SI Units & Quantities
Every measurement in physics is a comparison against an agreed standard. This lesson covers where those standards come from, and how the whole tangle of physical quantities you'll meet this year — forces, energies, fields, pressures — boils down to just a handful of base units.
Why bother with a system of units at all
Say a bridge is designed by an engineering team split across three continents. One group works in feet and pounds, another in metres and kilograms, a third in some local variant of either. History has shown what happens when that goes wrong — in 1999 a NASA orbiter was lost because one team's software output thrust in pound-force seconds while another's expected newton-seconds. Nobody made an arithmetic mistake. The numbers were just measured against different rulers.
Physics avoids this by insisting every quantity gets expressed in the Système International — the SI system — a small set of carefully defined base units the whole world's scientific community has agreed to share. Once you're fluent in it, a quantity's unit tells you almost as much as its number does.
The base quantities
Underneath every physical quantity in this course there are just a handful of genuinely independent ones — quantities that can't be built out of anything simpler. Everything else, no matter how complicated it looks, is some combination of these.
| Base quantity | SI unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
You don't need to memorise the formal definitions of these — how a second is fixed by a certain number of oscillations of a caesium atom, say — what matters is recognising them on sight and knowing nothing on this list can be broken down any further. (There's a seventh SI base unit, the candela, for luminous intensity, but it doesn't come up in this course.)
Building everything else: derived units
Almost every unit you'll write down in this course — newtons, joules, watts, pascals, ohms — is a derived unit: some combination of the base units above, built from a defining equation. This means you can reconstruct a unit from first principles any time you've forgotten it, rather than memorising a long list.
Homogeneity: checking an equation's units balance
Because every physical quantity carries a fixed combination of base units, a correct physics equation must have the same combination of units on both sides. This property is called homogeneity (or dimensional consistency), and it gives one of the fastest sanity checks in the course: if the units on the left of an equation don't match the units on the right, the equation cannot possibly be correct.
It's worth being honest about what this check can and can't do. It won't catch a stray factor of 2, or a missing $\frac{1}{2}$ — pure numbers with no units are invisible to a unit check. What it reliably catches is a structural mistake: multiplying where you should have divided, adding two quantities that don't measure the same kind of thing, or misremembering an equation's shape.
A homogeneity check is a one-way test. If the units don't match, the equation is definitely wrong. If the units do match, that's reassuring, but not proof — an equation can be dimensionally consistent and still have the wrong numerical constant, or the wrong power on a variable. Use it to rule equations out with confidence, not to rule them in with certainty.
SI prefixes
Physical quantities in this course range from the diameter of a nucleus (about $10^{-15}\,\text{m}$) to the distance to a star (perhaps $10^{18}\,\text{m}$). Writing every number out in full would be unreadable, so SI defines prefixes that shift the decimal point by a fixed power of ten, attached directly to a unit's name or symbol.
| Prefix | Symbol | Multiplier |
|---|---|---|
| tera | T | $10^{12}$ |
| giga | G | $10^{9}$ |
| mega | M | $10^{6}$ |
| kilo | k | $10^{3}$ |
| centi | c | $10^{-2}$ |
| milli | m | $10^{-3}$ |
| micro | μ | $10^{-6}$ |
| nano | n | $10^{-9}$ |
| pico | p | $10^{-12}$ |
| femto | f | $10^{-15}$ |
A resistor labelled 4.7 kΩ is $4.7\times10^3\,\Omega$. A capacitor labelled 100 nF is $100\times10^{-9}\,\text{F}$. Once these prefixes are second nature, reading a component's markings or a graph's axis label stops requiring conscious translation.
Two of these are easy to mix up under exam pressure: m on its own means milli ($10^{-3}$), but capital M means mega ($10^{6}$) — a factor of a billion apart. And k for kilo is always lower case; a capital K is reserved for the kelvin, a unit in its own right, not a prefix.
Alongside prefixes, exam answers are usually expected in standard form: a number written as $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer. Keeping exactly one non-zero digit before the decimal point means magnitudes can be compared at a glance just by reading off $n$. Standard form and SI prefixes are two notations for the same idea, so be fluent converting between them:
- $4.7\,\text{k}\Omega = 4.7\times10^{3}\,\Omega$ — the prefix kilo is absorbed into the power of ten.
- $250\,\text{nm} = 250\times10^{-9}\,\text{m} = 2.50\times10^{-7}\,\text{m}$ — the second step slides the leading digit back to satisfy $1 \le a < 10$.
- $3.2\times10^{6}\,\text{W} = 3.2\,\text{MW}$ — going the other way, matching the power of ten to the nearest prefix.
Converting between units of the same quantity
Energy is one of the few quantities you'll meet expressed in genuinely different-sized units depending on context — the joule for everyday physics, the electron-volt for particle-scale energies, and the kilowatt-hour for household electricity bills. Since they all measure the same underlying quantity, converting between them is just a matter of knowing or deriving the right multiplying factor.
The kilowatt-hour works the same way — power multiplied by time, using non-SI units of power and time instead of watts and seconds. $1\,\text{kWh} = (1000\,\text{W})\times(3600\,\text{s}) = 3.6\times10^{6}\,\text{J}$. In both cases the trick is identical: go back to the defining relationship, substitute the numbers you know in SI units, and let the arithmetic produce the conversion factor rather than memorising it.
Key takeaways
- SI units exist so that a measurement means the same thing to every physicist, everywhere — this is the entire point of the system.
- Six base quantities matter for this course: mass (kg), length (m), time (s), current (A), temperature (K), and amount of substance (mol). Nothing on this list can be broken down further.
- Every other unit is derived from these by substituting into a defining equation — the newton, joule and pascal are all built this way, and you can always reconstruct one if you forget it.
- A correct equation must be homogeneous — the same combination of base units on both sides. Checking this catches structural errors (multiplying instead of dividing, mismatched quantities) but can't catch a wrong pure-number constant, and matching units never proves an equation correct.
- SI prefixes (T, G, M, k, c, m, μ, n, p, f) shift the decimal point by a fixed power of ten — watch the difference between lower-case m (milli) and capital M (mega).
- Standard form writes a number as $a \times 10^n$ with $1 \le a < 10$ — fluently converting between standard form and prefixed units is an exam-essential skill.
- Converting between units of the same quantity — J and eV, J and kWh — comes from returning to the defining equation, not from memorising a conversion table.