Foundations — Significant figures — rules for operations

Before you can trust the rules in the parent topic, you need every single symbol and idea it quietly assumes. This page builds them from nothing, in the order they lean on one another. Never trust a rule whose pieces you can't name.

0. What is a "digit", really?

A digit is one of the ten marks $0,1,2,3,4,5,6,7,8,9$. When we write a number like $2.3$, each digit sits in a place — a slot that means "how many tens", "how many ones", "how many tenths", and so on.

Look at the figure: every position is worth ten times the one to its right. The decimal point (the dot) is just the fence that separates whole-number places (left) from fractional places (right). This "places" picture is the skeleton every later idea hangs on.

1. Measurement and the "first uncertain digit"

The parent note says a measurement is "known reliably plus the first uncertain digit". Let's see what that means.

The figure shows a ruler marked only in centimetres, with a pencil ending between the $2$ and $3$ marks. You can be sure it's past $2$ — that digit is reliable. But the tenths digit (is it $.2$? $.3$? $.4$?) is a guess by eye — this is the first uncertain digit.

This single idea — "digits are promises of knowledge" — is the seed of the whole topic. Everything else counts and protects these trustworthy digits. See Measurement & uncertainty for the fuller story.

2. Significant figures = the counted promises

We write it as $N$. So $2.3$ has $N=2$; the ruler measurement above carries two trustworthy promises.

The parent's counting rules are all about answering one question: "is this particular zero a promise, or just a spacer?"

The figure sorts three kinds of zero:

• Leading zeros (red, in $0.0042$) — pure spacers that just push the digits into their places. They promise nothing → not significant.
• Sandwiched zeros (green, in $2.05$) — trapped between real digits, so they must be real → significant.
• Trailing zeros after a decimal (blue, in $2.300$) — nobody writes them unless they measured them → significant.

3. Decimal places — a different count for a different job

Sig figs count trustworthy digits. But the addition rule needs a second, separate count.

Why does the topic need two different counts?

• Significant figures measure relative precision — "what fraction of the value do I know?"
• Decimal places measure absolute precision — "down to which physical place-slot do I know?"

Multiplication cares about fractions; addition cares about slots. Hold both counts in your head — the parent's two rules each use exactly one of them.

4. Absolute vs relative — the deepest split

This is the idea that justifies both rules, so it earns its own picture.

The symbol $\Delta$ (a Greek capital "delta") is universal shorthand for "a small amount of" or "the uncertainty in". In the figure, the same $\Delta A=0.1$ is a fat fraction of a small number but a thin fraction of a big one — that's exactly why one measurement can be absolutely precise yet relatively sloppy, or vice versa. This split is unpacked fully in Error propagation — relative vs absolute.

Why fewer digits always means a bigger wobble

The whole reason we can later use $\min$ is that the two counts are monotonic ladders: drop a rung and the uncertainty strictly grows.

Where the propagation formulas come from

We don't just assert that relative errors add for products — you can see it.

Note that multiplication and division share one rule because both combine relative uncertainties in the same way — dividing does not flip the sign of a relative wobble; the fractions still add.

5. The $\min$ symbol

Both rules use $\min$.

Why the topic needs it: Section 4 proved the strict link "fewer digits ⇒ worse wobble". So when errors add, the worst (largest) uncertainty dominates — and it always comes from the number with the fewest trustworthy digits/decimals. Picking that number is exactly picking the smallest count, so "the answer inherits the weakest input" is written compactly as a $\min$.

6. Scientific notation — the ambiguity fixer

The parent flags that $4500$ is ambiguous — are those trailing zeros promises or spacers? You can't tell. Rewriting as $4.50\times 10^3$ makes the count explicit ($N=3$): only the digits you wrote in the mantissa are significant. This is the tool from Scientific notation, and it exists precisely to kill trailing-zero ambiguity.

7. Exact numbers — infinite promises

Why it matters: an exact number can never be the "weakest input", so it never wins the $\min$ and never limits your answer. If you forget this, you'll wrongly shrink good results.

8. Edge case — adding wildly different sizes

The addition rule can do something surprising when the terms differ by many orders of magnitude.

Prerequisite map

The diagram below draws the dependency chain in one glance; here is the same content in words, in case it does not render. Digits and place value (Section 0) are the base. From them you build the idea of a measurement's first uncertain digit (Section 1), which is counted two different ways: as significant figures (Section 2) and as decimal places (Section 3). The absolute uncertainty $\Delta A$ (Section 4) is measured by decimal places; dividing it by the value $A$ gives the relative uncertainty $\Delta A/A$, which sig figs track. The strict "fewer ⇒ worse" link makes the worst input dominate, captured by the $\min$ operation (Section 5). Scientific notation (Section 6) feeds in to fix trailing-zero ambiguity, and exact numbers (Section 7) feed in as inputs that never limit the result. All of these converge on the sig-fig rules for operations — the parent topic.

Each arrow is a dependency: you literally cannot state the multiplication rule until you have sig figs and relative error and $\min$. Notice that relative uncertainty $\Delta A/A$ is built from the absolute uncertainty $\Delta A$ divided by the value $A$ — sig figs merely track how big that fraction is, they don't create it. This map is the reading order in one glance. Related toolkits: Orders of magnitude & estimation and Dimensional analysis.

Equipment checklist

Cover the right side; can you answer before revealing?