Can you represent the number 0.1 exactly? Remember that MATLAB does not store the exact representation of numbers. Instead, it uses a binary storage form. An exact numeric storage would quickly become impossible to achieve, because numbers would very quickly grow in size/length, and the symbolic computations would become impossible. This means you have no real choice but to use double precision arithmetic, or something like it, where only a reasonably finite number of bits are used to store any number. And doubles use an IEEE standard binary store scheme, where only 52 bits are used to store the mantissa. But in binary, just like you cannot represent the number 1/3 exactly as a decimal, you cannot represent the number 1/10 exactly.
The representation of 0.1 in binary is the infinite sequence...
0.00011001100110011001100110011001100110011001100110011010...
where the ones represent negative powers of 2. We could try this:
P = [-4 -5 -8 -9 -12 -13 -16 -17 -20 -21 -24 -25 -28 -29 -32 -33 -36 -37 -40];
And you see it is not exactly 0.1, any more than 0.33333333333 is exactly 1/3. And no matter how far out you go, you will never be able to stop. There is never a point where you will find a binary representation of 0.1. And this is true of almost all fractions you might write. The only fractions that are exactly representable in binary are simple things like
0.375 == 3/8 == 1/4 + 1/8
