Can zero be divided by zero?

Question

I understand that one cannot divide a non-zero number by zero because the operation will never be completed because no matter how many zeroes you add up you will never reach a non-zero number. Also it makes no sense to divide something into zero parts.

However, I am wondering about zero divided by zero.

If you take the definition of a/b as "how many times b goes into a" then, at first glance, 0/0 seems to be 1, because the numerator and the denominator are equal.

However, if you use an equation:

Let A = 0/0
Then A * 0 = 0

Every possible value of A satisfies this equation. Hence, you could say that 0/0 can equal any number. This also makes sense if you use the "a/b = how many times b goes into a" definition because zero can go as many times as it wants into zero because no matter how many times you add zero to itself it will always equal zero.

Answer 1

The definition of a/b is not the one you use, but it is defined as the inverse of the multiplicative operator: a^-1 is defined to be the number such that if a*b = c then b = a^-1*x. Definitions in math are chosen very carefully to avoid contradictions.

Your conclusion from your definition is correct: A can be any number. This is just why this is not considered a useful definition. It makes a special case in any proof, which can be avoided by simply defining anything divided by 0 to be undefined.

Have you studied limits? For instance, if you look at the limit of

x/sin^2(x), x^2/sin^2(x), x^3/sin^2(x), each limit is 0/0, but the limits are infinity (unlimited), 1, and 0, so if you allow 0/0 then you can prove that 0 = 1 = infinity.

There have been math systems written using division by 0. There is an exposition of calculus using "infinitesimals", which are numbers smaller than any positive number but greater than 0. These are defined as 1/(infinity), where infinity is 1/0.

One thing important to remember about math is there is not a single one correct math, there are lots of different ones, and you choose the one most useful in your physical system - for instance, the angles of a triangle sum to 180 on the plane, but on other surfaces they can sum to more or less.

Answer 2

Well, you make a good point, and I actually considered the idea that 0/0 could equal anything, too. Most of the time, you won't just be given x=0/0, solve for x; you'll usually only encounter 0/0 if you're considering a function of a variable with a fraction in it, like f(x) = (5x-5)/(x^2-1). If we were to say x=1, then f(x)=0/0. Now, 0/0 still has no meaning, and that would literally just be a hole in the graph. You can, however, use calculus to find what is called the limit as x goes to 1. This is the point that the graph is approaching when it gets close to 1. You will probablly learn about this in higher mathematics.

Answer 3

Such a division would make our counting system a bit inconsistent. However, The symbol 0/0 may be a number when we are talking limits. For example Sin(0)=0 and the limit of (Sin(x))/x with x approaching 0 is an example of "0/0" and equals 1. But here "0/0" is only a symbol, defining the mathematical situation in which we can use other means to find the limit. In this case, de'l Hospital'a rule for example.
But in the sense of plain division, no.

Answer 4

You can't divide anything by nothing.. this includes nothing... I think it's great you're trying to figure this out on your own... You're kind of following the path in which many many many years ago mathematicians followed in trying to figure out many things we know take for granted today (0 wasn't even known as a number until thousands of years after counting and very basic math came around).

Basically, the short answer to your question is no, 0 doesn't divide 0, I can't remember the exact proof behind this, but it doesn't go this way.... I do know that if you're taking the limit of a function that has a numerator and denominator and they both equal 0 when you take your limit, it means you have a lot more steps to take to determine you answer...

Answer 5

branch by skill of 0 is an operation for which you will no longer discover an answer, so it is disallowed. you could comprehend why in case you think of roughly how branch and multiplication are correct. 12 divided by skill of 6 is two because of the fact 6 situations 2 is 12 12 divided by skill of 0 is x could advise that 0 situations x = 12 yet no cost could paintings for x because of the fact 0 situations any extensive style is 0. So branch by skill of 0 would not paintings.

Answer 6

basically, the reason why any number(except zero)cannot be divided by zero is this:
when any number "k" is divided by a non zero number, the smalle the divisor gets the larger the qoutient will be...
as in 5/10=0.5
5/9= 0.555555....
5/6= 0.83333...
.....5/2= 2.5
.....5/0.5= 10
.....5/0.01= 500 to cut the story short, when divisor approaches zero, the quotient approaches infinity(a very large,undefined value).

in the case of 0/0, studies in caculus call this solution, "indeterminate"...which shows that no such solution exists and no possible values may be assumed unlike infinity , of which at least you have an idea.

i may not be able to answer your question but i hope this will help.

Answer 7

interesting theory. but it is impossible as there isn't anything to be divided in the 1st place. use a scientific calculater and the thing will show u math error. so don't go and confuse yourself. no number can be divided by zero let alone zero. it is like taking an empty cup and trying to divide the emptiness in it into infinity empty cups

Answer 8

The definition of a/b is not "how many times b goes into a". It is "a divided into b equal parts." Zero represents nothing, so it is meaningless to divide nothing into any number of equal parts.

Answer 9

Can you divide nothing? No, because it's not there to divide. So how can you divide zero by zero. Zero is nothing.

Answer 10

I admire your thinking it's a beautiful thing to see someone second guess things. But I assure you no zero can not be divided by zero. Think about it like elementary math. If you have no ducks and no pools how many ducks will go into how many pools?