This page assumes nothing. Before you touch the parent note Floating Point Gotchas, we build every symbol it uses, one brick at a time, each brick resting on the one before it.
A calculator screen has a fixed width. If it can show 4 digits, then 12340000 fits (as 1.234×107) but adding 1 changes nothing — the 1 falls off the right edge. A real computer is the same idea with more digits. Everything below is a precise version of "digits fall off the edge."
A bit is a single 0 or 1. Computers write numbers in base 2 (binary) the same way we write them in base 10 — but the place-values are powers of 2, not 10.
In base 10, 3.14=3+101+1004.
In base 2, (1.011)2=1+0⋅21+1⋅41+1⋅81=1.375.
WHY split it this way? Because this is scientific notation in binary. Storing "digits" and "scale" separately lets one small chunk of memory represent both 10−300 and 10+300. The topic needs this because the exponent decides the size of the gap — and gap size is the whole story.
Because the mantissa has only 52 bits, near the number 1.0 the smallest change you can make is flipping the last bit, worth 2−52. That is the size of the gap right after 1.0.
The picture shows the crucial fact: the gap grows with the number's magnitude. Near 1 the gap is ≈2.2×10−16; near 1016 the gap is about 2; near 1016 the number 1 is smaller than one gap and simply vanishes when added. That single visual explains the parent's "1+1016=1016" absorption.
WHY two? Being off by 1 is disastrous for x=2 (50%) but negligible for x=109 (a billionth). The computer's guarantee is stated in the relative ruler because that is what stays constant across all magnitudes.
HOW to read the symbols:
fl(x) ::: "float of x" — the nearest representable double to x.
δ (delta) ::: the tiny relative fudge, somewhere in [−u,+u].
u=21εmach≈1.1×10−16 ::: because rounding picks the nearer neighbour, you are never off by more than half a gap.
Everything the parent derives — the cancellation amplifier, the summation bound — is algebra applied to this one line.
Read it top-down: raw bits give the format, the format gives the gap, the gap gives εmach, and together with the relative/absolute distinction you get the rounding model — the single equation from which Catastrophic cancellation, Round-off error propagation, Kahan compensated summation and Quadratic formula numerical issues all follow. The whole chain lands in Numerical stability and conditioning.
