Significant Figures

Every time you measure something in a lab, there is a limit to how precisely you can read the instrument. Suppose you weigh an object on a basic platform balance and get $9.4 \text{ g}$. Then you weigh the same object on a high-precision analytical balance and get $9.4213 \text{ g}$. The analytical balance clearly gives more detail, but notice that both readings agree on the digit 9. The digit 4 in the platform balance reading, however, is where the uncertainty begins. How do scientists communicate exactly how much of a measured value they are confident about? That is where significant figures come in.

What Are Significant Figures?

Significant figures (often shortened to “sig figs”) are the meaningful digits in a measured or calculated value. They include all the digits that are known with certainty, plus one final digit that is estimated or uncertain.

Consider a volume reading of $11.2 \text{ mL}$. The “11” part is certain because the measurement clearly falls between 11 and 12. The “.2” is the best estimate of where the reading falls within that range. So the value has three significant figures: two certain digits and one uncertain digit.

Unless stated otherwise, every measurement carries an implied uncertainty of $\pm 1$ in its last digit. Writing $11.2 \text{ mL}$ means the true value lies somewhere between $11.1$ and $11.3 \text{ mL}$.

The Five Rules for Counting Significant Figures

Deciding which digits count as significant follows a clear set of rules. Let’s walk through each one.

Rule 1: Every Non-Zero Digit Is Significant

This is the simplest rule. Any digit from 1 through 9 always counts.

• $285 \text{ cm}$ has three significant figures
• $0.25 \text{ mL}$ has two significant figures (only the 2 and 5 count; the leading zero is addressed by Rule 2)

Rule 2: Leading Zeros Are Not Significant

Zeros that appear before the first non-zero digit serve only as placeholders to show where the decimal point sits. They carry no measurement information.

• $0.03$ has one significant figure (only the 3)
• $0.0052$ has two significant figures (only the 5 and 2)

Think of it this way: writing $0.0052$ is the same as writing $5.2 \times 10^{-3}$. The leading zeros disappear, leaving just the two meaningful digits.

Rule 3: Zeros Between Non-Zero Digits Are Significant

When a zero is sandwiched between two non-zero digits, it is part of the measured value and definitely counts.

• $2.005$ has four significant figures (2, 0, 0, and 5 are all significant)

Rule 4: Trailing Zeros Depend on the Decimal Point

This is the trickiest rule and the one that causes the most confusion:

• If a decimal point is present, trailing zeros after it are significant. They indicate that the measurement was precise enough to confirm those digits as zero. For example, $0.200 \text{ g}$ has three significant figures (the trailing zeros tell you the measurement was made to the nearest thousandth of a gram).

• If there is no decimal point, trailing zeros are not considered significant because it is ambiguous whether they were actually measured or are just placeholders. For example, $100$ has only one significant figure.

To remove this ambiguity, scientific notation is the best tool:

Expression	Significant figures
$100$	1
$100.$	3
$100.0$	4
$1 \times 10^2$	1
$1.0 \times 10^2$	2
$1.00 \times 10^2$	3

Notice how scientific notation lets you state exactly how many figures are significant by choosing how many digits to include in the digit term.

Rule 5: Exact (Counting) Numbers Have Infinite Significant Figures

When you count discrete objects, there is no measurement uncertainty at all. If you have 2 balls or 20 eggs, those are exact values. You could write them as $2.000000...$ or $20.000000...$ with as many zeros as you like. Counting numbers never limit the significant figures in a calculation.

Significant Figures in Scientific Notation

A useful consequence of scientific notation: when a number is already written in the form $N \times 10^n$, every digit in $N$ is significant. The power of 10 simply tells you the scale and does not affect the count.

• $4.01 \times 10^2$ has three significant figures
• $8.256 \times 10^{-3}$ has four significant figures

Precision vs Accuracy: Two Different Qualities

When scientists talk about how good their measurements are, they use two words that sound similar but mean very different things.

Precision describes how close your repeated measurements are to each other. If you measure the same thing three times and get nearly identical results each time, your measurements are precise. Precision reflects the consistency of your technique and instrument.

Accuracy describes how close your measurements are to the true or accepted value. You could be extremely consistent (precise) but consistently wrong (inaccurate), perhaps because your instrument is poorly calibrated.

Seeing the Difference in Practice

Suppose the true mass of an object is $2.00 \text{ g}$. Three students each measure it twice:

Student	Measurement 1 (g)	Measurement 2 (g)	Average (g)	Verdict
A	1.95	1.93	1.940	Precise but not accurate
B	1.94	2.05	1.995	Neither precise nor accurate
C	2.01	1.99	2.000	Both precise and accurate

Student A’s readings are tightly grouped (precise), but they sit well below the true value (not accurate). Student B’s readings are far apart from each other (not precise) and their average happens to be close to the true value only by chance (not reliably accurate). Student C’s readings are close together and centred on the true value, making them both precise and accurate.

The ideal in any experiment is to achieve both. Good precision with poor accuracy usually signals a systematic error (something consistently shifting your results in one direction), while poor precision points to random errors (inconsistencies in technique or conditions).

Arithmetic with Significant Figures

When you combine measurements in a calculation, the result cannot be more precise than the least precise input. The rules for how to handle this differ between addition/subtraction and multiplication/division.

Addition and Subtraction: Match the Decimal Places

For addition and subtraction, the result must not have more digits after the decimal point than the input with the fewest digits after the decimal.

Worked example:

$12.11 + 18.0 + 1.012 = 31.122$

Look at the decimal places in each input:

• $12.11$ has two digits after the decimal
• $18.0$ has one digit after the decimal
• $1.012$ has three digits after the decimal

The most limiting input is $18.0$, with just one decimal place. So the sum must be rounded to one decimal place:

$\text{Reported result} = 31.1$

Multiplication and Division: Match the Significant Figures

For multiplication and division, the result must not have more significant figures than the input with the fewest significant figures.

Worked example:

$2.5 \times 1.25 = 3.125$

Count the significant figures in each input:

• $2.5$ has two significant figures
• $1.25$ has three significant figures

The more limiting input is $2.5$ with two significant figures, so the product must be rounded to two significant figures:

$\text{Reported result} = 3.1$

Notice the key difference: addition and subtraction care about decimal places, while multiplication and division care about total significant figures. Mixing up these two rules is one of the most common mistakes students make.

Rounding Off: The Three Rules

Whenever you need to trim a calculated result to the correct number of significant figures, you need to round off. The process depends on the value of the rightmost digit being removed.

Rule 1: Removed Digit Greater Than 5: Round Up

If the digit you are dropping is 6, 7, 8, or 9, increase the preceding digit by one.

Example: Round $1.386$ to three significant figures. The digit being removed is 6 (which is greater than 5), so the 8 becomes 9:

$1.386 \rightarrow 1.39$

Rule 2: Removed Digit Less Than 5: Keep Unchanged

If the digit you are dropping is 1, 2, 3, or 4, the preceding digit stays as it is.

Example: Round $4.334$ to three significant figures. The digit being removed is 4 (which is less than 5), so the preceding 3 stays:

$4.334 \rightarrow 4.33$

Rule 3: Removed Digit Exactly 5: Round to Even

This is the special case. When the digit being removed is exactly 5, check the preceding digit:

• If the preceding digit is odd, round it up (increase by one)
• If the preceding digit is even, leave it as it is

This “round to even” convention (sometimes called banker’s rounding) prevents a systematic upward bias that would occur if you always rounded 5 up.

Example 1: Round $6.35$ to two significant figures. The removed digit is 5. The preceding digit is 3 (odd), so round up:

$6.35 \rightarrow 6.4$

Example 2: Round $6.25$ to two significant figures. The removed digit is 5. The preceding digit is 2 (even), so leave it unchanged:

$6.25 \rightarrow 6.2$

This rule ensures that over many rounding operations, the upward and downward roundings balance out, keeping your accumulated results free from rounding bias.