General Physics 1.2 — Significant Figures and Scientific Notation

Why Precision Has to Be Stated Explicitly

When you write down a number from a measurement, the digits you write carry an implicit claim: this is how precisely I know this quantity. Writing 12.3 cm is not the same as writing 12.30 cm — the second form claims you know the length to the nearest hundredth of a centimeter, the first only to the nearest tenth. The digits that carry this meaning are called significant figures (or significant digits).

The rules exist because calculators and computers will happily multiply 12.3 × 4.56 and give you 56.088 — five digits — even though neither input was known to five figures. Reporting 56.1 (three significant figures, matching the least precise input) is honest. Reporting 56.088 implies a precision you never had.

Counting Significant Figures

The rules for which digits are significant:

• Non-zero digits are always significant. 453 has three; 7.92 has three.
• Zeros between non-zero digits are significant. 4023 has four; 1.008 has four.
• Leading zeros (zeros before the first non-zero digit) are never significant — they are just placeholders. 0.0047 has two significant figures (4 and 7).
• Trailing zeros after a decimal point are significant. 12.00 has four; 0.500 has three.
• Trailing zeros in a whole number with no decimal point are ambiguous. 1500 could be 2, 3, or 4 significant figures — you can’t tell. Scientific notation resolves this: 1.5 × 10³ (two sig figs), 1.50 × 10³ (three), 1.500 × 10³ (four).

Arithmetic with Significant Figures

Multiplication and division: the result has as many significant figures as the input with the fewest significant figures.

6.38 × 2.1 = 13.398 → rounds to 13 (2 sig figs, limited by 2.1)

Addition and subtraction: the result is rounded to the least precise decimal position among the inputs.

12.52 + 0.3 = 12.82 → rounds to 12.8 (limited by 0.3, which is known only to tenths)

The logic differs between the two cases: multiplication/division combine relative uncertainties (percentages); addition/subtraction combine absolute uncertainties (the last decimal place).

Scientific Notation

Scientific notation writes any number as a coefficient between 1 and 10, multiplied by a power of ten:

N × 10ⁿ where 1 ≤ N < 10

Examples:

• 6,370,000 m (Earth’s radius) → 6.37 × 10⁶ m
• 602,214,076,000,000,000,000,000 (Avogadro’s number) → 6.022 × 10²³
• 0.000000000167 C (electron charge) → 1.67 × 10⁻¹⁰ C (approximate)
• 299,792,458 m/s (speed of light) → 2.998 × 10⁸ m/s

To convert: count how many places you move the decimal point. Move left → positive exponent. Move right → negative exponent. Alternatively: if the original number is greater than 1, the exponent is positive; if it is less than 1, the exponent is negative.

Why it matters:

1. Eliminates ambiguity about significant figures (the coefficient carries them, the exponent is exact).
2. Makes order-of-magnitude comparisons immediate — a quick scan of the exponent tells you the scale.
3. Makes arithmetic on very large or small numbers tractable: multiply the coefficients, add the exponents.

(3.0 × 10⁸) × (2.0 × 10⁻³) = 6.0 × 10⁵

Accuracy vs. Precision

These are often conflated but mean different things:

• Accuracy — how close a measurement is to the true value. A systematic error (miscalibrated instrument, wrong zero) degrades accuracy.
• Precision — how repeatable a measurement is; the spread among repeated measurements. A noisy instrument gives imprecise results even if the average is accurate.

You can be precise but inaccurate (consistently measuring 10.2 cm when the true value is 9.8 cm), or accurate but imprecise (measurements scatter around the true value with no systematic offset). Good measurement requires both.

Significant figures encode precision, not accuracy — they say nothing about whether your instrument was calibrated correctly.

Order of Magnitude

A related habit: order of magnitude estimation. Before doing a precise calculation, estimate the answer to the nearest power of ten. If your precise answer is off by more than one or two orders of magnitude, you’ve made an error somewhere — either in setup or arithmetic.

Fermi estimation relies on this: the number of piano tuners in a city, the mass of all humans on Earth, the number of heartbeats in a lifetime — all estimable to within an order of magnitude by chaining rough but honest numbers together. The point isn’t precision; it’s catching gross errors and building physical intuition.

What stuck: Significant figures are not a formatting rule — they are a statement about epistemic honesty. Every measurement has an uncertainty, and you are responsible for propagating it faithfully through your calculations rather than hiding it behind false precision.