infinity...

or... check out this is a lack of identity...

(12/0) * (0/12) =? 1

infinity * 0 =? 1
0=1... I don't think so... this is the reason... that dividing by zero... is "Undefined" rather than some value like Infinity.

Answer 7

The problem

x = a / 0

is by definition equal to finding the unique number x with the property

x * 0 = a

Now we know that x * 0 = 0 no matter what the value of x is. So if a is non-zero, there are no solutions to the problem. This proves that

a / 0 is undefined for all non-zero a

If a is zero, all numbers x solve the problem. In that case, there is no unique solution. This proves that

0 / 0 is undefined.

Answer 8

There is no way do assign a value to x/0 that keeps things consistant with the rest of arithmetic.

To see this all you have to do is look at the "proofs" that 0 = 1 that are posted here a couple of times a day. They all depend on dividing by 0.

Answer 9

Dividing by zero means you are saying put a certain number of things into groups of zero.Zero means there is nomthing at all there so you would be saying but a certain number of things in groups of nothing which would equal nothing.For more info about dividing by zero visit :

http://en.wikipedia.org/wiki/Division_by_zero

Answer 10

For instance...
If a/0 =b then a/b = 0 but then a must be 0. So 0/0 = b
If b is 0 is this possible?
I don't think even then 0/0 makes sense because:

If 0/0=0 then 0/0*5=0*5 or then 0 *(5/0) =0, but we just said that only 0/0 is possible, so the property of associativity is lost over multiplication. (It's been a while though)

Answer 11

As an exercise, sketch a graph of y = 1/x

When x is positive and as you approach zero, y gets larger and larger in a positive direction. When x is negative and as you approach zero, y gets larger and larger in a negative direction.

You might say that y approaches infinity coming from the right of zero and y approaches negative infinity coming from the left of zero.

The graph obviously has a discontinuity at x = 0.

x/0 is unknown.