Significant Figures Rules – How to Round Calculations for CSEC Physics

CSEC Essential Skill: Significant figures (sig figs) tell you about the precision of a measurement. The last digit of a value expressed with a certain degree of significance gives the maximum uncertainty. Mastering sig figs is crucial for CSEC Physics calculations and SBA reports.

What Are Significant Figures?

Significant figures are the digits in a measurement that carry meaning about its precision. They include all certain digits plus one uncertain (estimated) digit.

Certain

Significant
(trailing after decimal)

4 sig figs

The number 23.40 has 4 significant figures. The last digit (0) is uncertain but significant—it tells us the measurement was made to the nearest 0.01.

CSEC Insight: “The degree of significance shown in a result should reflect the degree of precision of the instrument used to obtain it.” If you use a metre rule (precision = 1 mm), your measurement should have 3 sig figs (e.g., 15.2 cm), not 4 (e.g., 15.23 cm would suggest false precision).

The 5 Rules for Counting Significant Figures

1 All non-zero digits are ALWAYS significant

Examples:

456 has 3 sig figs (4, 5, 6 are all significant)
2.34 has 3 sig figs (2, 3, 4 are all significant)
789.12 has 5 sig figs

2 Zeros BETWEEN non-zero digits ARE significant

Examples:

1002 has 4 sig figs (1, 0, 0, 2 are all significant)
5.008 has 4 sig figs (5, 0, 0, 8 are all significant)
30.04 has 4 sig figs

3 LEADING zeros (before first non-zero) are NEVER significant

Examples:

0.0056 has 2 sig figs (only 5 and 6 are significant)
0.00034 has 2 sig figs
0.080 has 2 sig figs? Wait for rule 5!

4 TRAILING zeros (after last non-zero) in a number WITH a decimal point ARE significant

Examples:

250. has 3 sig figs (the decimal point tells us the zero is significant)
12.00 has 4 sig figs
0.00450 has 3 sig figs (4, 5, and the last 0)

5 TRAILING zeros in a number WITHOUT a decimal point are AMBIGUOUS (usually not significant in CSEC Physics)

Examples:

2500 usually has 2 sig figs in CSEC (unless stated otherwise)
1200 usually has 2 sig figs
Use scientific notation to avoid ambiguity: 2.50 × 10³ clearly has 3 sig figs

Zeros that ARE Significant:

Between non-zero digits: 1002
After decimal point: 12.00
After non-zero digit with decimal: 250.

Zeros that are NOT Significant:

Before first non-zero: 0.005
Placeholders without decimal: 1200 (usually)
Leading zeros: 00234

📚 Practice Counting Sig Figs:

Number	Significant Figures	Rule(s) Applied
34.5	3	Rule 1: All non-zero digits
0.0087	2	Rule 3: Leading zeros not significant
400.0	4	Rule 4: Trailing zeros after decimal are significant
4000	1 or 4? (Usually 1 in CSEC)	Rule 5: Ambiguous, use scientific notation
0.05020	4	Rule 3 (first two zeros) + Rule 4 (last zero)

Why Significant Figures Matter in CSEC Physics

CSEC Reality: The place value of the final digit also tells the reader the maximum error or the maximum uncertainty of the value. For example:

15.2 cm means maximum error = ±0.05 cm (measured to nearest 0.1 cm)
15.20 cm means maximum error = ±0.005 cm (measured to nearest 0.01 cm)
15 cm means maximum error = ±0.5 cm (measured to nearest 1 cm)

The number of significant figures communicates the precision of your measurement!

Visualizing Rounding: Number Line Example

Rounding 2.436 to 3 significant figures:

2.43

2.435

2.44

2.436 is closer to 2.44 than to 2.43, so it rounds to 2.44 (3 sig figs).

Rounding Rules for Calculations

🎯 The Golden Rule for CSEC Calculations

“Your final answer should have no more significant figures than the least precise measurement used in the calculation.”

Rule A: Multiplication & Division

The result should have the same number of significant figures as the measurement with the fewest significant figures.

Example: Area of a rectangle

Length = 12.4 cm (3 sig figs), Width = 5.2 cm (2 sig figs)

Area = 12.4 × 5.2 = 64.48 cm² (calculator result)

Width has fewest sig figs = 2

Round 64.48 to 2 sig figs = 64 cm² (NOT 64.5 or 64.48)

Reason: The width measurement (5.2 cm) has the greatest uncertainty (±0.05 cm), so the area can’t be more precise than that.

Rule B: Addition & Subtraction

The result should have the same number of decimal places as the measurement with the fewest decimal places.

Example: Adding lengths

12.45 cm (2 decimal places) + 3.1 cm (1 decimal place) + 0.567 cm (3 decimal places)

12.45 + 3.1 + 0.567 = 16.117 cm (calculator)

3.1 cm has fewest decimal places = 1

Round 16.117 to 1 decimal place = 16.1 cm

Reason: The 3.1 cm measurement is uncertain in the tenths place, so the sum can’t be certain in the hundredths or thousandths places.

⚠️ Special Case: Exact Numbers

Some numbers are exact and have infinite significant figures. They don’t limit your final answer’s sig figs.

Counting numbers: “3 trials” has infinite sig figs
Definitions: 1 m = 100 cm (exact conversion)
Mathematical constants: π, e (use as many digits as needed)
Conversion factors: Sometimes exact (1 kg = 1000 g is exact in SI)

The Rule of Thumb

“A useful guiding principle (or a ‘rule of thumb’) in deciding how many significant figures to use is that the degree of significance of the result should be no greater than that of the least precise of the values used in the calculation. Therefore you should use the same number of significant figures as that in the least precise of the values used in the calculation.”

📖 Worked Example (Adapted):

Problem: Current = 5.05 A (3 sig figs), Time = 1200 s (2 sig figs, usually), Voltage = 12 V (2 sig figs). Calculate electrical energy.

Energy = V × I × t = 12 × 5.05 × 1200

12 V has fewest sig figs = 2

Calculator gives 72720 J

Round to 2 sig figs = 73,000 J or 7.3 × 10⁴ J

Note: If 1200 s is meant to have 4 sig figs, it should be written as 1200. or 1.200 × 10³.

Step-by-Step: Applying Sig Figs in CSEC Calculations

Step 1: Identify the sig figs in all measured values
Count significant figures in each measurement. Remember rules for zeros!

Step 2: Determine which measurement is least precise
For × or ÷: Fewest sig figs
For + or −: Fewest decimal places

Step 3: Do the calculation normally
Use calculator, keep all digits during intermediate steps

Step 4: Round final answer
Round to appropriate sig figs or decimal places
Use standard rounding rules: 5 or more rounds up

Step 5: Check if answer makes sense
Does the precision match your instruments?
Is it consistent with the “rule of thumb”?

Common CSEC Mistakes with Significant Figures

Mistake	Why It’s Wrong	Correct Approach
Reporting 15.236 cm from a metre rule	Metre rule has 1 mm precision (0.1 cm), so 0.001 cm precision is false	Report 15.2 cm (3 sig figs)
Using all calculator digits (e.g., 12.345678)	Calculator doesn’t know about measurement precision	Round to appropriate sig figs (e.g., 12.3 if 3 sig figs needed)
Rounding intermediate steps too early	Introduces rounding errors in multi-step calculations	Keep extra digits during calculation, round only final answer
Confusing decimal places with sig figs in +/− vs ×/÷	Different operations have different rules	+/−: fewest decimal places ×/÷: fewest sig figs
Forgetting exact numbers don’t limit sig figs	Treating “2” in formula as 1 sig fig	Exact numbers (counts, conversions) have infinite sig figs

CSEC Exam Practice

CSEC Exam Practice: Significant Figures

Question 1: How many significant figures are in each number?
(a) 0.00450
(b) 300.0
(c) 1200
(d) 1.020 × 10³

Answer:
(a) 3 sig figs (4, 5, and the last 0 after decimal)
(b) 4 sig figs (decimal point makes all zeros significant)
(c) Usually 2 sig figs in CSEC (ambiguous without decimal)
(d) 4 sig figs (1, 0, 2, 0 – scientific notation clarifies)

Explanation: Remember the rules: leading zeros not significant (a), trailing zeros with decimal are significant (b), trailing zeros without decimal are ambiguous (c), scientific notation removes ambiguity (d).

Question 2: A rectangle measures 12.4 cm by 5.25 cm. Calculate its area with appropriate significant figures.

Answer: 65.1 cm²

Explanation:
12.4 cm × 5.25 cm = 65.1 cm² (calculator gives 65.1 exactly)
12.4 has 3 sig figs, 5.25 has 3 sig figs
Both have 3 sig figs, so answer should have 3 sig figs
65.1 has 3 sig figs (6, 5, 1)

If calculator gave 65.100, we’d still round to 65.1 cm² for 3 sig figs.

Question 3: A student measures: 15.2 cm, 3.45 cm, and 0.567 cm. What is the total length with appropriate significant figures?

Answer: 19.2 cm

Explanation:
15.2 cm (1 decimal place)
3.45 cm (2 decimal places)
0.567 cm (3 decimal places)

Addition rule: Answer has same decimal places as measurement with fewest decimal places.
15.2 has fewest decimal places = 1
15.2 + 3.45 + 0.567 = 19.217 cm (calculator)
Round to 1 decimal place = 19.2 cm

Question 4: The density of an object is calculated using: mass = 12.5 g (3 sig figs), volume = 2.25 cm³ (3 sig figs). What is the density with correct significant figures?

Answer: 5.56 g/cm³

Explanation:
Density = mass ÷ volume = 12.5 ÷ 2.25 = 5.555555… g/cm³
Both measurements have 3 sig figs
Division rule: Answer has same sig figs as measurement with fewest sig figs (both have 3)
Round 5.5555… to 3 sig figs = 5.56 g/cm³

Note: The repeating 5s round up: 5.555… rounded to 3 sig figs is 5.56.

Question 5: A student uses a metre rule (precision = 1 mm) to measure a length as 15.2 cm. He then calculates an area using this length. How many significant figures should his area calculation have, and why?

Answer: 3 significant figures

Explanation:
The length measurement (15.2 cm) has 3 sig figs (1, 5, 2).
The metre rule can measure to 1 mm = 0.1 cm, so the last digit (2 in 15.2) is uncertain but significant.
“The place value of the final digit should tell us the precision of the instrument that was used.”
Any calculation using this measurement should have no more than 3 sig figs in the final answer, as the precision of the original measurement limits the precision of calculated results.

Question 6: Convert 0.000567 to scientific notation with 2 significant figures.

Answer: 5.7 × 10⁻⁴

Explanation:
0.000567 has 3 sig figs (5, 6, 7)
To write with 2 sig figs: keep 5 and 7, round the 6 (it’s followed by 7, which is ≥5, so 6 rounds up to 7)
0.000567 → 0.00057 (2 sig figs)
In scientific notation: 5.7 × 10⁻⁴

Remember: In scientific notation, the coefficient (5.7) contains the significant figures.

🎯 CSEC Sig Fig Summary

Count sig figs: All non-zero + zeros between + trailing zeros after decimal
Multiplication/Division: Fewest sig figs in measurements
Addition/Subtraction: Fewest decimal places
Exact numbers: Don’t limit sig figs (counts, conversions)
CSEC rule: Final answer sig figs = least precise measurement’s sig figs
Reporting: Match sig figs to instrument precision