title: the gaps between floats

panel 1: floating point numbers have to fit into 32 or 64 bits

This means there are only 2^64 64-bit floats, the same way there are only 2^64 64-bit integers

panel 2: this means floating point numbers have to be spread out

you can imagine them all spaced out on a number line, like this: (picture of a bunch of lines, with small gaps between them. The gaps are smaller on the left and bigger on the right)

panel 3: the gaps start small.

the next 64-bit float after 1.0 is 1 point (lots of 0s) 2

the gap between these two floats is 0 point (lots of 0s) 2, or 2^-52

gaps are always a power of 2

panel 4: the gaps get bigger as the numbers get bigger

the next 64-bit float after 1000000000000000000 is that number plus 16384.

so the gap is 16384, or 2^14!

panel 5: the gaps make calculations inaccurate

when you do math on floating point numbers, often you have to round the result to the nearest float

usually this doesn’t make a big difference, but small mistakes can add up

panel 6: this inaccuracy is inevitable

if you want math to be fast, you have to store the numbers in a fixed number of bits, like 64 bits. So you’re always going to have accuracy issues.