### the (64-bit) floating point number line Floating point numbers aren’t evenly distributed. Instead, they’re organized into windows: [0.25, 0.5], [0.5, 1], [1,2], [2,4], [4,8], [8,16], all the way up to [2^1023, 2^1024].

Every window has 2^52 floats in it.

• between -2 and -1
• between -1 and - 1/2
• between - 1/2 and - 1/4
• between - 1/4 and 0
• between 0 and 1/4
• between 1/4 and 1/2
• between 1/2 and 1
• between 1 and 2

the windows go from REALLY small to REALLY big

The window closest to 0 is [2-1023 2-1022].

This is TINY: a hydrogen atom weighs about 2^-76 grams.

The biggest window is [2^1023, 2^1024].

This is HUUUGE: the farthest galaxy we know about is about 2^90 meters away.

the gaps between floats double with every window

• window: [1, 2] gap: 2^-52
• window: [2, 4] gap: 2^-51
• window: [4, 8] gap: 2^-50
• window: [8, 16] gap: 2^-49

why does 10000000000000000.0 + 1 = 10000000000000000.0?

• In the window [2^n, 2^n+1], the gap between floats is 2^n-52
• 10000000000000000.0 is in the window [2^53, 2^54], where the gap is 2^1 (or 2)
• So the next float after 10000000000000000.0 is 10000000000000002.0