Significant Figures and Rounding

Significant figures decide how many digits your all-year numerical answers should carry — a silent mark-loser. Boards test the rules directly (1-mark: "How many significant figures in …") and expect correctly rounded final answers throughout.

Why it matters

A significant figure is a digit that carries real information about a measurement's precision. Reporting more digits than justified is false precision; too few loses information.

The rules

Rule	Example	Sig. figs
All non-zero digits count	234.5	4
Zeros between non-zeros count	2005	4
Leading zeros do not count	0.0025	2
Trailing zeros with a decimal count	2.300	4
Trailing zeros without a decimal are ambiguous	2300	2 (use scientific notation)

Scientific notation removes ambiguity: 2.30 × 10³ has 3 sig figs. The power of 10 and exact/counted numbers (like the 2 in 2πr) have infinite significant figures.

Arithmetic rules

• Multiplication/division: the result keeps as many sig figs as the factor with the fewest sig figs.
• Addition/subtraction: the result keeps as many decimal places as the term with the fewest decimal places.

Rounding rule

Round the last kept digit: drop <5 → keep; >5 → round up; exactly 5 → round to make the preceding digit even (the convention used in NCERT).

Exam Tricks & Tips

• 🎯 Multiply/divide → count sig figs; add/subtract → count decimal places. Different rules!
• 🎯 Trailing zeros in 4700 are ambiguous — write 4.7 × 10³ (2 s.f.) or 4.700 × 10³ (4 s.f.) to be clear.
• 🎯 Do rounding only at the final step; keep guard digits during intermediate calculation.
• 🎯 Exact numbers (counts, defined constants) have unlimited sig figs and never limit the answer.
• 🎯 A change of unit does not change the number of significant figures (2.30 m = 230 cm = 2300 mm, all 3 s.f.).
• ❌ Common mistake: counting leading zeros (0.00530 has 3 s.f., not 5) — leading zeros only fix the decimal point.

Expected exam pattern

1-mark: count sig figs in a given number, or state a result to correct sig figs. 2-mark: perform a multiplication/addition and report with proper significant figures/rounding.

Quick recap

Non-zero and sandwiched zeros count; leading zeros never; trailing zeros count only with a decimal. Multiply/divide → fewest sig figs; add/subtract → fewest decimal places. Round at the end (5 → nearest even).

Cover the answer, recall, then check. 11 cards on significant figures.

Q1. How many significant figures in 0.0025?
A1. Two — leading zeros do not count.

Q2. How many significant figures in 2.300?
A2. Four — trailing zeros after a decimal point are significant.

Q3. How many significant figures in 2005?
A3. Four — zeros between non-zero digits count.

Q4. Why is 2300 ambiguous, and how do you fix it?
A4. Trailing zeros without a decimal are ambiguous; write in scientific notation, e.g. 2.3 × 10³.

Q5. State the significant-figure rule for multiplication and division.
A5. The result has as many significant figures as the factor with the fewest.

Q6. State the rule for addition and subtraction.
A6. The result keeps as many decimal places as the term with the fewest decimal places.

Q7. How do you round when the digit to drop is exactly 5?
A7. Round so the preceding digit becomes even (e.g. 2.745 → 2.74, 2.735 → 2.74).

Q8. How many significant figures do exact/counted numbers have?
A8. Infinite — they never limit the precision of a result.

Q9. Does changing units change the number of significant figures?
A9. No — 2.30 m, 230 cm and 2300 mm all have three significant figures.

Q10. Round 3.137 to three significant figures.
A10. 3.14 (the dropped 7 > 5 rounds the 3 up to 4).

Q11. When should you round in a multi-step calculation?
A11. Only at the final answer — carry extra guard digits through intermediate steps.

A measurement of "2.50 cm" quietly claims more precision than "2.5 cm". Significant figures are how a number carries its own certificate of precision.

Definition / core idea

Significant figures are the digits in a measurement that are known reliably plus the first uncertain digit. They communicate how precise the measurement is.

Deep explanation

Beginner — the counting rules

1. All non-zero digits are significant (234 → 3 s.f.).
2. Zeros between non-zeros are significant (2004 → 4 s.f.).
3. Leading zeros are NOT significant (0.0023 → 2 s.f.) — they only fix the decimal point.
4. Trailing zeros after a decimal ARE significant (2.300 → 4 s.f.).
5. Trailing zeros in a whole number are ambiguous (500) — resolve with scientific notation.

Intermediate — scientific notation clears ambiguity

Write the number as a × 10ᵇ with 1 ≤ a < 10. Then every digit in a is significant. So 4.700 × 10³ clearly shows 4 s.f., removing the ambiguity of "4700".

Advanced — arithmetic with significant figures

• Multiplication/Division: the result keeps as many significant figures as the factor with the fewest. (1.2 × 3.45 = 4.14 → round to 4.1, 2 s.f.)
• Addition/Subtraction: the result keeps as many decimal places as the term with the fewest. (12.11 + 0.3 = 12.41 → round to 12.4, one decimal.)

Rounding rules

Round the last kept digit: if the next digit is >5 round up, <5 round down, exactly 5 → round to the even digit (banker's rounding, e.g. 2.5 → 2, 3.5 → 4) to avoid systematic bias. Exact/counted numbers (like 2 in 2πr) have infinite significant figures and never limit the result.

Worked example

A rectangular sheet measures length 4.234 m and breadth 1.05 m. Find the area with correct significant figures.
Area = 4.234 × 1.05 = 4.4457 m².
Least significant figures among factors = 3 (from 1.05). So area = 4.45 m².

Real-world application

A pharmacist reading a dose as 5.0 mg versus 5.00 mg is stating different tolerances; significant figures tell the next person exactly how tightly the value is known.

Exam tricks & shortcuts

• Multiply/divide → count significant figures; add/subtract → count decimal places.
• Leading zeros never count; trailing zeros after a decimal always count.
• Exact constants and pure counts do not limit significant figures.

Key formulas / rules

• Counting: all non-zero digits are significant; zeros between non-zeros are significant; leading zeros are not; trailing zeros after a decimal point are significant.
• Multiplication/division: result keeps the least number of significant figures among the operands.
• Addition/subtraction: result keeps the least number of decimal places among the operands.
• Rounding: if the dropped digit is >5 round up; <5 round down; =5 → round to make the preceding digit even.
• Scientific notation: a × 10ᵇ with 1 ≤ |a| < 10; the digits in a are the significant figures.

Example: 4.237 g / 2.51 cm³ = 1.69 g·cm⁻³ (3 sig figs, limited by 2.51).

Worked Example

Problem: A rectangular sheet has length 16.2 cm and breadth 10.1 cm. Compute its area and state the result to the correct number of significant figures. Then round 4.837 × 10³ to three significant figures.

Solution:
Step 1 — Multiply the measured values:
Area = 16.2 cm × 10.1 cm = 163.62 cm².

Step 2 — Apply the significant-figure rule for multiplication. In multiplication and division, the result carries as many significant figures as the factor with the fewest significant figures. Both 16.2 (3 s.f.) and 10.1 (3 s.f.) have three significant figures, so the answer must have three.

Step 3 — Round 163.62 to three significant figures. The fourth figure is 6 (≥ 5), so round the third figure up:
163.62 → 164 cm².

Step 4 — Round 4.837 × 10³ to three significant figures. Keep 4.83 and look at the next digit, 7 (≥ 5), so round up:
4.837 × 10³ → 4.84 × 10³.

Answer: Area = 164 cm² (3 s.f.); and 4.837 × 10³ ≈ 4.84 × 10³.