Quote results at the right precision using standard form and SI prefixes
AQA A-Level Physics 1
🎯Count significant figures correctly in any number
✂️Round answers to an appropriate number of significant figures
📐Write values in standard form (scientific notation)
🔤Use SI prefixes from pico to tera
⚖️Match precision of answer to least precise input
🚫Avoid false precision in physics calculations
What Are Significant Figures?
Significant figures (s.f.) express how precisely a value is known. Every digit that carries meaningful information about the size of the number is significant.
Rules for counting significant figures:
1. All non-zero digits are significant: 4.72 has 3 s.f.
2. Zeros between non-zero digits are significant: 5.04 has 3 s.f.
3. Leading zeros are NOT significant: 0.007 has 1 s.f.
4. Trailing zeros after a decimal point ARE significant: 2.50 has 3 s.f.
5. Trailing zeros in a whole number may or may not be significant — use standard form to be unambiguous: 3000 could be 1–4 s.f.; write 3.0 × 10³ to show 2 s.f.
Number
Significant Figures
0.0450
3 (4, 5, 0)
30700
ambiguous — use standard form
3.07 × 10⁴
3
1.000
4
0.00102
3
Rounding and Appropriate Precision
Your final answer should be given to the same number of significant figures as the least precise measurement used in the calculation. Giving more figures implies a precision you don't actually have — this is called false precision.
Rounding rule: if the next digit is 5 or above, round up; if below 5, round down.
Example: 3.745 rounded to 3 s.f. → 3.75 (the 4th digit is 5 → round up)
Only round at the final step. Rounding intermediate values causes accumulated rounding errors that can change your answer significantly.
In multi-step calculations, carry one or two extra significant figures throughout and round only the final answer.
Standard Form (Scientific Notation)
Standard form writes any number as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
General form: A × 10ⁿ
Example: 0.000 348 = 3.48 × 10⁻⁴
Example: 6 400 000 = 6.4 × 10⁶
Standard form removes ambiguity about significant figures and makes very large or very small numbers manageable. In physics, you will constantly work with values like the charge on an electron (1.60 × 10⁻¹⁹ C) or Avogadro's number (6.02 × 10²³ mol⁻¹).
To convert: count how many places the decimal point moves. Moving left gives positive power; moving right gives negative power.
SI Prefixes
SI prefixes let us express very large or very small quantities without standard form notation in everyday usage. You must know these for A-Level Physics.
Prefix
Symbol
Multiplier
Power of 10
tera
T
1 000 000 000 000
10¹²
giga
G
1 000 000 000
10⁹
mega
M
1 000 000
10⁶
kilo
k
1 000
10³
milli
m
0.001
10⁻³
micro
μ
0.000 001
10⁻⁶
nano
n
0.000 000 001
10⁻⁹
pico
p
10⁻¹²
10⁻¹²
Always convert prefixed units to base SI units before substituting into equations. E.g. 450 nm = 450 × 10⁻⁹ m = 4.50 × 10⁻⁷ m.
How many significant figures does 0.004 080 have? Write it in standard form.
1Leading zeros (0.00) are not significant — ignore them.
24, 0, 8, 0 remain. The zero between 4 and 8 is significant. The trailing zero after 8 is significant (it's after the decimal point).
3Count: 4 → 0 → 8 → 0 = 4 significant figures
4Standard form: 4.080 × 10⁻³
4 significant figures; standard form: 4.080 × 10⁻³
Calculate the speed of a wave: v = fλ where f = 3.2 × 10⁸ Hz and λ = 1.40 × 10⁻² m. Give your answer to an appropriate number of significant figures in standard form.
4Least precise input: f = 3.2 × 10⁸ Hz has 2 s.f. → round to 2 s.f.
v = 4.5 × 10⁶ m s⁻¹
Convert 780 nm to metres and express in standard form with appropriate significant figures.
1nm = nanometres → 1 nm = 1 × 10⁻⁹ m
2780 nm = 780 × 10⁻⁹ m
3Rewrite in standard form: 7.80 × 10⁻⁷ m
7.80 × 10⁻⁷ m (3 s.f., matching the original 780 nm)
A student calculates a resistance as 47.286 Ω from measurements of V = 12.3 V (3 s.f.) and I = 0.260 A (3 s.f.). How should the result be quoted?
1Both inputs have 3 significant figures → answer should be 3 s.f.
247.286 rounded to 3 s.f.: look at 4th digit → 8, round up → 47.3 Ω
R = 47.3 Ω (3 s.f.)
1. How many significant figures does 0.02300 have?
Leading zeros are not significant. The digits 2, 3, 0, 0 are all significant (the trailing zeros after the decimal point count). Answer: 4 s.f.
2. What is 1 μA in amperes in standard form?
μ (micro) = 10⁻⁶. So 1 μA = 1 × 10⁻⁶ A.
3. Round 0.006 753 to 2 significant figures.
The first two significant figures are 6 and 7. The next digit is 5, so round up: 0.0068 (= 6.8 × 10⁻³).
4. A calculation gives 9.998 × 10⁴. Rounded to 2 s.f., this is:
9.998 rounds to 10.0 → in standard form 1.00 × 10⁵. To 2 s.f.: 1.0 × 10⁵.
5. Express 45 GHz in Hz in standard form.
1. A student writes a calculated velocity as 12.475 836 m s⁻¹ based on distance = 45.2 m and time = 3.6 s. Identify the problem with this answer and write the correct version.
The student has given 8 significant figures, implying far greater precision than the data warrants. Distance has 3 s.f., time has 2 s.f. — the answer should be to 2 s.f. (the least precise). Correct answer: v = 45.2/3.6 = 12.555... ≈ 13 m s⁻¹ (2 s.f.).
2. Explain the difference between 3 × 10⁴, 3.0 × 10⁴, and 3.00 × 10⁴. Why does this matter in physics?
3 × 10⁴ = 1 significant figure (precision: ±5000). 3.0 × 10⁴ = 2 significant figures (precision: ±500). 3.00 × 10⁴ = 3 significant figures (precision: ±50). In physics, trailing zeros in standard form indicate measured precision. Writing more zeros claims higher precision. This matters because it tells other physicists how reliable the measurement is and determines how many s.f. to use in further calculations.
3. A student measures a current of 2.4 mA with a digital ammeter showing 2 decimal places in mA. Another student measures a voltage of 5.50 V. Calculate the resistance and quote it correctly.
I = 2.4 mA = 2.4 × 10⁻³ A (2 s.f.); V = 5.50 V (3 s.f.). R = V/I = 5.50 / (2.4 × 10⁻³) = 2291.67 Ω. Least precise: I has 2 s.f. → answer to 2 s.f. R = 2300 Ω = 2.3 × 10³ Ω.