java - Why does for loop using a double fail to terminate

Question

I'm looking through old exam questions (currently first year of uni.) and I'm wondering if someone could explain a bit more thoroughly why the following for loop does not end when it is supposed to. Why does this happen? I understand that it skips 100.0 because of a rounding-error or something, but why?

for(double i = 0.0; i != 100; i = i +0.1){
    System.out.println(i);
}

Answer 1

The number 0.1 cannot be exactly represented in binary, much like 1/3 cannot be exactly represented in decimal, as such you cannot guarantee that:

0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1==1

This is because in binary:

0.1=(binary)0.00011001100110011001100110011001....... forever

However a double cannot contain an infinite precision and so, just as we approximate 1/3 to 0.3333333 so must the binary representation approximate 0.1.

Expanded decimal analogy

In decimal you may find that

1/3+1/3+1/3
=0.333+0.333+0.333
=0.999

This is exactly the same problem. It should not be seen as a weakness of floating point numbers as our own decimal system has the same difficulties (but for different numbers, someone with a base-3 system would find it strange that we struggled to represent 1/3). It is however an issue to be aware of.

Demo

A live demo provided by Andrea Ligios shows these errors building up.