Significant Figures

Every measurement in science carries an inherent uncertainty — no ruler, scale, or clock is perfectly precise. Significant figures (often abbreviated as 'sig figs') are the digits in a measured or calculated value that carry meaningful information about its precision. They tell us how well we actually know a quantity, distinguishing trustworthy digits from those that are merely placeholders or guesses. Consider measuring a pencil with a ruler marked in millimetres. You might confidently read 14.3 cm and estimate one more digit, writing 14.32 cm. That final digit '2' is your best estimate — it is uncertain, but it still carries information. All four digits here are significant. By contrast, a number like 0.0047 m has only two significant figures: the leading zeros simply locate the decimal point and convey no precision about the measurement itself. Significant figures are not just a bookkeeping exercise — they prevent the illusion of false precision. If you measure two lengths each to the nearest centimetre and then report their sum to the nearest micrometre, you are claiming far more precision than your instruments ever provided. Mastering significant figures ensures that every number you write faithfully represents what was actually measured or calculated, and nothing more.

Key takeaways
  • Identifying Significant Figures
  • Significant Figures in Addition and Subtraction
  • Significant Figures in Multiplication and Division
  • Exact Numbers and Constants
  • Rounding Rules

Identifying Significant Figures

Apply these rules to any number: (1) All non-zero digits are always significant — 3.47 has 3 sig figs. (2) Zeros between non-zero digits are significant — 3.047 has 4 sig figs. (3) Leading zeros (before the first non-zero digit) are never significant — 0.0047 has 2 sig figs; the zeros only fix the decimal position. (4) Trailing zeros after a decimal point are significant — 3.470 has 4 sig figs, indicating the measurement was precise to the thousandths place. (5) Trailing zeros in a whole number are ambiguous without a decimal point — 1500 could have 2, 3, or 4 sig figs; writing 1.500×1031.500 \times 10^3 makes it unambiguously 4 sig figs. Scientific notation eliminates all ambiguity.

Significant Figures in Addition and Subtraction

When adding or subtracting, the result must be rounded to the same decimal place as the input with the fewest decimal places — not the fewest significant figures. For example: 12.11+18.0+1.013=31.12312.11 + 18.0 + 1.013 = 31.123, but 18.0 is only known to the tenths place, so the answer rounds to 31.131.1. Think of it as aligning numbers in a column: your answer cannot be more precise than your least-precise addend.

Significant Figures in Multiplication and Division

When multiplying or dividing, round the result to the same number of significant figures as the measurement with the fewest significant figures. For example: 4.56×1.4=6.3844.56 \times 1.4 = 6.384, but 1.4 has only 2 sig figs, so the answer is 6.46.4. This rule reflects that the relative uncertainty of a product or quotient is dominated by the least precise factor involved.

Exact Numbers and Constants

Some quantities are exact — they have infinite significant figures and never limit your answer. These include defined conversions (exactly 100 cm in 1 m), counted quantities (12 eggs in a dozen), and mathematical constants used in formulas (the '2' in E=12mv2E = \frac{1}{2}mv^2 is exact). Fundamental constants like c=2.998×108c = 2.998 \times 10^8 m/s are measured, however, and do carry significant figures that can limit precision.

Rounding Rules

To round a number: identify the last significant digit you wish to keep. Look at the next digit. If it is less than 5, leave the kept digit unchanged (round down). If it is 5 or greater, increase the kept digit by 1 (round up). Example: 3.48513.4851 rounded to 3 sig figs is 3.493.49 (the digit after '8' is 5, so round up). In multi-step calculations, carry extra digits through intermediate steps and round only the final answer to avoid cumulative rounding errors.

The Key Equation

$$N_{\text{sig}} = \text{digits counted by the five rules above}
- **$N_{\text{sig}}$** — Number of significant figures (dimensionless): The total count of digits in a number that convey real precision about the measurement. - **$\delta x$** — Absolute uncertainty (same as measured quantity): The uncertainty in the last significant figure, typically $\pm 1$ in the last recorded digit for a single measurement. - **$\frac{\delta x}{x}$** — Relative (fractional) uncertainty (dimensionless): Uncertainty expressed as a fraction of the measured value; roughly $10^{-(N_{\text{sig}}-1)}$ for a quantity with $N_{\text{sig}}$ significant figures. ## Worked Example **Problem.** A student measures the sides of a rectangular metal plate: length $L = 24.3$ cm (3 sig figs) and width $W = 8.52$ cm (3 sig figs). She also measures its mass as $m = 47.8$ g (3 sig figs) and its thickness as $t = 0.30$ cm (2 sig figs). (a) Calculate the area $A = L \times W$ to the correct number of significant figures. (b) Calculate the volume $V = L \times W \times t$ to the correct number of significant figures. (c) Calculate the density $\rho = m/V$ to the correct number of significant figures.

A=207 cm2,V=62 cm3,ρ=0.77 g/cm3A = 207 \text{ cm}^2,\quad V = 62 \text{ cm}^3,\quad \rho = 0.77 \text{ g/cm}^3

∗∗Answer.∗∗Area=207cm2,Volume=62cm3,Density=0.77g/cm3**Answer.** Area = 207 cm², Volume = 62 cm³, Density = 0.77 g/cm³
Figure
Diagram
Identifying Significant Figures — Visual GuideNumber: 0.004 70→ Leading zeros: NOT significant (grey) | '4', '7', '0' ARE significant → 3 sig figs0.004 7 0Number: 3 0 4 . 0 5→ All digits significant (zeros between non-zeros, trailing zero after decimal) → 6 sig figs3 0 4 . 0 5Number: 1500 (ambiguous) vs 1.500 × 10³ (unambiguous)→ Without decimal: could be 2–4 sig figs. Scientific notation → clearly 4 sig figsAddition/Subtraction Rule (align decimal places) 12.11 ← 2 decimal places 18.0 ← 1 decimal place ← LEAST PRECISE+ 1.013 ← 3 decimal places= 31.1 ← round to 1 decimal place
Origins · 1 minCarl Friedrich Gauss and how this was discovered
CF
Carl Friedrich Gauss
1777–1855 · German

Gauss developed the method of least squares (1809) and the theory of measurement errors, laying the mathematical foundation for understanding uncertainty and precision in experimental data. His work on the normal distribution (the 'Gaussian' curve) explains why random measurement errors cluster symmetrically around the true value — the bedrock of modern uncertainty analysis and significant-figure conventions. Gauss reportedly corrected his father's payroll arithmetic at age three. He later said mathematics was so natural to him that he learned to calculate before he learned to read.

  • Measurement Uncertainty & Error — Significant figures are a practical shorthand for expressing uncertainty; understanding random and systematic errors gives the deeper physical reason why measured digits become unreliable beyond a certain place.
  • Scientific Notation — Scientific notation is the standard way to express numbers of any magnitude without ambiguity about how many digits are significant, making it inseparable from proper sig-fig practice.
  • Dimensional Analysis — Checking that units and dimensions balance in an equation is a companion skill to significant figures — together they guard against both unit errors and false precision in reported results.
  • Propagation of Uncertainty — The sig-fig rules for multiplication/division and addition/subtraction are simplified versions of formal error propagation, which precisely tracks how uncertainties combine through calculations.