IB Physics SL Tool 3 — Mathematics Paper 1 & 2 SI units & prefixes ~8 min read

Base & Derived Units

Every measurement in physics, from the mass of an electron to the distance across a galaxy, is built out of just seven base units. Everything else — the newton, the joule, the volt — is a derived unit, assembled from those seven like words from an alphabet. Learn how the building blocks fit together and you can check any equation, decode any unit, and never be thrown by an unfamiliar one again.

📘 What you need to know

There are 7 SI base units; IB Physics uses six of them (not the candela)
Prefixes (kilo, milli, micro…) scale a unit up or down by powers of ten
Derived units are combinations of base units, found from a quantity’s defining equation
The newton is kg m s⁻²; the joule is kg m² s⁻²; the pascal is kg m⁻¹ s⁻²
Some quantities use convenient non-SI units (eV, kWh, light year, parsec, AU) that you must be able to convert
The prefix and unit tables are in the data booklet — you don’t memorise them, you use them

The seven SI base units

These are the foundation. Each measures one physical quantity and can’t be broken down into anything simpler within the SI system.

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

A tiny exam-saver: the candela is the odd one out — it’s a genuine SI base unit, but IB Physics doesn’t use it. So when a question asks how many base units you’ll actually meet in the course, the answer is six, not seven.

Prefixes: scaling units up and down

When numbers get very large or very small, prefixes let you keep them tidy. Each prefix stands for a power of ten — a kilowatt is 10³ watts, a milligram is 10⁻³ grams. Here are the ones that come up most in physics.

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

Building derived units

A derived unit is just base units multiplied and divided together. The trick to finding one is to start from the quantity’s defining equation and swap each symbol for its base units. Take the newton, the unit of force. Force is mass × acceleration, and acceleration is metres per second squared, so:

Swap each quantity for its base units, then multiply through: the newton is kilogram-metre-per-second-squared.

The same recipe unpacks every derived unit. Energy is ½mv², so the joule is kg × (m s⁻¹)² = kg m² s⁻². Pressure is force ÷ area, so the pascal is kg m s⁻² ÷ m² = kg m⁻¹ s⁻².

🧭 Finding a derived unit

Write the defining equation for the quantity (from the data booklet if needed)
Replace each symbol with its base units
Multiply and divide the units, collecting powers of each base unit
Simplify to the tidiest form — that’s your answer

Common derived units

These come up again and again. It’s worth being able to recognise the base-unit form, since a “state the fundamental units of…” question is a quick, easy mark.

Derived unit	Quantity	In base SI units
newton (N)	force	kg m s⁻²
pascal (Pa)	pressure	kg m⁻¹ s⁻²
joule (J)	energy	kg m² s⁻²
watt (W)	power	kg m² s⁻³
hertz (Hz)	frequency	s⁻¹
coulomb (C)	charge	A s
volt (V)	potential difference	kg m² s⁻³ A⁻¹
ohm (Ω)	resistance	kg m² s⁻³ A⁻²

Quick recap: seven base units (six used in IB); prefixes scale by powers of ten; derived units come from a defining equation — newton = kg m s⁻², joule = kg m² s⁻², pascal = kg m⁻¹ s⁻².

Handy non-SI units

Sometimes an SI unit is awkwardly big or small, so physicists use a convenient alternative. You need to know what each one means and how to convert it.

Unit	Used for	Conversion
electronvolt (eV)	tiny energies	1 eV = 1.60 × 10⁻¹⁹ J
kilowatt-hour (kWh)	household energy	1 kWh = 3.60 × 10⁶ J
light year (ly)	stellar distance	1 ly = 9.46 × 10¹⁵ m
parsec (pc)	stellar distance	1 pc = 3.26 ly
astronomical unit (AU)	Solar-System distance	1 AU = 1.50 × 10¹¹ m

WE 1

(a) A household uses 3200 kWh of electricity in a year. Express this energy in joules. (b) A star lies 40 pc from Earth. Express this distance in metres. Use 1 kWh = 3.60 × 10⁶ J, 1 pc = 3.26 ly and 1 ly = 9.46 × 10¹⁵ m.

Part (a) — energy in J 1 kWh = 3.60 × 10⁶ J 3200 × (3.60 × 10⁶) = 1.152 × 10¹⁰ ≈ 1.2 × 10¹⁰ J Part (b) — distance in m first pc → ly, then ly → m 40 × 3.26 × (9.46 × 10¹⁵) ≈ 1.2 × 10¹⁸ m Chain the conversions one step at a time — parsecs to light years, light years to metres — so no factor gets dropped.

WE 2

Show that the pascal, the SI unit of pressure, is equivalent to kg m⁻¹ s⁻² in base units.

Start from the definition of pressure and substitute the base units of force and area.

Definition pressure = force ÷ area, so Pa = N ÷ m² Substitute base units N = kg m s⁻² (force = mass × acceleration) Pa = (kg m s⁻²) ÷ m² Pa = kg m⁻¹ s⁻² Dividing by m² drops the length power from +1 to −1 — that’s where the m⁻¹ comes from.

💡 Top tips

Start from the defining equation for any “fundamental units” question — it hands you the base units directly
Track powers carefully: dividing by a unit makes its exponent more negative, not smaller in size
Chain conversions one step at a time (pc → ly → m) rather than trying to leap in one go
The tables live in the data booklet — practise using them quickly rather than memorising them

⚠ Common mistakes

Calling the kilogram a prefixed gram — kg is the base unit itself, not a “prefixed” unit
Forgetting to square the prefix: 1 cm² is 10⁻⁴ m², not 10⁻² m²
Mixing up sign of the exponent when a unit is on the bottom of a fraction
Assuming there are seven base units in the IB course — the candela isn’t used, so it’s six

Up next: Using Dimensional Analysis — now that you can break any unit into base units, we’ll use that skill to check whether an equation is even possible, by testing that the units balance on both sides.

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Base & Derived Units

📘 What you need to know

The seven SI base units

Prefixes: scaling units up and down

Building derived units

🧭 Finding a derived unit

Common derived units

Handy non-SI units

💡 Top tips

⚠ Common mistakes

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