Here's a breakdown of significant figures—what they are, why they matter, and how to use them.
What Are Significant Figures?
Think of significant figures (or "sig figs") as a way of showing how precise a measurement is.
Imagine you're measuring water in a graduated cylinder. The markings are for every 1 mL. The water level is past the 42 mL mark but not quite at the 43 mL mark. You estimate it's at 42.5 mL.
The "4" and "2" are certain digits (you know it's at least 42).
The ".5" is your estimated digit (it's your best guess).
Your measurement, 42.5 mL, has 3 significant figures.
The Rule: Significant figures in a measurement include all the digits you know for sure, plus one estimated digit.
They're important because they tell you (and other scientists) the quality of the tool you used. Measuring 42.5 mL (3 sig figs) is very different from measuring 42.500 mL (5 sig figs), which would require a much more expensive, high-precision instrument.
How to Count Significant Figures (The Rules)
This is the part you usually have to memorize. Here are the rules for counting them in a given number.
1. Non-Zero Digits (1-9):
Rule: All non-zero digits are always significant.
Example: 22.7 g (has 3 sig figs)
2. "Captive" Zeros:
Rule: Zeros between two non-zero digits are always significant.
Example: 305.1 m (has 4 sig figs)
Example: 5007 L (has 4 sig figs)
3. "Leading" Zeros:
Rule: Zeros at the beginning of a number (before any non-zeros) are never significant. They are just placeholders to show where the decimal point is.
Example: 0.0025 kg (has 2 sig figs: the 2 and the 5)
Example: 0.04 s (has 1 sig fig: the 4)
4. "Trailing" Zeros:
Rule: Zeros at the end of a number. This is the trickiest one, and it depends on the decimal point.
A) If a decimal point is present (anywhere): Trailing zeros are significant. They were measured!
Example: 12.50 mL (has 4 sig figs)
Example: 300. m (has 3 sig figs - the decimal point is key!)
Example: 0.500 g (has 3 sig figs)
B) If there is no decimal point: Trailing zeros are not significant (they are considered ambiguous placeholders).
Example: 1200 cm (has 2 sig figs)
Example: 50,000 people (has 1 sig fig)
5. Scientific Notation:
Rule: This is the best way to avoid ambiguity! All digits shown in the coefficient (the number part) are significant.
Example: 1.20 x 10^3 cm (has 3 sig figs)
Example: 5 x 10^4 people (has 1 sig fig)
6. Exact Numbers:
Rule: Numbers that are counted or are definitions have unlimited sig figs.
Example: "12 students" (You can't have 12.1 students. It's an exact count).
Example: 1 m = 100 cm (This is a definition).
When you use these in a calculation, they never limit your final answer's sig figs.
How to Use Significant Figures in Calculations
This is where the "weakest link" idea comes in. Your answer can't be more precise than your least precise measurement.
There are two different rules, one for multiplication/division and one for addition/subtraction.
Rule 1: Multiplication and Division
The Rule: Your final answer must have the same number of sig figs as the measurement with the fewest sig figs.
Example:
4.52 cm x 1.2 cm = ?4.52 has 3 sig figs.
1.2 has 2 sig figs.
Your "weakest link" is 1.2 (only 2 sig figs). Your answer must be rounded to 2 sig figs.
Calculator says: 5.424 cm^2
Correct Answer: 5.4 cm^2 (rounded to 2 sig figs)
Rule 2: Addition and Subtraction
The Rule: This one is different! It's not about the count of sig figs, but the place value. Your final answer must be rounded to the same decimal place as the measurement with the fewest decimal places.
Example:
12.52 g + 3.1 g = ?12.52 is precise to the hundredths place (2 decimal places).
3.1 is only precise to the tenths place (1 decimal place).
Your "weakest link" is 3.1. Your answer must be rounded to the tenths place.
Calculator says: 15.62 g
Correct Answer: 15.6 g (rounded to the tenths place)
A Quick Tip on Rounding:
If the digit you're dropping is 5 or greater, round the last digit up. (15.68 rounded to 3 sig figs is 15.7)
If the digit you're dropping is 4 or less, leave the last digit as is. (15.62 rounded to 3 sig figs is 15.6)
Summary
What: Sig figs show the precision of a measurement (all certain digits + one estimate).
Counting: Learn the 4 rules for zeros (Captive = Yes, Leading = No, Trailing = Only with a decimal).
Calculating:
Multiplication and division: Fewest total sig figs.
Addition and subtraction: Fewest decimal places.
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