Significant Figures — All Rules

Why Significant Figures Matter

Every measurement carries uncertainty. When we write a length as 5.27 cm, we’re saying we measured to the nearest 0.01 cm — three digits we trust. Significant figures are how physics tracks that trust through calculations. Round wrong, and your answer claims more precision than the data deserves.

This shows up in every NEET and JEE numerical: an answer of “9.8 m/s²” is fine when given “20 cm in 0.20 s”, but writing “9.8023 m/s²” would be wrong — the data doesn’t support four digits of precision.

We’ll cover all the counting rules, the operations rules (which differ for addition vs multiplication), scientific notation, and the rounding conventions. By the end, you’ll know exactly how many digits to report on any answer.

Key Terms & Definitions

Significant figures (sig figs): The digits in a measured value that carry meaningful information about the precision of the measurement.

Exact numbers: Counts (12 students) or defined constants (1 km = 1000 m). Have infinite sig figs and don’t limit precision.

Precision: How finely a measurement is reported (number of sig figs). Different from accuracy (how close to the true value).

Order of magnitude: The power of 10 that captures the rough size of a number. $5.2 \times 10^{-3}$ has order of magnitude $-3$.

The Counting Rules

Rule 1: All non-zero digits are significant

$237$ has 3 sig figs. $4.829$ has 4 sig figs.

Rule 2: Zeros between non-zero digits are significant

$1002$ has 4 sig figs. $30.07$ has 4 sig figs.

Rule 3: Leading zeros are NOT significant

$0.0042$ has 2 sig figs (only the 4 and 2). The leading zeros just place the decimal point.

Rule 4: Trailing zeros after a decimal point ARE significant

$2.500$ has 4 sig figs. $0.0030$ has 2 sig figs.

Rule 5: Trailing zeros in a whole number are ambiguous (without context)

$1500$ could be 2, 3, or 4 sig figs. Use scientific notation to disambiguate:

• $1.5 \times 10^3$ → 2 sig figs
• $1.50 \times 10^3$ → 3 sig figs
• $1.500 \times 10^3$ → 4 sig figs

The Operations Rules

The trickiest part: addition/subtraction follow a different rule from multiplication/division.

Multiplication and Division

The result has as many sig figs as the input with the fewest sig figs.

$2.5 \times 3.42 = 8.55 \to 8.6$ (2 sig figs)

$2.5$ has 2 sig figs, $3.42$ has 3, so the answer keeps 2.

Addition and Subtraction

The result has as many decimal places as the input with the fewest decimal places.

$12.34 + 1.5 = 13.84 \to 13.8$ (1 decimal place)

The fewer-decimal-places rule beats the fewer-sig-figs rule for addition.

Mixed Operations

Apply each rule at each step. But don’t round intermediate results — keep extra digits during the calculation, round only at the end.

Scientific Notation Rules

Numbers in scientific notation are written as $a \times 10^n$ where $1 \leq |a| < 10$.

Sig figs are counted in the mantissa $a$:

• $6.02 \times 10^{23}$ → 3 sig figs
• $6.022 \times 10^{23}$ → 4 sig figs
• $1.0 \times 10^{-5}$ → 2 sig figs

Operations:

• Multiplication: multiply mantissas, add exponents
• Division: divide mantissas, subtract exponents
• Addition: must have same exponent first, then add mantissas

Rounding Conventions

When the digit to be dropped is…

• Less than 5: round down. $3.14 \to 3.1$.
• Greater than 5: round up. $3.16 \to 3.2$.
• Exactly 5: round to even (banker’s rounding). $3.15 \to 3.2$, $3.25 \to 3.2$.

The “round to even” rule keeps cumulative bias near zero in long calculations. NCERT uses this convention.

Solved Examples

Easy (CBSE)

Express the result of $4.327 + 2.1$ to the correct sig figs.

$4.327 + 2.1 = 6.427$. The number $2.1$ has 1 decimal place, so the answer rounds to $6.4$.

Easy (NCERT)

How many sig figs in $0.04050$?

Leading zeros don’t count, internal and trailing zeros after decimal do count. So digits 4, 0, 5, 0 = 4 sig figs.

Medium (JEE Main)

The radius of a sphere is measured as $2.1$ cm. Volume to correct sig figs?

$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(2.1)^3 = 38.79\text{ cm}^3$

Input has 2 sig figs, so $V = 39$ cm³.

Hard (JEE Advanced)

A length is measured as $5.207 \pm 0.005$ m and a time as $2.34 \pm 0.01$ s. Find the speed and percentage uncertainty.

$v = 5.207 / 2.34 = 2.225$ m/s. Round to 3 sig figs (limited by the time): $v = 2.22$ m/s.

% uncertainty: $\Delta v/v = \Delta L/L + \Delta T/T = 0.005/5.207 + 0.01/2.34 = 0.00096 + 0.00427 = 0.00523 \approx 0.5\%$.

So $v = 2.22 \pm 0.01$ m/s.

Exam-Specific Tips

CBSE Boards

• The 1-mark “how many sig figs” questions are easy marks. Don’t lose them on rounding off.
• Express answers in scientific notation when the magnitude is large or small. Reduces ambiguity.
• $\pi$ in calculations: use 3.14 if the data has 3 sig figs, $22/7$ if the data has 2 sig figs.

JEE Main

• Numerical-type answers are checked to specific precision. Always carry one extra digit through calculations and round at the very end.
• Error propagation questions are common. Memorise the rules:
  • $z = x + y$: $\Delta z = \Delta x + \Delta y$
  • $z = xy$ or $x/y$: $\Delta z/z = \Delta x/x + \Delta y/y$
  • $z = x^n$: $\Delta z/z = n \Delta x/x$

NEET

• Sig fig questions appear in the units & measurements chapter, usually 1-2 questions per paper. Easy 4 marks.

Common Mistakes to Avoid

1. Counting leading zeros as significant: $0.0042$ has 2 sig figs, not 4. Leading zeros are placeholders.

2. Rounding intermediate steps: Always keep at least one extra digit until the final answer. Premature rounding compounds errors.

3. Mixing decimal-places rule with sig-fig rule: For + and −, count decimal places. For × and ÷, count sig figs.

4. Trailing zeros without context: $1500$ is ambiguous. Use $1.5 \times 10^3$ or $1.500 \times 10^3$ to be clear.

5. Treating exact numbers as limited: A count of 12 books is exact. The 12 doesn’t limit your sig figs; only measured quantities do.

Practice Questions

Q1. How many sig figs in 0.000302?

Q2. Round $2.345$ to 3 sig figs (using round-to-even).

Q3. Compute $2.5 \times 3.42 \times 4$ to correct sig figs.

Q4. $52.16 - 2.7 = ?$ in correct sig figs.

Q5. The radius of a circle is 2.1 cm. Find area in correct sig figs.

Q6. Convert 0.05670 to scientific notation.

Q7. The mass of an object is $25.45 \pm 0.05$ g. What is the percentage error?

Q8. Why is $0.500$ different from $0.5$ in significant figures?

FAQs

Are sig figs the same as decimal places? No. $25$ has 2 sig figs, $0$ decimal places. $0.025$ has 2 sig figs, 3 decimal places.

How does $\pi$ affect sig figs? $\pi$ is exact. Use as many digits of $\pi$ as needed; the data limits the result.

What’s the difference between accuracy and precision? Accuracy = closeness to the true value. Precision = closeness of repeated measurements (or, in sig fig context, how finely a value is reported).

Should I round each step or only at the end? Only at the end. Carry one or two extra digits through calculations, round once.

Does $\log$ or $\sin$ affect sig figs? For $\log x$, the number of decimal places in the result equals the sig figs in $x$. For $\sin x$, the result has the same sig figs as $x$ (when $x$ is in radians and not too close to a zero).

How are sig figs different from uncertainty? Sig figs are a quick proxy for uncertainty. A measurement of $5.27$ implies uncertainty of about $\pm 0.01$. Explicit uncertainty (like $5.27 \pm 0.03$) is more precise but also more verbose.