What is 0/0?

Question

if anything over itself is 1
if anything over 0 is underfined
if 0 over anything is zero
what is zero over zero

Answer 1

it is an undefined quantity. Any number, even zero, divided by zero is undefined.

it is not infinity, it is not one, and it is not zero. it is undefined. period.

Answer 2

0/0 is undefined yes, but in certain circumstances the limit may be evaluated.
There's a helpful relation between limits in mathematical analysis known as L'Hopital's rule(sorry about the missing accent)
this basically states that if,
lim f(x) / g(x) = 0 / 0
x->a
That is f(a) = 0 and g(a)=0
f(a) / g(a) = f'(a) / g'(a)

That is the ratio of the functions equal the respective ratio of their derivatives. The proof is quite elementary and a simple search will give you a myriad of links.
The most common application is that as x->0 sin x / x = cos x / 1
= 1
The most important feature here is that the proof works even when for x->a, a=0, infinity, or a complex number.
This solves the issue as to 'from where' the function approaches the value that would yield a zero over zero.

eg.
in it's simplest form
as x->0
x/x = 1/1= 1
but -x/x= -1/1 = -1

Zero is a pivotal value somewhat like (if I may speak abstractly) a pole in the number system, standard definitions don't work there.
Also the result of a zero over a zero may be a value other that 1,-1...eg.
lim x-> 45
(sin x - cos x) / x - 45 = 2V2

where x is in degrees.

Answer 3

Division is usually taken to be the opposite of multiplication. So the question is, what number multiplied by 0 is equal to 0? Or algebraically, solve 0 x a = 0. But *every* number is a solution to this! It doesn't make sense to pick a particular value, so we don't define one.

I want to make a remark for people making limit arguments.

If a is any number, then the limit of ax/x as x goes to 0 is a.

Also x^2/x goes to 0, x/x^2 goes to infinity, and -x/x^2 goes to negative infinity.

These are all the "0/0" indeterminate form, and you see you can get any number, including the infinities. You can even get "complex infinities", for example using ix/x^2.

But you wouldn't expect calculus to resolve such a purely algebraic question.

Answer 4

The answer to your question lies in college level calculus classes. The answer is relative to the equation which produces that situation. That is, for f(x)/(g(y) where f(x) = 0 and g(y) = 0.

In most cases, 0/0 is 1, as both factors are approaching 0 from the same "direction." This means that f(x) is approaching 0 from a positive value, and g(y) is similarly approaching 0 from a positive value. Likewise, if both factors are increasing towards 0 from negative values.

Sometimes, however, the answer is, in fact, -1. In that case, both functions are approaching 0 from opposite values. For example, f(x) approaches 0 from a positive value, and g(y) approaches 0 from a negative value.

Answer 5

Maths deals with quantities or magnitudes zero is just a place holder for our number system. Or it can mean no difference.
To have a zero result in a term often means a wrong assumption or priority.
I would love to see a proof that a^0 = 1 that doesn't use circular reasoning or convention (it is a good convention though)

Answer 6

In mathematical analysis, and in particular in elementary calculus certain expressions are indeterminate forms, and must be treated as symbolic only, until more careful discussion has taken place. 0/0 form has no definite meaning, as division by zero is not a meaningful operation in arithmetic.

Answer 7

undefined, but the limit as n->0 of 0/n is zero

Answer 8

Any number divided by zero, including zero itself is still undefined.

Answer 9

It's amazing how many times this question gets asked and how many people answer when they just don't have a clue.

0/0 is UNDEFINED. It is a NO-NO. It is NOT ALLOWED.

Does that make it plain enough?


Doug

Answer 10

0/0 is infinity

Answer 11

The correct logic is:

Anything over itself is 1, EXCEPT 0.

etc. etc.

0 over itself is still 0, because you're basically asking what is nothing divided by nothing.

Ask the reverse... what is infinity over infinity? Or plus infinity? Or multiplied? Same thing ;-)

The answer to your question = 0.

-Daniel