Why 1/3 as a double is 0.33333333333333331
The closest way to represent 1/3 in binary is like this:
0.0101010101…
That’s the same as the series 1/4 + (1/4)^2 + (1/4)^3 + (1/4)^4…
Of course, this is limited by the number of bits you can store in a double. A double is 64 bits, but one of those is the sign bit and another 11 represent the exponent (think of it like scientific notation, but in binary). So the rest, which is called the mantissa or significand is 52 bits. Assume a 1 to start and then use two bits for each subsequent power of 1/4. That means you can store:
1/4 + 1/4^2 + … + 1/4 ^ 27
which is 0.33333333333333331
Why multiplying by 3 rounds this to 1
So 1/3 represented in binary and limited by the size of a double is:
0.010101010101010101010101010101010101010101010101010101
I’m not saying that’s how it’s stored. Like I said, you store the bits starting after the 1, and you use separate bits for the exponent and the sign. But I think it’s useful to consider how you’d actually write it in base 2.
Let’s stick with this “mathematician’s binary” representation and ignore the size limits of a double. You don’t have to do it this way, but I find it convenient. If we want to take this approximation for 1/3 and multiply by 3, that’s the same as bit shifting to multiply by 2 and then adding what you started with. This gives us 1/3 * 3 = 0.111111111111111111111111111111111111111111111111111111
But can a double store that? No, remember, you can only have 52 bits of mantissa after the first 1, and that number has 54 ones. So we know that it’ll be rounded, in this case rounded up to exactly 1.
Why for decimal you get 0.9999999999999999999999999999
With decimal, you get 96 bits to represent an integer, with additional bits representing the exponent up to 28 powers of 10. So even though ultimately it’s all stored as binary, here we’re working with powers of 10 so it makes sense to think of the number in base 10. 96 bits lets us express up to 79,228,162,514,264,337,593,543,950,335, but to represent 1/3 we’re going to go with all 3’s, up to the 28 of them that we can shift to the right of the decimal point: 0.3333333333333333333333333333.
Multiplying this approximation for 1/3 by 3 gives us a number we can represent exactly. It’s just 28 9’s, all shifted to the right of the decimal point: 0.9999999999999999999999999999. So unlike with double’s there’s not a second round of rounding at this point.