Why is 0 divided by 0 an error?

Question

I have come across this problem in a calculation I do in my code, where the divisor is 0 if the divident is 0 too. In my code I return 0 for that case. I am wondering, while div

Answer 1

When you type in zero divided by zero, there's an error because whatever you multiply zero from will be zero so it could be any number.

Answer 2

Let's say:

0/0 = x

Now, rearranging the equation (multiplying both sides by 0) gives:

x * 0 = 0

Now do you see the problem? There are an infinite number of values for x as anything multiplied by 0 is 0.

Answer 3

Another explanation of why 0/0 is undefined is that you could write:

0/0 = (4 - 4)/0 = 4/0 - 4/0

And 4/0 is undefined.

Answer 4

Here's a full explanation:

http://en.wikipedia.org/wiki/Division_by_zero

( Including the proof that 1 = 2 :-) )

You normally deal with this in programming by using an if statement to get the desired behaviour for your application.

Answer 5

You may want to look at Dr. James Anderson's work on Transarithmetic. It isn't widely accepted.

Transarithmetic introduces the term/number 'Nullity' to take the value of 0/0, which James likens to the introduction 'i' and 'j'.

Answer 6

The problem is with the denominator. The numerator is effectively irrelevant.

10 / n
10 / 1 = 10
10 / 0.1 = 100
10 / 0.001 = 1,000
10 / 0.0001 = 10,000
Therefore: 10 / 0 = infinity (in the limit as n reaches 0)

The Pattern is that as n gets smaller, the results gets bigger. At n = 0, the result is infinity, which is a unstable or non-fixed point. You can't write infinity down as a number, because it isn't, it's a concept of an ever increasing number.

Otherwise, you could think of it mathematically using the laws on logarithms and thus take division out of the equation altogther:

    log(0/0) = log(0) - log(0)

BUT

    log(0) = -infinity

Again, the problem is the the result is undefined because it's a concept and not a numerical number you can input.

Having said all this, if you're interested in how to turn an indeterminate form into a determinate form, look up l'Hopital's rule, which effectively says:

f(x) / g(x) = f'(x) / g'(x)

assuming the limit exists, and therefore you can get a result which is a fixed point instead of a unstable point.

Hope that helps a little,

Tony Breyal

P.S. using the rules of logs is often a good computational way to get around the problems of performing operations which result in numbers which are so infinitesimal small that given the precision of a machine’s floating point values, is indistinguishable from zero. Practical programming example is 'maximum likelihood' which generally has to make use of logs in order to keep solutions stable