Foundations — Significant figures and rounding rules

Before you can trust the parent note on significant figures, you need every tool it quietly assumes. This page builds each one from nothing: plain words → a picture → why the topic can't live without it. Read top to bottom; each idea leans on the one above it.

0 · What is a "digit", really?

Plain words. A digit is one of the ten symbols $0,1,2,3,4,5,6,7,8,9$. A number is just a row of digits, and where a digit sits tells you its size.

The picture. Think of a number as boxes on a ruler of powers of ten. Each box is worth ten times the box on its right.

Why the topic needs it. Significant figures are a rule about which digits carry real information. You can't reason about "which digits" until you can name every digit and its place value — the box it sits in. Look at the figure: the digit $3$ in the "tens" box means $30$, but the same $3$ in the "hundredths" box means $0.03$. Same symbol, different worth. That "place" idea is the whole game.

0.5 · The sign symbols $+$ and $-$

Plain words. Before we go further, two tiny symbols do a lot of work:

• The plus sign $+$ marks a number as positive (right of zero on the number line) and, between two numbers, means "add them."
• The minus sign $-$ marks a number as negative (left of zero) and, between two numbers, means "subtract."

When a sign is glued to a lone number ($-2$, $+3$) it tells you which side of zero you are on. When it sits between two numbers ($5-2$) it is an operation. Same mark, two jobs — exactly like the two jobs of a digit.

The multiplication mark $\times$. The symbol $\times$ between two numbers means "multiply" — repeated addition, e.g. $3\times 4=4+4+4=12$. We will use it constantly in $N\times 10^n$, so it needs a name now.

The transformation arrow $\to$. When you see $a\to b$ in these notes it simply means "$a$ becomes / is rewritten as $b$" — the same value shown in a new form, with nothing about it changed in worth. It is a shorthand for "turns into," not a calculation.

Why the topic needs it. Exponents can be negative ($10^{-2}$), values can be negative, and scientific notation is built on the $\times$ mark. You cannot read a single formula below without these three symbols nailed down.

1 · The decimal point and the number line

Plain words. The decimal point is the marker . that separates the whole part (ones and up) from the fractional part (tenths and down).

The picture. Every number is a single point on a horizontal line. Zero sits in the middle; positive ($+$) numbers go right, negative ($-$) go left. A measurement like $12.3$ is a dot at a spot on that line — but a real dot has a fuzzy edge, because we are never infinitely sure.

Why the topic needs it. The parent note keeps saying a digit is "uncertain." On the number line that means: the true value lies somewhere inside a small band around our dot. The width of that band is what significant figures secretly track. See Accuracy vs precision for how wide vs. off-centre that band can be.

2 · Zero — the trickiest character

Plain words. Zero means "nothing in this box." But that innocent job makes zero do two very different things:

1. A measured zero — you looked, and the box really held nothing. This carries information.
2. A placeholder zero — it only exists to push other digits into the right box. This carries no precision, only position.

The picture. In $0.0042$, the two zeros after the point are doing nothing but sliding the $4$ and $2$ into the thousandths/ten-thousandths boxes. In $4.20$, that last zero says "I checked the hundredths box and it was empty" — a real observation.

Why the topic needs it. The single hardest sig-fig rule ("leading zeros don't count, trailing decimal zeros do") is entirely about telling these two zeros apart. Master this figure and rules 3–5 in the parent note become obvious.

3 · Powers of ten and $10^n$

Plain words. The notation $10^n$ means "multiply ten by itself $n$ times." The little raised number $n$ is the exponent; it just counts how many times you multiply.

$10^3=10\times 10\times 10=1000,10^0=1,10^{-2}=\frac{1}{10\times 10}=0.01$

What can $n$ be? The exponent $n$ can be any integer — that means any of $\dots ,-3,-2,-1,0,1,2,3,\dots$: the positive whole numbers, zero, and the negatives. We write this as $n\in \mathbb{Z}$. Here the symbol $\in$ means "is a member of / belongs to" (picture $n$ as one ball dropped into a bucket), and $\mathbb{Z}$ is just shorthand for the bucket holding all the integers. So $n\in \mathbb{Z}$ reads aloud as "$n$ is one of the integers." A positive $n$ multiplies (makes big numbers); $n=0$ does nothing and lands on $1$; a negative $n$ divides (makes small numbers). All three appear above, so $n$ is deliberately not limited to the non-negative "whole numbers."

Why $10^n$ and not just writing the number out? Because it lets you separate which digits are real from where the decimal sits. That is exactly the surgery significant figures need. When you write $1.50\times 10^3$, the part $1.50$ holds only real measured digits, and $10^3$ holds only the position. The two jobs of a digit — value and placeholder — get split into two different places. That is why the parent note says scientific notation "removes all ambiguity."

Why does sliding the decimal by $k$ places equal multiplying/dividing by $10^k$? Here $k$ is just a name for how many places you slide — a plain counting number ($k=1,2,3,\dots$). Moving a digit one box to the left multiplies its worth by ten and one box to the right divides it by ten — that is the definition of place value from Section 0. Sliding the decimal point right by $k$ places shifts every digit $k$ boxes left, so every digit's worth is multiplied by ten, $k$ times over — i.e. by $10^k$. Slide left by $k$ and every digit drops $k$ boxes, dividing the whole number by $10^k$, which is the same as multiplying by $10^{-k}$. The slide is the multiplication, digit by digit.

A negative exponent ($-$, from Section 0.5) just means "divide" — it slides the decimal left instead of right. So $10^{-2}$ is "how many hundredths," a shrinking machine. Full treatment lives in Scientific notation and orders of magnitude.

4 · Scientific notation $N\times 10^n$

First, the vertical bars $|\dots |$. The two upright strokes in $|N|$ mean absolute value — "the size of the number, ignoring its sign," i.e. its distance from zero on the number line. So $|-7|=7$ and $|7|=7$; both sit $7$ steps from zero. We use $|N|$ so the rule below works whether $N$ is positive or negative.

Plain words. Any non-zero number can be written as $\underset{\text{the real digits}}{\underbrace{N}}\times \underset{\text{the position}}{\underbrace{10^n}}$ where $N$ is a number with exactly one non-zero digit before the decimal point (so its size obeys $1\le |N|<10$), and $n$ is any integer ($n\in \mathbb{Z}$ — positive, zero, or negative, exactly as in Section 3). The $\times$ is the multiply mark from Section 0.5.

The picture (mental). Grab the decimal point and slide it until only one non-zero digit sits to its left. Count the slides: that count (with a sign) is $n$. Slide left → $n$ positive; slide right → $n$ negative. By Section 3, each slide is exactly one multiply/divide by ten, so the value never changes — you have only repackaged it. (Recall from Section 0.5 that the arrow $\to$ just means "becomes.")

Why the topic needs it. The parent note's rule 5 ("trailing zeros in a whole number are ambiguous") is cured only by this notation. It is the topic's escape hatch.

5 · Fractions, division, and the bar $\frac{a}{b}$

Plain words. The bar $\frac{a}{b}$ means "$a$ shared into $b$ equal parts" — it is division written vertically. The top ($a$) is the numerator, the bottom ($b$) the denominator.

The picture. Density in the parent note is $\rho =\frac{\text{mass}}{\text{volume}}$: how much stuff sits in each unit of space. Cut the mass into as many equal shares as there are millilitres — one share per mL is the density.

Why the topic needs it. Example 5 in the parent note divides a 4-sig-fig mass by a 2-sig-fig volume. To apply the min-sig-figs rule you must first recognise the bar as a division and know its top and bottom are separate measurements with separate precisions. Division and its precision-tracking cousin live in Density and derived quantities and Units and dimensional analysis.

6 · Comparing quantities: $<$, $>$, $=$, and $\min$

Plain words.

• $a<b$ means "$a$ is left of $b$ on the number line" (smaller).
• $a>b$ means "$a$ is right of $b$" (bigger).
• $a=b$ means "same point."
• $\min (a,b,\dots )$ means "pick the smallest of the listed numbers."

Why the topic needs it. The rounding rule compares the dropped digit $d$ against $5$ using $<$ and $>$. The multiplication rule is literally $\text{answer sig figs}=\min (\text{each factor’s sig figs})$ — "the weakest link wins." You cannot read either rule without these three symbols.

7 · Rounding — turning "know" into "honestly claim"

Plain words. To round is to replace a number with a nearby simpler one that carries fewer digits. You look at the first digit you are about to throw away — call it the decider $d$ — and decide whether the kept part goes up or stays.

The full decider rule. Using $<$ and $>$ from Section 6:

• If $d<5$: round down — leave the kept digit unchanged (throw the rest away).
• If $d>5$: round up — add $1$ to the last kept digit.
• If $d=5$ with non-zero digits after it: round up (you are past halfway).
• If $d=5$ with nothing (or only zeros) after it: round to the nearest even kept digit (banker's rounding — the fair tie-breaker).

The picture. On the number line, rounding snaps your fuzzy dot to the nearest tick mark (Section 1) at your chosen precision — and remember, the tick spacing is the place value you decided to keep. If the dot is exactly halfway between two ticks (the $d=5$ case), you need a fair tie-breaker; that is why the tie sends half of all cases up and half down, so no bias creeps in.

Why the topic needs it. Every worked example ends in a rounding step; this is the mechanical act that enforces the honesty promise.

Prerequisite map

Read it top-down: the plainest ideas (a digit, its box, the sign marks) feed the decimal point and powers of ten; those meet in scientific notation; comparison and division join in — and all of it pours into significant figures and rounding.

Equipment checklist

Test yourself — cover the right side and answer each before revealing.

Once every line reveals a match, you are ready for the parent note.