How to Use Significant Figures — Rules & Rounding

Introduction

In scientific measurement and calculation, precision matters—not just the numerical result, but how honestly it reflects the limitations of your instruments and data. This is where significant figures (often called "sig figs") come in.

Why Significant Figures Matter

Significant figures are the language of scientific integrity, ensuring that:

Measurement precision is accurately communicated
Data reliability is properly represented
Calculation results reflect true accuracy
Scientific honesty is maintained in reporting
Instrument limitations are acknowledged

Academic and Professional Importance

Understanding sig figs is essential for:

GCSE and A-Level science examinations
University lab reports and research
Engineering specifications and tolerances
Financial reporting with consistent rounding
Quality control in manufacturing

Calculator vs Manual Understanding

A significant figures calculator automates the process, but mastering the underlying rules empowers you to:

Interpret data critically and spot errors
Avoid common pitfalls in measurement
Communicate results with clarity and honesty
Handle ambiguous cases with confidence
Understand uncertainty in real-world measurements

What You'll Learn

This guide explains how to:

Count significant figures accurately
Round numbers using proper rules
Apply sig figs in all arithmetic operations
Handle ambiguous cases (trailing zeros)
Use scientific notation to eliminate confusion

These rules exist not as arbitrary conventions, but as practical tools for managing uncertainty in real-world measurements.

The Four Foundational Rules for Counting Significant Figures

Before you can round or calculate, you must correctly identify how many significant figures a number contains. Follow these rules in order:

1. All Non-Zero Digits Are Significant

Digits 1 through 9 always count.

Example: 347 has 3 sig figs; 0.892 has 3.

2. Zeros Between Non-Zero Digits Are Significant

These “sandwiched” zeros are part of the measured value.

Example: 202 has 3 sig figs; 4.005 has 4.

3. Leading Zeros Are Never Significant

Zeros before the first non-zero digit only locate the decimal point.

Example: 0.0042 has 2 sig figs (just 4 and 2); 0.0007030 has 4 (7, 0, 3, 0).

4. Trailing Zeros Are Significant Only If a Decimal Point Is Present

This is the most nuanced rule—and the reason scientific notation exists.

With decimal: 350.0 has 4 sig figs; 8.200 has 4.
Without decimal: 3500 is ambiguous—it could have 2, 3, or 4 sig figs.
Solution: Use scientific notation:
- 3.5 × 10³ → 2 sig figs
- 3.50 × 10³ → 3 sig figs
- 3.500 × 10³ → 4 sig figs

Special Cases

Exact numbers (e.g., “5 trials”, “12 inches in a foot”) have infinite sig figs—they don’t limit your calculation.
Defined constants (like π or e) should be used with at least one more digit than your least precise measurement.

Rounding to a Specified Number of Significant Figures

Once you know how many sig figs you need, follow this process:

Identify the first significant digit (leftmost non-zero).
Count to the desired number of sig figs.
Look at the next digit (the “decider”):
- If ≥ 5, round up the last kept digit.
- If below 5, leave it unchanged.
Replace extra digits with zeros or use scientific notation to preserve place value.

Example: Round 0.0045678 to 3 sig figs.

First sig fig: 4
Count 3: 4, 5, 6
Decider: 7 (≥5) → round 6 up to 7
Result: 0.00457

Rules for Arithmetic: Propagating Uncertainty

The result of a calculation cannot be more precise than the least precise input.

Multiplication & Division

→ Round to the fewest number of sig figs among all inputs.

Example:
12.3 (3 sf) × 0.45 (2 sf) = 5.535 → 2 sig figs → 5.5

Addition & Subtraction

→ Round to the fewest number of decimal places among all inputs.

Example:
14.25 (2 dp) + 3.1 (1 dp) = 17.35 → 1 decimal place → 17.4

Mixed Operations

→ Apply rules step by step, but keep extra digits during intermediate steps to avoid rounding error. Only round the final answer.

Example:
(8.2 × 3.1416) + 0.5

Multiply: 8.2 (2 sf) × 3.1416 ≈ 25.761 (keep all digits)
Add: 25.761 + 0.5 (1 dp) = 26.261
Final rounding: 1 decimal place → 26.3

Pro Tips for Real-World Application

Always use scientific notation for large or small numbers to avoid ambiguity.
Never round intermediate results in multi-step calculations—use your calculator’s memory or “ANS” key.
When in doubt, under-report precision. It’s better to say “≈ 2.5 g” than “2.5034 g” if your balance only reads to 0.1 g.
In lab reports, state the precision of your instruments (e.g., “mass measured to ±0.01 g”) to justify your sig figs.

Worked Examples & Practice Problems

Example 1: Counting Sig Figs

Determine the number of significant figures:

0.007050 → Leading zeros (×3) don’t count; trailing zero after decimal does → 4 sig figs
4200 → Ambiguous! Assume 2 sig figs unless specified otherwise.
9.000 × 10⁻² → Mantissa has 4 sig figs

Example 2: Rounding Practice

Round 298.765 to:

2 sig figs: 3.0 × 10² (not 300—ambiguous!)
4 sig figs: 298.8
6 sig figs: 298.765 (already there)

Example 3: Calculation with Proper Rounding

Problem: A rectangle is 12.4 cm long and 5.6 cm wide. Find its area.

Calculation: 12.4 × 5.6 = 69.44 cm²
Sig figs: 12.4 (3 sf), 5.6 (2 sf) → answer must have 2 sig figs
Final answer: 69 cm² (or 6.9 × 10¹ cm²)

Example 4: Addition with Decimal Places

Problem: Add 100.25 g and 3.4 g.

Sum: 103.65 g
Decimal places: 100.25 (2 dp), 3.4 (1 dp) → round to 1 dp
Final answer: 103.7 g

Practice Problems (Try These!)

How many sig figs in 0.040050?
Round 0.006789 to 2 sig figs.
Calculate (2.50 × 4.1) – 1.23 and round correctly.
Express 5600 with 3 sig figs in scientific notation.

Answers:

5 (4,0,0,5,0)
0.0068
2.50×4.1=10.25; 10.25–1.23=9.02; 4.1 has 2 sf → 9.0
5.60 × 10³

Why are significant figures important in science?

They prevent false precision. Reporting a result with more digits than your instrument can measure misleads others about the reliability of your data. Sig figs ensure honesty and reproducibility in scientific communication.

How do I handle trailing zeros without a decimal point?

Treat them as ambiguous. In formal work, always use scientific notation to clarify. For example, write 1.20 × 10⁴ instead of 12000 if you mean 3 sig figs.

Can a number have zero significant figures?

No. Even 0.000 implies a measurement precise to the thousandths place, so it has 1 sig fig (the final zero after the decimal). The number 0 itself is a special case—it has 1 sig fig by convention when it represents a measured null value.

What if my calculator shows more digits than I need?

Ignore them until the final step. Use all digits during calculation, then round the final answer according to sig fig rules. Never round intermediate values.

Do percentages follow sig fig rules?

Yes. A percentage like 25% (from 25/100) has 2 sig figs. If it came from 25.0/100.0, it would be 25.0% (3 sig figs).

How do I apply sig figs to logarithms?

For log(x), the number of decimal places in the result equals the number of sig figs in x.

Example: log(2.5 × 10³) → 2.5 has 2 sig figs → log = 3.39794... → round to 2 decimal places → 3.40

Is there a difference between “significant figures” and “decimal places”?

Yes!

Sig figs count all meaningful digits, regardless of decimal position.
Decimal places count digits after the decimal point.
They are used in different contexts: sig figs for multiplication/division, decimal places for addition/subtraction.

What if two numbers in a calculation have the same number of sig figs but different magnitudes?

The rule still applies. For example, 100. (3 sf) × 0.00200 (3 sf) = 0.200 (3 sf)—scientific notation (1.00 × 10² × 2.00 × 10⁻³ = 2.00 × 10⁻¹) makes this clearer.