# 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

Mar 28, 2016

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

#### Explanation:

If we attempt to make $\frac{0}{0}$ 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 $\frac{0}{0} = 1$, then, using the usual definition of multiplication, we get

$2 \times \frac{0}{0} = \frac{2 \times 0}{0} = \frac{0}{0} = 1$ and also

$2 \times \frac{0}{0} = 2 \times 1 = 2$, 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 $\frac{0}{0} = 1$ (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 $\frac{0}{0}$ evaluate to $1$ unless we are prepared to lose tangent lines and rates of change. Limits of difference quotients would all evaluate to $1$.

Mar 28, 2016

$\frac{0}{0}$ is undefined

#### Explanation:

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

Here are some spurious proofs for illustration/consideration:

$\textcolor{w h i t e}{}$
"Proof" 1

$\frac{x}{x} = 1$ for any number $x$, so surely $\frac{0}{0} = 1$ for consistency.

Division by $0$ is always undefined.

$\textcolor{w h i t e}{}$
"Proof" 2

${x}^{0} = 1$ for any number $x$

So $\frac{x}{x} = {x}^{1} \cdot {x}^{- 1} = {x}^{1 - 1} = {x}^{0} = 1$ for any $x$, in particular $0$

This is a thinly disguised attempt to justify $\frac{0}{0} = 1$ using the convention that ${0}^{0} = 1$, just like ${x}^{0} = 1$ for any $x \ne 0$

Mar 29, 2016

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

$\frac{a}{b} = c$ for all values of $b$ except for $b = 0$.
We must remember that $a , b \mathmr{and} c$ 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. Mar 31, 2016

I agree that $\frac{0}{0}$ 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 $c \times b = a$
then
$\textcolor{w h i t e}{\text{XXX}} a \div 0$ for $a \ne 0$
is quite different from
$\textcolor{w h i t e}{\text{XXX}} a \div 0$ for $a = 0$

If $a \ne 0$ then there is no possible value that could be defined for $c$ such that
$\textcolor{w h i t e}{\text{XXX}} c \times 0 = a$
however if $a = 0$ we could define $c$ to be some (perhaps arbitrary) value and maintain consistency.

Sep 14, 2016

$\frac{0}{0}$ is undefined

#### Explanation:

$\frac{0}{0}$ is undefined..

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

"Zero divided by anything is equal to 0"

$\frac{0}{5} = 0 , \text{ "0/25 = 0, " } \frac{0}{- 14} = 0$ etc

"Anything divided by itself is equal to 1.

$\frac{5}{5} = 1 , \text{ "37/37 = 1, " } \frac{- 12}{- 12} = 1$ 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 $\frac{0}{0}$ is undefined.