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

Question

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

5 Answers

Impact of this question

5625 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-28T11:53:06

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

Explanation:

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

$2 \times \frac{0}{0} = \frac{2 \times 0}{0} = \frac{0}{0} = 1$ $2xx0/0 = (2xx0)/0 = 0/0 = 1$ and also

$2 \times \frac{0}{0} = 2 \times 1 = 2$ $2xx0/0 = 2xx1 = 2$ , so we can prove that $1 = 2$ $1=2$ . This is not a useful result.

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

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

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

Answer 2 · 2016-03-28T13:57:44

$\frac{0}{0}$ $0/0$ 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

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

Division by $0$ $0$ is always undefined.

$color(white)()$
"Proof" 2

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

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

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

Answer 3 · 2016-03-29T02:02:15

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 $c$ $c$ times $b$ $b$ equals $a$ $a$ , can be written symbolically as

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

$\frac{a}{b} = c$ $a/b = c$ for all values of $b$ $b$ except for $b = 0$ $b=0$ .
We must remember that $a, b and c$ $a, b and c$ 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.
enter image source here

Answer 4 · 2016-03-31T15:51:13

I agree that $\frac{0}{0}$ $0/0$ 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 $a \div b = c$ $a div b = c$ means $c \times b = a$ $cxxb=a$
then
$XXX a \div 0$ $color(white)("XXX")a div 0$ for $a \neq 0$ $a!=0$
is quite different from
$XXX a \div 0$ $color(white)("XXX")a div 0$ for $a = 0$ $a=0$

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

Answer 5 · 2016-09-14T17:01:19

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

Explanation:

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

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

"Zero divided by anything is equal to 0"

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

"Anything divided by itself is equal to 1.

$\frac{5}{5} = 1, \frac{37}{37} = 1, \frac{- 12}{- 12} = 1$ $5/5 = 1, " "37/37 = 1, " " (-12)/(-12) = 1$ 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? $Is \frac{0}{0} = 0 ?, Is \frac{0}{0} = 1 ? Is \frac{0}{0} = infinity ?$ $"Is" 0/0 =0?," Is" 0/0 = 1?" Is" 0/0= "infinity"?$

No-one knows, so it best to just say that $\frac{0}{0}$ $0/0$ is undefined.

Please refer to the link given below/

Science

Math

Humanities

... and beyond

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

5 Answers

Explanation:

Explanation:

Explanation:

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

I just want to make sure about a concept : What is 000/0? Usually anything divided by 00 is undefined, but a past student of mine reckons they were taught it is 11. Is there any theorem or proof of this? Thanks

5 Answers

Explanation:

Explanation:

Explanation:

Explanation:

Explanation:

Related questions

Impact of this question

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