Significant Figures and Scientific Notation: The Rules Engineering Students Get Wrong in Lab Reports

How many digits should you report? A practical guide to sig figs, rounding, scientific notation and error propagation — with the exact rules graders check.

A student measures a beam with a ruler marked in millimetres, computes a stress in Excel, and reports 147.3928571 MPa. Every grader on Earth circles it. The ruler knew four digits; the answer claims ten. Significant figures are the honest way to say how much you actually know.

Which digits are significant?

1. Non-zero digits are always significant: 3.14 has 3.
2. Zeros between non-zeros are significant: 1002 has 4.
3. Leading zeros are never significant: 0.0025 has 2.
4. Trailing zeros after a decimal point are significant: 2.500 has 4.
5. Trailing zeros in a whole number are ambiguous: 1500 could be 2, 3 or 4. This is exactly why scientific notation exists.

1500      → ambiguous
1.5 × 10³ → 2 sig figs
1.50 × 10³ → 3 sig figs
1.500 × 10³ → 4 sig figs

Scientific notation: the two things to get right

Form: a × 10ⁿ with 1 ≤ |a| < 10. Two habits prevent most errors:

• Moving the decimal left increases the exponent. 0.000 032 → 3.2 × 10⁻⁵ (moved 5 places right, so exponent −5).
• Calculator E-notation is not multiplication. 3.2E-5 means 3.2 × 10⁻⁵. Typing 3.2 × 10 ^ -5 on a calculator without parentheses can silently give you (3.2 × 10)^−5. Use the EXP/EE key or write 3.2*10^(-5).

Engineering notation is the same idea restricted to exponents that are multiples of 3, so numbers map directly onto SI prefixes: 47 × 10³ Ω = 47 kΩ; 4.7 × 10⁻⁶ F = 4.7 µF.

Rounding rules that survive a calculation

Multiplication and division → count sig figs

The result keeps as many significant figures as the least precise input.

Force  = 2.5 kg × 9.81 m/s²  = 24.525 → 25 N   (2.5 has 2 sig figs)
Stress = 1250 N / 0.0331 m²  = 37 764.4 → 3.78 × 10⁴ Pa (3 sig figs)

Addition and subtraction → count decimal places

The result keeps as many decimal places as the least precise input.

12.11 + 0.3 + 1.005 = 13.415 → 13.4   (0.3 has 1 decimal place)

Exact numbers don't count

Counting numbers (3 bolts), defined conversions (1 in = 25.4 mm exactly) and integers in formulas (the 2 in ½mv²) have infinite sig figs and never limit the result.

Round only at the end

Keep at least one extra guard digit through intermediate steps. Rounding 2.35 → 2.4 then squaring gives 5.76; keeping 2.35 gives 5.52. That is a 4% error introduced by tidiness.

What sig figs are really approximating: uncertainty

Sig figs are a shorthand. The real thing a lab report wants is x ± u. The core propagation rules for independent uncertainties:

Worked example. Density from m = 45.2 ± 0.1 g and V = 12.0 ± 0.5 mL:

ρ = 45.2 / 12.0 = 3.7667 g/mL
relative u = √((0.1/45.2)² + (0.5/12.0)²) = √(4.9e-6 + 1.74e-3) = 0.0418
u_ρ = 3.7667 × 0.0418 = 0.157
Report: ρ = 3.77 ± 0.16 g/mL

Notice the volume dominates the error. That is the actionable insight — buy a better graduated cylinder, not a better balance — and you only see it by propagating, not by counting digits.

Reporting conventions graders check

• Round the uncertainty to 1 significant figure (2 if it starts with 1), then round the value to the same decimal place: 3.77 ± 0.16, not 3.7667 ± 0.157.
• Value and uncertainty share units and exponent: (2.45 ± 0.03) × 10⁵ Pa.
• Percent uncertainty is fine in text; absolute uncertainty is required in tables.
• Don't let spreadsheet defaults set your precision. Format cells deliberately.