·102 min read·foundation
Floating Point: Designing a Number System from 32 Bits
Start with a blank 32-bit register and one question: how do you encode real numbers in binary? Each piece of IEEE 754 float32, the sign bit, biased exponent, implicit leading 1, subnormals, arrives as the only reasonable fix to a concrete problem. Then connect representation to arithmetic: the alignment step that erases small terms during addition, rounding that breaks associativity, catastrophic cancellation, and why 0.1 + 0.2 returns 0.30000000000000004.