Floating-point numbers cannot represent all real numbers. This is known as floating point inaccuracy.
There are infinitely many floating points numbers and they can be infinitely long (e.g.
π), thus being able to represent them perfectly would require infinitely amount of memory. Seeing this was a problem, a special representation for "real number" storage in computer was designed, the IEEE 754 standard. In short, it describes how computers store this type of numbers, with an exponent and mantissa, as,
floatnum = sign * 2^exponent * mantissa
With limited amount of bits for each of these, only a finite precision can be achieved. The smaller the number, smaller the gap between possible numbers (and vice versa!). You can try your real numbers in this online demo.
Be aware of this behavior and try to avoid all floating points comparison and their use as stopping conditions in loops. See below two examples:
>> 0.1 + 0.1 + 0.1 == 0.3 ans = logical 0
It is poor practice to use floating point comparison as shown by the precedent example. You can overcome it by taking the absolute value of their difference and comparing it to a (small) tolerance level.
Below is another example, where a floating point number is used as a stopping condition in a while loop:**
k = 0.1; while k <= 0.3 disp(num2str(k)); k = k + 0.1; end % --- Output: --- 0.1 0.2
It misses the last expected loop (
0.3 <= 0.3).
x = 0.1 + 0.1 + 0.1; y = 0.3; tolerance = 1e-10; % A "good enough" tolerance for this case. if ( abs( x - y ) <= tolerance ) disp('x == y'); else disp('x ~= y'); end % --- Output: --- x == y
Several things to note:
yare treated as equivalent.
Nis some problem-specific number. A reasonable choice for
N, which is also permissive enough, is
1E2(even though, in the above problem
N=1would also suffice).
See these questions for more information about floating point inaccuracy: