recall and use the following prefixes and their symbols to indicate decimal submultiples or multiples of both base and derived units: pico (p), nano (n), micro ( μ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T)

1 Physical Quantities and Units (Cambridge International AS & A Level Physics 9702)

1.1 Physical quantities

  • Quantity – a property of a system that can be measured (e.g. length, mass, time).

  • Quantity type – the kind of quantity (e.g. length, force, energy).

  • Magnitude + unit – every measured quantity must be expressed as a number together with its unit.

  • Estimation – students should be able to make reasonable estimates of a quantity (e.g. “estimate the mass of a textbook”).

  • Scalar vs. vector

    • Scalar: described by magnitude only (speed, temperature).

    • Vector: described by magnitude and direction (displacement, velocity, force).

1.2 SI base units

The seven fundamental units from which all other units are derived are:

Base unitSymbolQuantity
metremlength
kilogramkgmass
secondstime
ampereAelectric current
kelvinKthermodynamic temperature
molemolamount of substance

1.3 SI prefixes – decimal multiples and sub‑multiples

Prefixes are placed directly before a unit symbol to express very large or very small quantities in a compact form. Each prefix represents a power of ten.

PrefixSymbolFactorExample (base unit)Example (mole)
picop10‑121 pF = 1 × 10‑12 F1 pmol = 1 × 10‑12 mol
nanon10‑91 nL = 1 × 10‑9 L1 nmol = 1 × 10‑9 mol
microµ10‑61 µm = 1 × 10‑6 m1 µmol = 1 × 10‑6 mol
millim10‑31 ms = 1 × 10‑3 s1 mmol = 1 × 10‑3 mol
centic10‑21 cm = 1 × 10‑2 m–
decid10‑11 dm = 1 × 10‑1 m–
kilok1031 km = 1 × 103 m1 kmol = 1 × 103 mol
megaM1061 MJ = 1 × 106 J1 Mmol = 1 × 106 mol
gigaG1091 GW = 1 × 109 W1 Gmol = 1 × 109 mol
teraT10121 TB = 1 × 1012 bytes1 Tmol = 1 × 1012 mol

How to apply a prefix

  1. Identify the quantity and its numerical value.

  2. Choose a prefix that brings the number into the convenient range 0.1 – 1000.

  3. Divide (for a sub‑multiple) or multiply (for a multiple) by the corresponding power of ten.

  4. Write the result using the prefix symbol directly before the unit symbol.

Worked examples

Example 1 – Length

Convert 0.00045 m to an appropriate SI prefix.

  1. 0.00045 m = 4.5 × 10‑4 m.

  2. The nearest power of ten that matches a prefix is 10‑3 (milli).

  3. Divide by 10‑3: (4.5 × 10‑4 m) ÷ 10‑3 = 0.45 mm.

  4. Result: 0.45 mm.

Example 2 – Energy

Express 3.2 × 106 J using a suitable prefix.

  1. 106 corresponds to the prefix mega (M).

  2. Divide by 106: (3.2 × 106 J) ÷ 106 = 3.2 MJ.

  3. Result: 3.2 MJ.

Example 3 – Power (derived unit)

Convert 0.025 W to a more convenient form.

  1. 0.025 W = 2.5 × 10‑2 W.

  2. Using the milli‑prefix (10‑3): (2.5 × 10‑2 W) ÷ 10‑3 = 25 mW.

  3. Result: 25 mW.

1.4 Derived units and prefix usage

Derived units are formed from the base units. Prefixes are applied in exactly the same way as for base units.

Derived unitBase‑unit expressionCommon prefixed form
newton (N)kg·m·s‑2kN = 103 N (kilonewton)
joule (J)N·m = kg·m2·s‑2MJ = 106 J (megajoule)
watt (W)J·s‑1 = kg·m2·s‑3mW = 10‑3 W (milliwatt)
pascal (Pa)N·m‑2 = kg·m‑1·s‑2kPa = 103 Pa (kilopascal)
volt (V)kg·m2·s‑3·A‑1mV = 10‑3 V (millivolt)
microgram (µg)10‑6 gµg = 10‑6 g

Scientific notation with prefixes

Instead of writing 4.5 × 10‑6 F you may write 4.5 µF. Both are equivalent; the prefixed form is preferred in laboratory work and exam answers.

Worked example – Converting a derived quantity

Convert 3 kN·m to joules.

  1. 1 N·m = 1 J, so 1 kN·m = 103 J.

  2. 3 kN·m = 3 × 103 J = 3 kJ.

  3. Result: 3 kJ.

1.5 Errors, uncertainties & significant figures (AO2)

  • Absolute uncertainty – the ± value expressed in the same units as the measurement.

  • Relative (fractional) uncertainty – Δx / x.

  • Percentage uncertainty – (Δx / x) × 100 %.

  • Significant figures

    • All non‑zero digits are significant.

    • Zeros between non‑zero digits are significant.

    • Leading zeros are not significant; trailing zeros are significant only if a decimal point is present.

Uncertainty propagation with prefixes (AO2)

Suppose a voltage is measured as 12.3 ± 0.2 V. Express the result in millivolts and give the uncertainty.

  1. 1 V = 103 mV ⇒ 12.3 V = 12 300 mV.

  2. Uncertainty: 0.2 V = 0.2 × 103 mV = 200 mV.

  3. Result: 12 300 ± 200 mV.

1.6 Checking dimensional consistency (AO1)

Every term in a physical equation must have the same dimensions. Prefixes do not affect dimensions; they only scale the numerical value.

Mini‑exercise

Show that the equation for mechanical work, W = F d, is dimensionally consistent when F = 3 kN and d = 2 m. Express the final answer in joules.

  1. Convert the force: 3 kN = 3 × 103 N.

  2. Multiply: W = (3 × 103 N)(2 m) = 6 × 103 N·m.

  3. Since 1 N·m = 1 J, W = 6 kJ = 6 × 103 J.

  4. Dimensions: kg·m2·s‑2 on both sides – the equation is consistent.

1.7 Practice questions (linked to syllabus outcomes)

  1. AO1 – Knowledge & understanding: Write 7.8 × 10‑9 F using an appropriate SI prefix.
    Answer: 7.8 nF.

  2. AO1: Convert 5 km to metres and then express the result using the most suitable prefix.
    Answer: 5 km = 5 000 m = 5 km (the original prefix is already optimal).

  3. AO2 – Application of concepts: A power rating is 2.5 × 103 W. Express this in kilowatts.
    Answer: 2.5 kW.

  4. AO2: Express 0.00012 g in micrograms.
    Answer: 0.00012 g = 1.2 × 10‑4 g = 120 µg.

  5. AO2 – Uncertainty propagation: A length is measured as 3.45 ± 0.03 cm. Write the result in millimetres.
    Answer: 34.5 ± 0.3 mm.

  6. AO1 – Dimensional consistency: Convert 3 kN·m to joules and state the dimensions of each term.
    Answer: 3 kJ; dimensions = kg·m2·s‑2 for both sides.

Suggested diagram: a vertical ladder showing the hierarchy of SI prefixes from pico (10⁻¹²) at the bottom to tera (10¹²) at the top, with arrows indicating a factor of ten between successive steps.