Significant Figures: The Art of Honest Measurement
Imagine you're measuring the length of a table with a ruler that has marks every millimeter. You see the table edge falls somewhere between 152.3 cm and 152.4 cm. You estimate it as 152.35 cm. But here's the truth: you're certain about 152.3, pretty sure about the 0.05, and guessing about anything beyond that. Significant figures are simply a way to communicate how much of that number you actually know.
The Core Idea
Every measurement has uncertainty. Significant figures (or "sig figs") are the digits in a number that carry meaningful information about its precision. They include all the digits you're sure of, plus one more that you estimate.
Note
A digit is "significant" if removing it would change the precision of the measurement. Zeros can be tricky — they might just be placeholders.
The Rules (Memorize These)
1. Non-zero digits are always significant
123.45 has 5 sig figs. Simple.
2. Zeros between non-zero digits are significant
1002 has 4 sig figs. The zeros are "sandwiched" — they're part of the measurement.
3. Leading zeros are never significant
0.00123 has 3 sig figs. Those zeros just tell you where the decimal point is.
4. Trailing zeros are significant only if there's a decimal point
1200 has 2 sig figs (no decimal — zeros are placeholders)
1200. has 4 sig figs (decimal tells us those zeros were measured)
1200.0 has 5 sig figs
5. Exact numbers have infinite sig figs
If you count 5 apples, that's exactly 5 — no uncertainty. Conversion factors like 1 m=100 cm are exact by definition.
Tip
When in doubt, write the number in scientific notation. 1.20×103 clearly has 3 sig figs, while 1.2×103 has 2.
Why This Matters: Calculations
When you multiply or add measurements, the uncertainty propagates. You can't claim more precision than your least precise measurement.
Multiplication and Division
The result should have the same number of sig figs as the measurement with the fewest sig figs.
3.14×2.5=7.85 but you report 7.9 (2 sig figs, because 2.5 has only 2)
Addition and Subtraction
The result should have the same decimal places as the measurement with the fewest decimal places.
12.11+18.0=30.11 but you report 30.1 (one decimal place, because 18.0 has one) …
Why this formula?
Significant Figures: Why the Rules Work
Let’s start with the core idea: significant figures (sig figs) are a way to honestly report how precise a measurement is. The rules for addition/subtraction and multiplication/division aren’t arbitrary — they come directly from how uncertainty propagates through calculations.
1. The Fundamental Idea: Uncertainty is the Key
Every measurement has an uncertainty (error). When we say a length is 12.3 cm, we mean:
The true value lies somewhere between 12.25 cm and 12.35 cm (assuming ±0.05 cm uncertainty).
The last digit (3) is uncertain; the digits before it (1 and 2) are certain.
Why this matters: When we combine measurements, the uncertainty in the result depends on the uncertainties of the inputs. Sig fig rules are a shortcut for this uncertainty propagation.
2. Rule for Addition and Subtraction
Statement: The result should have the same number of decimal places as the measurement with the fewest decimal places.
Example:
12.3+4.56=16.86 → round to 16.9 (one decimal place, like 12.3)
Why this holds
Consider two measurements:
A=12.3±0.05 (uncertainty in the tenths place)
B=4.56±0.005 (uncertainty in the hundredths place)
When we add:
Certain digits: 12.3 has certainty up to the tenths place. 4.56 has certainty up to the hundredths place.
The weaker link: The tenths place of A is uncertain. So in the sum, the hundredths place (from B) is meaningless — because we don’t even know the tenths place of A exactly.
Mathematically, the absolute uncertainty in the sum is:
Δ(A+B)=(ΔA)2+(ΔB)2≈0.052+0.0052≈0.0502
This uncertainty is ~0.05, which affects the tenths place. So reporting the hundredths place is false precision.
Key takeaway: The result’s last significant digit is in the same decimal place as the least precise measurement’s last digit.
3. Rule for Multiplication and Division
Statement: The result should have the same number of significant figures as the measurement with the fewest significant figures.
Example:
12.3×4.56=56.088 → round to 56.1 (three sig figs, like both inputs)
Why this holds
Let’s use relative uncertainty (percentage error):
For multiplication, relative uncertainties add (approximately):
A×BΔ(A×B)≈(AΔA)2+(BΔB)2
Plugging in:
≈0.004072+0.001102≈0.00422 (0.422%)
Now, the absolute uncertainty in the product:
Δ(A×B)≈0.00422×(12.3×4.56)≈0.00422×56.088≈0.237
This uncertainty (~0.2) affects the tenths place of the result. So the result 56.088 has uncertainty in the first decimal — meaning only three digits (5, 6, and the uncertain 1) are meaningful. That’s three sig figs, matching the input with fewer sig figs (both have three here).
Key takeaway: The number of sig figs in the result is limited by the least precise measurement’s number of sig figs, because relative uncertainty is dominated by the measurement with the largest relative error.
To round 19.95 to three significant figures, look at the digit being dropped (the last 5); since the preceding digit (9) is odd, the 'round half to even' rule raises it, car …
Rounding 19.95 to three significant figures using the 'round half to even' convention gives 20.0.
19.95 has four significant figures (1, 9, 9, 5). To round to three significant figures, we must decide the fate of the third significant digit (the second 9), based on the digit being dropped (a 5, with nothing after it).
Rule for rounding off when the digit to be dropped is exactly 5: if the preceding digit is even, it is left unchanged; if the preceding digit is odd, it is increased by 1 (this is the 'round half to even' convention used for significant figures, and it avoids a systematic upward bias).