Significant figures — rules for operations: Visual walkthrough

We only need one new idea, and we draw it first.

Step 1 — A measurement is a number with a fuzzy edge

WHAT. Before any rule, we redraw what a measured number is. When you read " $4.6 cm$ " off a ruler, you are not pointing at one exact spot — you are pointing at a little fuzzy band on the number line. The true value lives somewhere inside that band.

WHY. Every rule we derive is just bookkeeping of how these fuzzy bands behave when we combine numbers. If we never draw the fuzz, the rules look like arbitrary magic. So the fuzz is the whole story.

PICTURE. Look at the figure. The solid orange tick is the number you wrote down; the pale band is the range the true value could be in. We give the half-width of that band a name:

$x = x_{0} \pm Δ x$

$x_{0}$ — the number you actually wrote down (centre of the band).
$Δ x$ — the half-width of the fuzzy band, called the absolute uncertainty. Small $Δ x$ = sharp measurement.
the " $\pm$ " — "the truth is anywhere from $x_{0} - Δ x$ up to $x_{0} + Δ x$ ".

Figure — Significant figures — rules for operations

Step 2 — Significant figures = a picture of relative fuzz

WHAT. Here we tie "number of significant figures" to the relative band width, so we never have to guess later.

WHY. The parent claimed "a 3-sig-fig number is known to about $0.1$ – $1%$ ." We need to see why, because Rule 1 leans entirely on it.

PICTURE. Two numbers on the same relative scale: $4.6$ (2 sig figs) and $4.63$ (3 sig figs). The last written digit is always the uncertain one, so its band is roughly $\pm$ half a unit of that last place.

$4.6$ : last place is tenths, band $\approx \pm 0.05$ , so $\frac{0.05}{4.6} \approx 1%$ .
$4.63$ : last place is hundredths, band $\approx \pm 0.005$ , so $\frac{0.005}{4.63} \approx 0.1%$ .

Each extra significant figure shrinks the relative band by about ten. That is the key sentence of the page:

Step 3 — Multiply two bands: the relative fuzzes add

WHAT. Take $A = A_{0} \pm Δ A$ and $B = B_{0} \pm Δ B$ and form the product. We track the widest and narrowest possible product.

WHY. We use multiplication (not addition) here because Rule 1 is the multiplication/division rule — and we want to discover what property of $A$ and $B$ survives into $A \times B$ .

PICTURE. The rectangle picture. Area $= A \times B$ . Give $A$ the horizontal side and $B$ the vertical side. The nominal rectangle is $A_{0} B_{0}$ . Nudge each side outward by its band and the extra area is the shaded L-shaped strip.

$(A_{0} + Δ A) (B_{0} + Δ B) = A_{0} B_{0} + A_{0} Δ B + B_{0} Δ A + tiny — drop it Δ A Δ B$

$A_{0} B_{0}$ — the rectangle you meant to draw.
$A_{0} Δ B$ — the thin strip added along the top.
$B_{0} Δ A$ — the thin strip added along the side.
$Δ A Δ B$ — the little corner square. Two smalls multiplied → negligibly small, so we discard it. (Why allowed: if each fuzz is $1%$ , the corner is $0.01%$ .)

Step 4 — Divide by $A_{0} B_{0}$ : sig figs pop out

WHAT. Divide the whole growth $A_{0} Δ B + B_{0} Δ A$ by the nominal area $A_{0} B_{0}$ to get the relative growth.

WHY. Step 2 told us significant figures track the relative band. So to connect to sig figs we must convert absolute growth into relative growth — that is exactly dividing by $A_{0} B_{0}$ .

PICTURE. The two strips of Step 3, each re-labelled as a fraction of the whole rectangle.

$\frac{Δ ( A B )}{A B} = \frac{A _{0} Δ B + B _{0} Δ A}{A _{0} B _{0}} = \frac{Δ A}{A _{0}} + \frac{Δ B}{B _{0}}$

$\frac{Δ ( A B )}{A B}$ — relative fuzz of the answer.
$\frac{Δ A}{A _{0}}$ — relative fuzz of $A$ (its side-strip over its own side).
$\frac{Δ B}{B _{0}}$ — relative fuzz of $B$ .

Relative fuzzes simply add. Now read it through Step 2's lens: the answer's relative fuzz is at least as big as the worst input's relative fuzz. The worst relative fuzz = the fewest significant figures. So:

Step 5 — Add two bands: the absolute fuzzes add

WHAT. Now stack the numbers instead of multiplying. Slide $A$ 's band and $B$ 's band along the same number line and read off the combined band.

WHY. Addition/subtraction is a totally different geometric operation — sliding, not scaling. We must check whether the same "relative" logic survives. Spoiler: it does not.

PICTURE. Two bands laid end-to-end. The widest possible sum uses both upper edges; the narrowest uses both lower edges. The combined half-width is the two half-widths added:

$Δ (A + B) = Δ A + Δ B$

$Δ (A + B)$ — half-width of the sum's band.
$Δ A, Δ B$ — the two input half-widths, in the same units, laid tip-to-tip.

Notice: nowhere did we divide. The thing that adds is the absolute width in physical units — which on the number line is the decimal place of the last trustworthy digit. No sig figs in sight.

Step 6 — Why decimal places, not sig figs, for sums

WHAT. Translate "the biggest absolute band wins" into a rule you can apply by eye.

WHY. We want a shortcut that needs no $\pm$ arithmetic — just look at the written numbers.

PICTURE. A vertical stack aligned on the decimal point (like the parent's addition figure). The term whose last digit sits furthest left (fewest decimals) has the fattest absolute band. Everything to the right of that column is already noise, so the answer must stop there.

Step 7 — The degenerate cases (never get ambushed)

WHAT. Four edge situations the two rules must survive.

WHY. The contract: the reader must never meet a scenario we didn't show.

PICTURE. Four mini-panels, one band-picture each.

Exact / counting numbers. The " $2$ " in $2 π r$ , or " $n = 6$ trials". Its band has zero width ( $Δ = 0$ ). In Step 4 it contributes $0$ to the relative sum; in Step 5 it contributes $0$ to the absolute sum. So it never limits the answer — infinite sig figs.
Subtraction of near-equal numbers ("catastrophic cancellation"). $5.03 - 5.01 = 0.02$ . Absolute bands ( $\pm 0.005$ each) add to $\pm 0.01$ — half the size of the answer itself! Rule 2 keeps 2 decimals ( $0.02$ ) but relative precision collapsed. Flag it: this is where you switch to real error propagation.
Multiplying by something $\approx 0$ . If $A_{0} \to 0$ , the relative fuzz $Δ A / A_{0}$ blows up. The product is essentially all noise; report $0$ with a warning, not a tidy sig-fig count.
Trailing-zero ambiguity in a sum result. If a sum lands on $115.0$ , the trailing zero is significant (it sits at the last honest decimal). Use Scientific notation if you must move it.

The one-picture summary

Two operations, two geometries, two "what adds":

Multiply → scale a rectangle → relative fuzzes add → count SIG FIGS.
Add → slide along a line → absolute fuzzes add → count DECIMAL PLACES.

Recall Feynman: the whole walkthrough in plain words

Every measured number is really a little smudge on a ruler, not a perfect dot — a centre with a fuzzy band around it. When you multiply two smudges you're really building a rectangle out of two fuzzy sides, and the percentage fuzz of each side just piles up in the area. Percentage fuzz is exactly what "significant figures" counts, so the answer inherits the sloppiest side's sig-fig count. When you add two smudges you slide them along the same line, so the widths in real units pile up instead — and width-in-real-units is what "decimal places" means, so the answer stops at the fattest smudge's last decimal. Same fuzz, two different games: scaling versus sliding. Exact counting numbers have zero width, so they sit out both games entirely.

Recall Quick self-check

Why does ×/÷ use sig figs but +/− use decimal places? ::: ×/÷ makes relative fuzzes add (rectangle area), and sig figs track relative fuzz; +/− makes absolute fuzzes add (sliding on a line), and decimal places track absolute fuzz. What happens to an exact number like the "2" in $2 π r$ ? ::: Its band has zero width, so it contributes nothing to either fuzz sum — infinite sig figs, never limits the answer. When does $+ / -$ silently ruin precision? ::: Subtracting near-equal numbers: absolute bands stay the same size but the answer shrinks, so relative precision collapses (catastrophic cancellation).

Connections

Significant figures — rules for operations — the parent note whose two rules we just built.
Error propagation — relative vs absolute — the exact $\pm$ arithmetic behind every band picture here.
Measurement & uncertainty — where the fuzzy band $x_{0} \pm Δ x$ comes from.
Scientific notation — the clean fix for ambiguous trailing zeros (Step 7, case 4).
Orders of magnitude & estimation — when only the biggest band matters at all.
Dimensional analysis — checks what you computed; sig figs check how precisely.