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
Mar 28, 2016

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.

Mar 28, 2016

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

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

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.
enter image source here

Mar 31, 2016

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.

Sep 14, 2016

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/