What is the reason of inaccuracy of operations on float numbers?

I wonder why in JavaScript

0.1 + 0.2  // return 0.30000000000000004

4%0.1 // return 0.09999999999999978

http://jsbin.com/oHISAfU/1/edit (Example)

In C the math.h library fmod function

printf("%f", fmod(4.0,0.1));  // print 0.100000

http://ideone.com/RG5Wyv (Example)

And in Spotlight (search feature in the Mac OS X ~ I already submit bug report) that support math operation

4%0.1 = 0.1

It's because 0.1 = 1 / 10 = 1 / (2 × 5) cannot be represented exactly in binary floating point. This happens in C too:

printf("%.20f", fmod(4.0,0.1));

prints 0.09999999999999978351.

Specifically, the only numbers that binary floating point can represent exactly are dyadic fractions of the form a / 2b, where a and b are integers. Even more specifically, for IEEE single precision floating point numbers, a must be between -224 and 224 and b must, in effect, be between -151 and 104. (Or at least approximately so; there are some tricky special cases like denormalized numbers.) For double precision floats, the ranges are wider, but even so, a fraction like 1 / 10 cannot be represented exactly, because the denominator is not a power of two.

StackExchange Computer Science Q#13810, answer score: 15