# 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

##### 5 Answers

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

#### Explanation:

If we attempt to make

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

If

In fact we can use any number

In fact any attempt to define division by

In fact we cannot even say that limits of the form

#### Explanation:

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

Here are some spurious proofs for illustration/consideration:

**"Proof" 1**

Division by

**"Proof" 2**

So

This is a thinly disguised attempt to justify

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

then

We must remember that

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.

I agree that **undefined**

but it raises the question of whether it is (arbitrarily) definable.

#### Explanation:

If division is defined as the opposite of multiplication

so that

then

is quite different from

If

however if

#### Explanation:

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

"Zero divided by anything is equal to 0"

"Anything divided by itself is equal to 1.

"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?

No-one knows, so it best to just say that

Please refer to the link given below/