Why can't I just report all the digits my calculator shows?

Because those extra digits are fiction. If you divided a mass known to three digits by a volume known to three digits, the machine still prints eight or ten digits, but no measurement in the problem justified them. Reporting them claims a level of certainty your equipment never delivered, which in a real lab would mislead anyone reading your data.

When do I use the decimal-place rule instead of the significant-figure rule?

Use decimal places for addition and subtraction, and total significant figures for multiplication and division. In a problem with both, apply each rule at the step where that operation happens, in normal order of operations. The precision that survives one step becomes the input to the next.

Do conversion factors and constants limit my significant figures?

Defined conversions such as 1\ kg = 1000\ g or 1\ min = 60\ s are exact and never limit the answer. Counted objects are also exact. Measured constants and experimentally determined factors do count, so use enough of their digits that they are not the limiting value — carrying one extra digit for a constant is standard practice.

Is 4500 two significant figures or four?

As written it is ambiguous, and most courses read it as two, since there is no decimal point to make the trailing zeros meaningful. To state the precision clearly, use scientific notation: 4.5 × 10^3 for two significant figures, 4.50 × 10^3 for three, or 4.500 × 10^3 for four. Writing 4500. with a trailing decimal point is another way to signal four.

CHEM-1.3

Measurement, SI Units & Significant Figures

Master SI base units, metric prefixes, and significant figure rules for multiplication, division, addition, and subtraction in high-school chemistry.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Measurement, SI Units & Significant Figures, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Measurement, SI Units & Significant Figures free

What this lesson covers

Every number a chemist writes carries two pieces of information: how big the quantity is and how well it was measured. A mass reported as 12 g and a mass reported as 12.000 g describe very different laboratory situations, even though both round to the same value. This lesson gives you the vocabulary and the rules to say exactly what your instruments actually told you.

You will learn the seven SI base units and the metric prefixes that scale them, how to count significant figures in any measurement, and the two different rounding rules that apply to multiplication/division versus addition/subtraction. These skills are not busywork — in the next lesson on density and dimensional analysis, every answer you calculate must be reported with the right unit and the right number of digits, and in later units the same rules govern molar mass, concentration, and gas law calculations.

SI Base Units and Metric Prefixes

The International System of Units (SI) defines a small set of base units from which all other units are built. Chemistry uses these constantly:

Quantity	Base unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Electric current	ampere	A
Luminous intensity	candela	cd

Derived units combine base units: volume in cubic meters (

\text{m}^3

), density in

\text{g/cm}^3

, energy in joules (

1\ \text{J} = 1\ \text{kg}\cdot\text{m}^2/\text{s}^2

). In the lab you will also meet the liter (

1\ \text{L} = 1000\ \text{cm}^3

) and the Celsius degree, which are accepted alongside SI.

Prefixes rescale a unit by powers of ten so you never have to write awkward numbers:

Prefix	Symbol	Factor
giga	G	$10^{9}$
mega	M	$10^{6}$
kilo	k	$10^{3}$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	$\mu$	$10^{-6}$
nano	n	$10^{-9}$
pico	p	$10^{-12}$

Two traps show up repeatedly. First, the SI base unit of mass is the kilogram, not the gram — the prefix is already built in. Second, students reverse the direction of a conversion: since

1\ \text{mm} = 10^{-3}\ \text{m}

, a length of

5\ \text{mm}

0.005\ \text{m}

, a smaller number, because millimeters are smaller units and it takes many of them to make one meter. Always ask whether the new unit is bigger or smaller and check that the number moves the opposite way.

Precision, Accuracy, and Where Significant Figures Come From

Accuracy is how close a measurement lies to the true value. Precision is how closely repeated measurements agree with one another, and it reflects the fineness of the instrument. A balance that reads to 0.01 g is more precise than one reading to 0.1 g; neither is automatically accurate, since a miscalibrated balance can give five tightly clustered readings that are all wrong.

Significant figures are the written record of precision. When you read an analog instrument such as a graduated cylinder or a ruler, you record every digit you know with certainty plus one final estimated digit. If a cylinder is marked every 1 mL and the meniscus sits a bit past the 24 mark, you write 24.3 mL: the 24 is certain, the 3 is your estimate. That last uncertain digit is significant and must be included — dropping it throws away real information about the measurement.

Counting rules:

All nonzero digits are significant. Zeros between nonzero digits (captive zeros) are significant: 5008 has four. Leading zeros are never significant; they only place the decimal point, so 0.0042 has two. Trailing zeros are significant only if a decimal point is present: 4.500 has four, while 4500 is ambiguous and is usually read as two.

Scientific notation removes that ambiguity entirely. Writing

4.5 \times 10^{3}

shows two significant figures and

4.500 \times 10^{3}

shows four. When a measurement's precision matters, use scientific notation.

Exact numbers — counted objects and defined relationships such as

1\ \text{km} = 1000\ \text{m}

or 12 items in a dozen — have infinite significant figures and never limit an answer.

The Two Rounding Rules

Calculated results cannot be more precise than the measurements that produced them, but the rule you apply depends on the operation.

For multiplication and division, the answer keeps the same number of significant figures as the measurement with the fewest significant figures.

4.72\ \text{cm} \times 2.1\ \text{cm} = 9.912 \to 9.9\ \text{cm}^2

The factor 2.1 has only two significant figures, so the product gets two.

For addition and subtraction, the answer keeps the same number of decimal places as the measurement with the fewest decimal places. You count place value here, not total digits.

12.11\ \text{g} + 8.9\ \text{g} + 0.235\ \text{g} = 21.245 \to 21.2\ \text{g}

The 8.9 is known only to the tenths place, so the sum stops at tenths — even though 8.9 has fewer significant figures than the other two values.

Operation	Rule based on	Example	Reported
$\times$ or $\div$	fewest significant figures	$6.02 \times 1.1$	$6.6$
$+$ or $-$	fewest decimal places	$6.02 + 1.1$	$7.1$

Three habits prevent most errors. Round only at the very end of a multi-step problem; rounding intermediate values propagates error. In a mixed calculation, apply each rule at the step where that operation occurs, tracking the allowed precision as you go. And round properly: look at the first digit being dropped, round up if it is 5 or greater, down otherwise — 21.245 rounded to tenths is 21.2 because the dropped digits begin with 4.

Subtraction of close values destroys precision dramatically:

50.32\ \text{g} - 50.28\ \text{g} = 0.04\ \text{g}

, a result with only one significant figure from two four-figure measurements.

Applying the Rules in Lab Work

In a real procedure you rarely get one clean calculation. Suppose you find the mass of an empty flask, add liquid, find the new mass, and then divide by a measured volume to get density. The mass of liquid comes from a subtraction, so the decimal-place rule sets its precision. The density comes from a division, so the significant-figure rule takes over using the digits that survived the subtraction.

When a problem gives you a conversion factor, decide whether it is exact or measured. Definitions such as

1\ \text{L} = 1000\ \text{mL}

1\ \text{m} = 100\ \text{cm}

, and

1\ \text{kg} = 1000\ \text{g}

are exact by definition and impose no limit. A measured or experimentally determined factor does count.

Units are part of the answer. Write them at every step and cancel them algebraically; if the surviving unit is not the one the question asked for, the setup is wrong no matter how tidy the arithmetic looks. This is exactly the habit that makes dimensional analysis work in the next lesson.

Common places students go wrong:

Reporting a calculator's full display, such as 12.4666667, when the data supported three digits. Applying the significant-figure rule to a sum. Counting leading zeros in values like 0.0250 (that value has three significant figures: 2, 5, and the trailing zero). Rounding partway through and then again at the end. Dropping a trailing zero that carries meaning, so that 3.50 g becomes 3.5 g and quietly claims less precision than the balance provided.

A clear answer states the number, the unit, and no more digits than the measurements justify.

Key terms

SI units.: The International System of Units, the standard set of base units (meter, kilogram, second, kelvin, mole, ampere, candela) used for scientific measurement worldwide.
Metric prefix.: A symbol placed before a unit that multiplies it by a power of ten, such as kilo ( $10^{3}$ ), centi ( $10^{-2}$ ), milli ( $10^{-3}$ ), or nano ( $10^{-9}$ ).
Significant figures.: All digits in a measurement that are known with certainty plus one final estimated digit; they communicate the precision of the instrument used.
Precision.: How closely repeated measurements of the same quantity agree with each other; reflected in the number of digits an instrument can provide.
Accuracy.: How close a measured value is to the accepted or true value.
Exact number.: A counted quantity or defined relationship, such as 24 test tubes or $1\ \text{m}=100\ \text{cm}$ , treated as having infinite significant figures so it never limits a result.
Scientific notation.: Writing a value as a coefficient between 1 and 10 times a power of ten, which shows significant figures unambiguously, as in $4.500 \times 10^{3}$ .
Derived unit.: A unit built from combinations of base units, such as $\text{g/cm}^3$ for density or the joule for energy.

Worked example

A student masses an empty beaker at 45.62 g. After pouring in a salt solution, the beaker and contents have a mass of 78.4 g. The student measures the solution's volume in a graduated cylinder as 31.5 mL. Calculate the density of the solution in grams per milliliter, and express the volume in liters, both to the correct number of significant figures.

Step 1 — Find the mass of the solution by subtraction.

78.4\ \text{g} - 45.62\ \text{g} = 32.78\ \text{g}

This is addition/subtraction, so the decimal-place rule applies. The value 78.4 is known only to the tenths place, while 45.62 goes to hundredths. The answer must stop at the tenths place: 32.8 g. Notice this leaves three significant figures.

Step 2 — Divide mass by volume to get density.

\frac{32.8\ \text{g}}{31.5\ \text{mL}} = 1.041269...\ \text{g/mL}

This is division, so the significant-figure rule applies. The mass has three significant figures and the volume has three, so the answer gets three: 1.04 g/mL.

Step 3 — Convert the volume to liters.

31.5\ \text{mL} \times \frac{1\ \text{L}}{1000\ \text{mL}} = 0.0315\ \text{L}

The relationship

1\ \text{L} = 1000\ \text{mL}

is a definition, so it is exact and does not limit precision. The leading zeros are not significant, so 0.0315 L still has three significant figures — the same information, just a different unit. In scientific notation this is

3.15 \times 10^{-2}\ \text{L}

.

Final answers: density = 1.04 g/mL, volume = 0.0315 L. The most common error here is skipping Step 1's rounding rule and carrying 32.78 g into the division, which would suggest four significant figures the balance never provided.

Practice questions

How many significant figures are in the measurement 0.030400 kg?

Three
Four
Five
Six

Answer: Five

Leading zeros (the 0 before the decimal and the single zero right after it) only locate the decimal point and are not significant. Counting from the first nonzero digit: 3, 0, 4, 0, 0. The captive zero between 3 and 4 is significant, and the two trailing zeros are significant because a decimal point is present. That gives five significant figures. A common wrong answer is three, from mistakenly dropping the meaningful trailing zeros.

Evaluate and report to the correct precision:

(15.7\ \text{cm} + 2.34\ \text{cm}) \times 0.60\ \text{cm}

Answer: $11\ \text{cm}^2$

Work the parentheses first. The sum

15.7 + 2.34 = 18.04

, but addition follows the decimal-place rule: 15.7 is known only to tenths, so the sum is 18.0 cm — three significant figures. Now multiply:

18.0 \times 0.60 = 10.8

. Multiplication follows the significant-figure rule, and 0.60 has only two significant figures (the leading zero does not count, the trailing zero does), so the product is reported with two: 11 cm². Writing 10.8 cm² claims precision the 0.60 measurement cannot support.

Explain why the trailing zero in 6.30 g is significant but the leading zeros in 0.0063 g are not, and rewrite each value in scientific notation.

Answer: The trailing zero in 6.30 g reports that the hundredths place was actually measured, so the value has three significant figures: $6.30 \times 10^{0}$ g. The leading zeros in 0.0063 g only position the decimal point and carry no measurement information, so that value has two significant figures: $6.3 \times 10^{-3}$ g.

Significance is about whether a digit came from the instrument. A balance reading 6.30 g had to resolve the hundredths place to know that digit is zero rather than 1 through 9, so that zero is real data. In 0.0063 g, no instrument produced those zeros; they exist only because we chose to write the number in decimal form rather than scientific notation. Converting to scientific notation makes this visible, since only the meaningful digits appear in the coefficient.

FAQ

Why can't I just report all the digits my calculator shows?: Because those extra digits are fiction. If you divided a mass known to three digits by a volume known to three digits, the machine still prints eight or ten digits, but no measurement in the problem justified them. Reporting them claims a level of certainty your equipment never delivered, which in a real lab would mislead anyone reading your data.
When do I use the decimal-place rule instead of the significant-figure rule?: Use decimal places for addition and subtraction, and total significant figures for multiplication and division. In a problem with both, apply each rule at the step where that operation happens, in normal order of operations. The precision that survives one step becomes the input to the next.
Do conversion factors and constants limit my significant figures?: Defined conversions such as $1\ \text{kg} = 1000\ \text{g}$ or $1\ \text{min} = 60\ \text{s}$ are exact and never limit the answer. Counted objects are also exact. Measured constants and experimentally determined factors do count, so use enough of their digits that they are not the limiting value — carrying one extra digit for a constant is standard practice.
Is 4500 two significant figures or four?: As written it is ambiguous, and most courses read it as two, since there is no decimal point to make the trailing zeros meaningful. To state the precision clearly, use scientific notation: $4.5 \times 10^{3}$ for two significant figures, $4.50 \times 10^{3}$ for three, or $4.500 \times 10^{3}$ for four. Writing 4500. with a trailing decimal point is another way to signal four.

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