### 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 252 floats in it.

The windows [-2, -1], [-1, -¹⁄₂], [-¹⁄₂, -¹⁄₄], [-¹⁄₄, 0], [0, ¹⁄₄], [¹⁄₄, ¹⁄₂], [¹⁄₂, 1], and [1, 2], each have 2^52 numbers. [2, 4] has 2^52 numbers. [4, 8] has 2^52 numbers.

Illustration of a horizontal line, with the windows plotted out on it, showing that each window doubles in size as it moves away from zero.

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