Units, prefixes and scientific notation

This is the part of the course nobody revises and almost everybody loses marks on, because it is assumed rather than taught. The IB has specific conventions for how quantities and units are written, and they are marked.

Quantities and units

A quantity is something that can be measured — time, length, electric current.

A unit is what you measure it in — second, metre, ampere.

How the IB expects you to write them

Read this section even if you think you know it. All four of these appear in mark schemes.

Quantities are written in italics: v = Δs / Δt
Units are written in Roman upright type: s, hr, ft, m, A
A unit written out in full is always lower case, even when it is named after a person: metre, newton, pascal, ampere, second
A unit symbol takes a capital first letter if it is named after a person — only the first letter: N (newton), A (ampere), Pa (pascal), K (kelvin), Hz (hertz). Pa, never PA.

The seven SI base units

Everything else in physics is built from these seven, so they have to be known rather than looked up.

QUANTITY UNIT NAME SYMBOL

length metre m
time second s
mass kilogram kg
electric current ampere A
thermodynamic temperature kelvin K
amount of substance mole mol
luminous intensity candela cd

Derived units

Derived units are built from the base units. The IB writes them with negative indices and spaces, not with a slash:

m s⁻¹ not m / s

kg m⁻³ not kg / m³

The method never changes: replace every quantity in the formula with its base units, then simplify.

ρ = m / V → kg / m⁻³ = kg m⁻³

v = Δs / Δt → m / s = m s⁻¹

a = Δv / Δt → (m s⁻¹) / s = m s⁻²

F = ma → kg × m s⁻² = kg m s⁻², so 1 N = 1 kg m s⁻²

SI prefixes

PREFIX SYM FACTOR PREFIX SYM FACTOR
peta P 10¹⁵ deci d 10⁻¹
tera T 10¹² centi c 10⁻²
giga G 10⁹ milli m 10⁻³
mega M 10⁶ micro μ 10⁻⁶
kilo k 10³ nano n 10⁻⁹
hecto h 10² pico p 10⁻¹²
deca da 10¹ femto f 10⁻¹⁵

Capital M is mega (10⁶). Lower-case m is milli (10⁻³). They are a factor of 10⁹ apart, and this is the most expensive typo in the whole topic.

Converting with prefixes

Turn the prefix into its power of ten first, then adjust the coefficient to sit between 1 and 10.

300 mm = 300 × 10⁻³ m = 3.00 × 10⁻¹ m

368 μm = 368 × 10⁻⁶ m = 3.68 × 10⁻⁴ m

200 MV = 200 × 10⁶ V = 2.00 × 10⁸ V

0.000000056 m = 0.056 × 10⁻⁶ m = 0.056 μm = 56 nm

10 m ÷ 200 mm = 10 / 0.200 = 50 (a ratio — no units)

Areas and volumes. A prefix inside a squared or cubed unit is raised to that power too. 1 cm² = (10⁻² m)² = 10⁻⁴ m², and 1 cm³ = 10⁻⁶ m³ — not 10⁻² and 10⁻³.

Scientific notation and order of magnitude

c = 3.00 × 10⁸ m s⁻¹

Coefficient 3.00 — must satisfy 1 ≤ coefficient < 10
Base 10
Exponent 8

The order of magnitude is the power of ten nearest the number. Write the number in scientific notation and read the exponent off.

a sheet of paper, mass 3.8 g = 3.8 × 10⁻³ kg
→ order of magnitude 10⁻³ kg

Where marks are actually lost

Writing m/s instead of m s⁻¹.
Capitalising a unit written out in full — "Newton" instead of "newton".
Writing PA for pascal instead of Pa.
Confusing M (mega) with m (milli).
Forgetting to square or cube the prefix in cm² and cm³.
Leaving a coefficient outside 1–10 in scientific notation.
Giving units to a ratio.

Notes and practice

Lecture notes — Units & Prefixes .pdf