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

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