Question

I just want to make sure about a concept : What is `0/0`? Usually anything divided by `0` is undefined, but a past student of mine reckons they were taught it is `1`. Is there any theorem or proof of this? Thanks

Answer 1

Your student either mis-remembers or was mis-taught. (Both things happen.)

Explanation:

If we attempt to make `0/0` equal to one (by definition or whatever), then we lose our number system.

Using what we know about multiplication, we can prove that there is only one number.

If `0/0 = 1`, then, using the usual definition of multiplication, we get

`2xx0/0 = (2xx0)/0 = 0/0 = 1` and also

`2xx0/0 = 2xx1 = 2`, so we can prove that `1=2`. This is not a useful result.

In fact we can use any number `x` in place of `2` to show that: if `0/0=1` (and we keep our definition of multiplication) then `x=1`.

In fact any attempt to define division by `0` for any numerator will result in destroying the number system.

In fact we cannot even say that limits of the form `0/0` evaluate to `1` unless we are prepared to lose tangent lines and rates of change. Limits of difference quotients would all evaluate to `1`.

Answer 2

`0/0` is undefined

Explanation:

Jim has covered this quite well, so I will add little more.

Here are some spurious proofs for illustration/consideration:

`color(white)()`
"Proof" 1

`x/x = 1` for any number `x`, so surely `0/0 = 1` for consistency.

Division by `0` is always undefined.

`color(white)()`
"Proof" 2

`x^0 = 1` for any number `x`

So `x/x = x^1 * x^(-1) = x^(1-1) = x^0 = 1` for any `x`, in particular `0`

This is a thinly disguised attempt to justify `0/0 = 1` using the convention that `0^0 = 1`, just like `x^0 = 1` for any `x != 0`

Answer 3

The question asks to prove/disprove something which was never defined.

Explanation:

This is how I was taught.

The problem has been posed with the assumption that dividing by 0 is a legitimate operation.

By definition, division operation is opposite of multiplication operation. e.g.,

If `c` times `b` equals `a`, can be written symbolically as

`c times b = a`
then `a` divided by `b` equals `c`, can be written as

`a/b = c` for all values of `b` except for `b=0`.
We must remember that `a, b and c` are unique numbers.

Division with 0 was never defined.

Moreover, the answer resides in the question itself.
Usually anything divided by 0 is undefined

and 'anything' includes all numbers including 0

The ancient Samskrit text which is treated as definition of zero also did/does not talk of division by zero.

Answer 4

I agree that `0/0` is undefined
but it raises the question of whether it is (arbitrarily) definable.

Explanation:

If division is defined as the opposite of multiplication
so that `a div b = c` means `cxxb=a`
then
`color(white)("XXX")a div 0` for `a!=0`
is quite different from
`color(white)("XXX")a div 0` for `a=0`

If `a!=0` then there is no possible value that could be defined for `c` such that
`color(white)("XXX")cxx0=a`
however if `a=0` we could define `c` to be some (perhaps arbitrary) value and maintain consistency.

Answer 5

`0/0` is undefined

Explanation:

`0/0` is undefined..

One could explain this by saying there are 3 rules in place here.

"Zero divided by anything is equal to 0"

`0/5 =0," "0/25 = 0, " "0/(-14) = 0` etc

"Anything divided by itself is equal to 1.

`5/5 = 1, " "37/37 = 1, " " (-12)/(-12) = 1` etc

"Division by 0 is not permissible/undefined"
Dividing by 0 actually gives infinity as the answer, but infinity is not a number.

So, we have 3 possible answers using valid maths concepts.

So which is it? `"Is" 0/0 =0?," Is" 0/0 = 1?" Is" 0/0= "infinity"?`

No-one knows, so it best to just say that `0/0` is undefined.

Please refer to the link given below/