Significant figures (often called "sig figs" or "significant digits") are one of the most fundamental concepts in science, engineering, and mathematics. They tell us how precise a measurement is and ensure that calculations don't falsely imply more accuracy than the original data supports.
Whether you are a student working through a chemistry problem set, an engineer verifying tolerances, or a researcher reporting lab results, understanding significant figures is essential. This guide covers everything you need: the five rules for counting sig figs, how to handle sig figs in arithmetic, common pitfalls, real-world applications, and worked examples.
Need to check your work quickly? Try our free Sig Figs Calculator to count significant figures, round numbers, and evaluate expressions with proper sig fig rules applied automatically.
Table of Contents
What Are Significant Figures and Why Do They Matter?
Every measurement has a degree of uncertainty. When you read a thermometer that shows 98.6 °F, the "98" is certain, but the ".6" is an estimate between the markings. Significant figures are all of the digits in a measurement that are known with certainty, plus one final estimated digit.
Significant figures matter because they communicate precision. If you report a measurement as 12.3 cm, you are telling the reader that the value is precise to the tenths place. Reporting it as 12.30 cm implies even greater precision, to the hundredths place. Getting this right is critical in science and engineering because it prevents misleading conclusions.
Key Principle
A calculated result can never be more precise than the least precise measurement used to obtain it. Significant figures enforce this principle by limiting how many digits you report in your answer.
Note for advanced students: Significant figures are a practical shorthand for measurement uncertainty. In research and professional science, uncertainty is expressed more rigorously using error propagation and confidence intervals (e.g., 9.81 ± 0.02 m/s²). These sig fig conventions follow guidelines from IUPAC recommendations and NIST measurement standards used in university laboratory manuals worldwide.
Consider this example: You measure the length of a table as 1.52 m (3 significant figures) and the width as 0.8 m (1 significant figure). If you multiply to find the area, your calculator shows 1.216 m². But reporting all four digits would be misleading because your width measurement was only precise to one significant figure. The correct answer is 1 m² (rounded to 1 sig fig).
The 5 Rules for Counting Significant Figures
These five rules let you determine how many significant figures any number has. Master these and you will handle sig figs with confidence.
Rule 1: All Non-Zero Digits Are Significant
Every digit from 1 through 9 always counts as significant, no matter where it appears in the number.
| Number | Sig Figs | Why |
|---|---|---|
| 1234 | 4 | All four digits are non-zero |
| 56.78 | 4 | All four digits are non-zero |
| 3.14159 | 6 | All six digits are non-zero |
Rule 2: Zeros Between Non-Zero Digits (Captive Zeros) Are Significant
Any zero sandwiched between non-zero digits is significant. These are sometimes called "captive zeros" or "trapped zeros."
| Number | Sig Figs | Why |
|---|---|---|
| 1002 | 4 | Zeros are between 1 and 2 |
| 3.0072 | 5 | Zeros are between 3 and 7 |
| 50.09 | 4 | Both zeros lie between non-zero digits 5 and 9 — both are captive |
Rule 3: Leading Zeros Are Never Significant
Zeros that appear before the first non-zero digit serve only as placeholders and do not count as significant figures. They simply indicate the position of the decimal point.
| Number | Sig Figs | Why |
|---|---|---|
| 0.0045 | 2 | Leading zeros (0.00) are placeholders; only 4 and 5 count |
| 0.123 | 3 | Leading zero is a placeholder; 1, 2, and 3 count |
| 0.000008 | 1 | Five leading zeros; only 8 is significant |
Rule 4: Trailing Zeros After the Decimal Point Are Significant
Zeros at the end of a number that appear after the decimal point are significant. They indicate that the measurement was precise enough to include those digits.
| Number | Sig Figs | Why |
|---|---|---|
| 2.50 | 3 | Trailing zero after decimal indicates precision |
| 100.0 | 4 | Decimal point present; all four digits are significant |
| 0.00340 | 3 | Leading zeros don't count; 3, 4, and trailing 0 are significant |
Rule 5: Trailing Zeros in Whole Numbers Without a Decimal Point Are Ambiguous
This is the rule that causes the most confusion. Trailing zeros in a whole number (like 1500) may or may not be significant, depending on how the number was measured. Without additional context, they are considered ambiguous.
| Number | Sig Figs | Interpretation |
|---|---|---|
| 1500 | 2, 3, or 4 | Ambiguous without context |
| 1500. | 4 | Trailing decimal point makes all zeros significant |
| 1.500 × 10³ | 4 | Scientific notation removes ambiguity |
| 1.5 × 10³ | 2 | Only 2 sig figs clearly indicated |
Tip: When writing your own measurements, always use scientific notation or a trailing decimal point to remove ambiguity about trailing zeros. Many science courses and textbooks assume trailing zeros in whole numbers are not significant unless marked otherwise.
How to Round Significant Figures
Once you know how many significant figures your answer should have, use these rounding rules:
- Digit after the cut-off is < 5: Round down (leave the last kept digit unchanged).3.742 rounded to 3 sig figs → 3.74
- Digit after the cut-off is > 5: Round up (increase last kept digit by 1).3.746 rounded to 3 sig figs → 3.75
- Digit after the cut-off is exactly 5: Most science courses use “round half up”—always round up when the dropped digit is exactly 5.3.745 rounded to 3 sig figs → 3.75
Note on professional practice: Introductory science courses teach round-half-up, but professional scientific computing (and IEEE floating-point standards) often use “round half to even” (banker's rounding) — where a trailing 5 rounds to the nearest even digit (e.g., 2.45 → 2.4, but 2.35 → 2.4). This eliminates the cumulative upward bias that repeated round-half-up introduces across large datasets. Our Sig Figs Calculator uses round-half-up to match textbook expectations.
Special Cases: Leading, Trailing, and Captive Zeros
Zeros are the source of almost every significant figures mistake. Here is a complete summary of how every type of zero is handled, along with additional special cases.
| Type of Zero | Example | Significant? | Sig Figs |
|---|---|---|---|
| Leading zeros | 0.0056 | Never | 2 |
| Captive zeros | 20.08 | Always | 4 |
| Trailing zeros (with decimal) | 8.100 | Always | 4 |
| Trailing zeros (no decimal) | 4500 | Ambiguous | 2, 3, or 4 |
| Trailing zeros (with decimal point) | 4500. | Always | 4 |
Exact Numbers and Defined Quantities
Some numbers are considered exact and have unlimited significant figures. These numbers never limit the precision of a calculation. Exact numbers include:
- Counted quantities: 12 eggs, 3 trial runs, 24 students (you counted them exactly)
- Defined relationships: 1 foot = 12 inches exactly, 1 kilogram = 1000 grams exactly, 1 inch = 2.54 cm exactly
- Integers in formulas: The 2 in C = 2πr, the 4 in A = 4πr²
Example: If you measure the diameter of a circle as 4.5 cm (2 sig figs) and compute the circumference using C = πd, the answer has 2 significant figures because π and the formula's implicit factor are exact. The measurement (4.5 cm) is the limiting factor, not the constants.
Significant Figures in Calculations
Different arithmetic operations follow different sig fig rules. This is one of the most tested topics in introductory science courses.
Addition and Subtraction: Use Decimal Places
For addition and subtraction, round the result to the same number of decimal places as the measurement with the fewest decimal places. Notice that this rule is about decimal places, not significant figures.
12.52 (2 decimal places)
+ 1.7 (1 decimal place) ← fewest decimal places
+ 0.158 (3 decimal places)
= 14.378 → rounded to 14.4 (1 decimal place)
The value 1.7 has only 1 decimal place, so the final answer must also have exactly 1 decimal place.
Multiplication and Division: Use Significant Figures
For multiplication and division, round the result to the same number of significant figures as the measurement with the fewest significant figures.
4.56 × 1.4 = 6.384
4.56 has 3 sig figs
1.4 has 2 sig figs ← fewest
Answer: 6.4 (rounded to 2 sig figs)
0.5893 ÷ 0.046 = 12.8108...
0.5893 has 4 sig figs
0.046 has 2 sig figs ← fewest
Answer: 13 (rounded to 2 sig figs)
Mixed Operations: Handle Each Step Separately
When a calculation involves both addition/subtraction and multiplication/division, apply the correct rule at each step. Important: keep extra guard digits during intermediate steps and only round the final answer.
Problem: (12.5 + 1.37) × 3.6
Step 1 (Addition): 12.5 + 1.37 = 13.87
The sum should have 1 decimal place (matching 12.5) → conceptually 13.9, but keep the unrounded value 13.87 as a guard digit for the next step.
Step 2 (Multiplication): 13.87 × 3.6 = 49.932
13.87 (guard digit, effectively 3 sig figs from Step 1 rule) × 3.6 (2 sig figs) → round to 2 sig figs.
Final answer: 50. (2 sig figs)
Guard Digits: Many chemistry and physics instructors recommend keeping at least 1-2 extra digits during intermediate calculations, then rounding only the final result. This prevents accumulation of rounding errors. Our sig figs calculator handles this automatically.
Logarithms and Sig Figs
Logarithms have a special rule. The number of significant figures in the original number determines the number of digits after the decimal point in the logarithm (these digits are called the "mantissa"). The digit before the decimal (the "characteristic") only indicates the order of magnitude.
log(3.45 × 104) = 4.538
3.45 has 3 sig figs → log result has 3 decimal places
The "4" is the characteristic; "538" is the mantissa (3 digits).
This is particularly important for pH calculations in chemistry: pH = -log[H+]
Scientific Notation and Significant Figures
Scientific notation is the clearest way to express significant figures because it removes all ambiguity about trailing zeros. In scientific notation, every digit in the coefficient is significant.
A number in scientific notation is written as: a × 10n, where 1 ≤ a < 10 and n is an integer.
| Standard Form | Scientific Notation | Sig Figs | Clarity |
|---|---|---|---|
| 4500 | 4.5 × 10³ | 2 | Ambiguity resolved |
| 4500 | 4.500 × 10³ | 4 | All four digits clearly significant |
| 0.0032 | 3.2 × 10-3 | 2 | Leading zeros eliminated |
| 0.003200 | 3.200 × 10-3 | 4 | Trailing zeros clearly significant |
| 100 | 1.00 × 10² | 3 | Exactly 3 sig figs, no ambiguity |
When in doubt, converting to scientific notation is the best way to make the number of significant figures absolutely clear. This is standard practice in scientific journals and lab reports.
Common Mistakes Students Make
After years of teaching and building calculator tools, these are the most frequent sig fig errors we see. Avoiding these will immediately improve your accuracy.
Mistake 1: Counting Leading Zeros as Significant
Students often count the zeros in 0.0052 and say it has 4 sig figs. It actually has 2. Leading zeros are never significant.
Fix: Think of 0.0052 as 5.2 × 10-3. Only the 5 and 2 are in the coefficient.
Mistake 2: Using the Sig Figs Rule for Addition/Subtraction
Many students apply the "fewest sig figs" rule to addition and subtraction. This is wrong. Addition and subtraction use the decimal places rule.
Example: 12.52 + 1.7 = 14.22 → 14.2 (1 decimal place, matching 1.7). If you wrongly applied the sig figs rule, you would look at 1.7 (2 sig figs, the fewest) and round 14.22 to 2 sig figs → 14. That is clearly wrong—the decimal places rule gives the correct precision.
Mistake 3: Rounding at Every Intermediate Step
Rounding at each step of a multi-step calculation introduces compounding rounding error. The final answer can be off by a significant amount.
Fix: Keep at least 1-2 extra guard digits in intermediate steps. Only round the final answer.
Mistake 4: Ignoring Trailing Zeros After the Decimal
Students sometimes think 2.50 and 2.5 are identical. They are numerically equal but express different precisions. Writing 2.50 means you measured to the hundredths place (3 sig figs), while 2.5 means you measured only to the tenths (2 sig figs).
Fix: Never drop trailing zeros after the decimal point unless you intend to change the precision.
Mistake 5: Treating Exact Numbers as Limited
When converting 5.00 inches to centimeters using 1 inch = 2.54 cm (exact), some students say the answer should have 3 sig figs (limited by 2.54). But 2.54 is exact by definition, so it has unlimited sig figs.
Fix: 5.00 in × 2.54 cm/in = 12.7 cm (3 sig figs, limited only by 5.00).
Mistake 6: Confusing Precision and Accuracy
Significant figures reflect precision (how reproducible or detailed a measurement is), not accuracy (how close the measurement is to the true value). A poorly calibrated scale might consistently give 12.345 g (5 sig figs, very precise) when the true mass is 15.000 g (very inaccurate).
Real-World Applications of Significant Figures
Significant figures are not just academic exercises. Professionals in many fields rely on them daily to ensure their work is reliable and their results are properly communicated.
Chemistry Lab Work
When titrating a solution, a chemist records a burette reading of 23.45 mL. The balance reads 4.052 g of a solid reagent. Every calculation from these measurements (molarity, moles, percent yield) must carry forward the correct number of significant figures.
pH calculations: If [H+] = 2.3 × 10-4 M (2 sig figs), then pH = 3.64 (2 digits after the decimal). Reporting pH = 3.6383 would imply a precision that the original measurement does not support.
Engineering Tolerances
A mechanical engineer designs a shaft with a diameter of 25.00 mm ± 0.05 mm. The four significant figures in 25.00 communicate that the measurement must be precise to the hundredths of a millimeter. Writing 25 mm would imply a much looser tolerance.
In aerospace engineering, components often require 5-6 significant figures in their specifications. One extra digit of precision can mean the difference between a safe part and a failure.
Physics Experiments
In a physics experiment measuring the acceleration due to gravity, a student finds g = 9.83 m/s² (3 sig figs). The accepted value is 9.807 m/s² (4 sig figs). The student's equipment and technique limited precision to 3 sig figs, which is acceptable for a teaching lab.
Particle physics experiments at CERN routinely report results to 4–5 significant figures. The Higgs boson mass has been measured as 125.20 ± 0.11 GeV (PDG 2024)—5 significant figures of precision achieved through billions of collision events at the LHC.
Medicine and Pharmacology
Drug dosages must be calculated with appropriate precision. A prescription for 0.125 mg of digoxin (3 sig figs) is very different from 0.13 mg (2 sig figs) or 0.1 mg (1 sig fig). The trailing digits matter when lives are at stake.
Blood test results are reported with specific sig figs. A glucose level of 100 mg/dL is different from 100. mg/dL or 1.00 × 10² mg/dL in terms of measurement confidence.
Worked Example: Step by Step
Let us work through a realistic chemistry problem step by step, applying all the sig fig rules we have discussed.
Problem
A student measures the mass of a beaker and sample as 45.823 g. The empty beaker has a mass of 30.07 g. The sample is dissolved in enough water to make 250.0 mL of solution. Calculate the concentration in g/mL.
Step 1: Find the Mass of the Sample (Subtraction)
45.823 g − 30.07 g = 15.753 g
Rule applied: Subtraction → use decimal places.
- 45.823 has 3 decimal places
- 30.07 has 2 decimal places (fewest)
- Round to 2 decimal places: 15.75 g
But we keep 15.753 as a guard value for the next step.
Step 2: Calculate Concentration (Division)
15.753 g ÷ 250.0 mL = 0.063012 g/mL
Rule applied: Division → use significant figures.
- 15.753 has 5 sig figs (guard value from Step 1—not the limiting factor)
- 250.0 has 4 significant figures (limiting factor)
- Round to 4 sig figs: 0.063012 → 0.06301
Final Answer
Concentration = 0.06301 g/mL (4 significant figures)
Using the guard value 15.753 instead of the prematurely rounded 15.75 gives 0.06301, not 0.06300—a difference in the 4th significant figure. In scientific notation: 6.301 × 10-2 g/mL.
Check your work: Use our free Sig Figs Calculator to verify sig fig counts and perform calculations with proper rounding automatically applied.
Try the Free Sig Figs Calculator
Count significant figures, round numbers to any number of sig figs, and evaluate expressions with the addition/subtraction decimal rule and multiplication/division sig figs rule applied automatically. Supports log, ln, and multi-step expressions.
Use Sig Figs Calculator →Frequently Asked Questions
What are significant figures?
Significant figures (also called significant digits or sig figs) are the digits in a measured number that carry meaningful information about the precision of that measurement. They include all certain digits plus one estimated digit. For example, if a ruler measures 12.3 cm, there are 3 significant figures: the 1 and 2 are certain, and the 3 is estimated.
How many significant figures does 0.00340 have?
0.00340 has 3 significant figures. The leading zeros (0.00) are not significant because they are just placeholders. The digits 3 and 4 are significant, and the trailing zero after the 4 is significant because it comes after the decimal point, indicating the measurement was precise to the hundred-thousandths place.
What is the difference between the addition/subtraction rule and the multiplication/division rule?
For addition and subtraction, round to the fewest number of decimal places among the measurements. For multiplication and division, round to the fewest number of significant figures among the measurements. These are different rules, and mixing them up is the most common sig fig mistake in science courses.
Are trailing zeros significant?
It depends on context. Trailing zeros after a decimal point are always significant (e.g., 2.50 has 3 sig figs). Trailing zeros in a whole number without a decimal point are ambiguous (e.g., 1500 could have 2, 3, or 4 sig figs). To remove ambiguity, use scientific notation: 1.5 × 10³ has 2 sig figs, while 1.500 × 10³ has 4 sig figs.
Do exact numbers have significant figures?
Exact numbers have unlimited (infinite) significant figures and never limit the precision of a calculation. Exact numbers include counted quantities (12 eggs), defined conversion factors (1 foot = 12 inches exactly), and integers in mathematical formulas (the 2 in C = 2πr).
How do you handle significant figures in multi-step calculations?
In multi-step calculations, keep extra digits (guard digits) in intermediate steps and only round the final answer. Rounding at each step introduces rounding errors that can accumulate. Many instructors recommend keeping at least one or two extra digits during intermediate calculations, then rounding the final result to the correct number of significant figures.
How do significant figures apply to logarithms and pH?
For logarithms, the number of significant figures in the original number equals the number of digits after the decimal point in the logarithm (the mantissa). The digit(s) before the decimal point (the characteristic) only indicate order of magnitude. For pH: if [H+] = 2.5 × 10-3 M (2 sig figs), then pH = 2.60 (2 digits after the decimal).
Quick Reference Summary
| Rule | Key Point | Example |
|---|---|---|
| Non-zero digits | Always significant | 247 → 3 SF |
| Captive zeros | Always significant | 1.003 → 4 SF |
| Leading zeros | Never significant | 0.0056 → 2 SF |
| Trailing zeros (decimal) | Always significant | 8.100 → 4 SF |
| Trailing zeros (no decimal) | Ambiguous | 1500 → 2-4 SF |
| Add/Subtract | Round to fewest decimal places | 12.5 + 1.37 → 13.9 |
| Multiply/Divide | Round to fewest sig figs | 4.56 × 1.4 → 6.4 |
| Exact numbers | Unlimited sig figs | 1 ft = 12 in |
Conclusion
Significant figures are the scientific community's standard for communicating measurement precision. Mastering the five rules for counting sig figs, understanding the difference between the addition/subtraction rule and the multiplication/division rule, and knowing how to handle special cases like exact numbers and logarithms will serve you well throughout any science, engineering, or mathematics course.
The key takeaways are: non-zero digits always count, captive zeros always count, leading zeros never count, trailing zeros after a decimal always count, and trailing zeros in whole numbers are ambiguous. For calculations, addition and subtraction use decimal places while multiplication and division use significant figures. And always keep guard digits in intermediate steps.
These conventions are consistent with IUPAC recommendations, NIST measurement standards, and the lab reporting guidelines used at universities worldwide. If you go on to research-level work, you will also encounter formal uncertainty analysis — expressing results as value ± uncertainty with defined confidence intervals — which is the rigorous extension of what significant figures approximate.
Practice makes perfect. Use our free Sig Figs Calculator to check your work, build confidence, and save time on homework and lab reports. It counts sig figs, rounds numbers, and evaluates expressions with proper sig fig rules applied automatically.