Before a single law is written, a physicist settles what can be counted. Every quantity is a number times a unit; every unit is built from seven base standards; and the exponents that assemble it — its dimensions — quietly police every equation you will ever write. Meet the seven base quantities and how each is defined, build a dimensional formula, test an equation for homogeneity, convert between systems, size the universe in powers of ten, and carry honest error bars.
§01
The seven base quantities
A handful of quantities are chosen to be independent of one another — the fundamental (base) quantities — and everything else is a derived quantity, built from them by multiplication and division. It turns out only seven are needed. Their SI units are the fundamental units; the units of derived quantities (m·s⁻¹, kg·m·s⁻², …) are the derived units.
A good standard must be invariable (never drifting) and reproducible anywhere — which is why the modern definitions pin each base unit to an unchanging constant of nature. Tap a row to read how each one is fixed.
Base quantitySI unitSymbolDimension symbol
Plus two supplementary units — the radian (rad) for plane angle and the steradian (sr) for solid angle — both dimensionless ratios.
Fundamental vs derived
Define a unit of length and the unit of area (length²) comes free; add a unit of time and speed (length/time) follows. Every derived unit is a product of powers of the seven base units — exactly the pattern §02 turns into a dimensional formula.
SI prefixes stride by 10³
T 10¹² · G 10⁹ · M 10⁶ · k 10³ m 10⁻³ · µ 10⁻⁶ · n 10⁻⁹ · p 10⁻¹² · f 10⁻¹⁵
§02
The dimension machine
Any quantity is a product of powers of the base quantities; each exponent is its dimension in that base. Pick a quantity and watch it resolve into its dimensional formula, one link of the derivation at a time.
Bench · dimensional formula
{{ dmUnit }}
quantity
{{ dmName }} {{ dmSym }}
{{ r.base }}
{{ r.exp }}
dimensional formula
[{{ dmSym }}] = {{ p.base }}{{ p.exp }}
the derivation, link by link
{{ step.n }}{{ step.t }}
{{ dmNote }}
§03
Dimension finder
A working lookup for the whole syllabus. Type any physical quantity — by name or symbol — and read off its dimensional formula, a defining relation and its SI unit. Because dimensions cut across every chapter, this same tool lives on the chapter home as a cross-topic connector.
§04
The homogeneity check
Terms that are added, subtracted or equated must carry identical dimensions. Feed the bench three candidate kinematics equations — it resolves each term and glows green only if every one agrees.
Bench · principle of homogeneity
{{ hgEqn }}
{{ t.expr }}
{{ t.dim }}
{{ t.mark }}
{{ hgVerdict }}
{{ hgExplain }}
§05
Converting between systems
A derived unit changes with its base units exactly as its dimensional formula dictates: n₂ = n₁ (M₁M₂)ᵃ (L₁L₂)ᵇ (T₁T₂)ᶜ. Pick a quantity and a target system; the factor assembles itself from the powers.
Write any length as a×10ᵇ and the exponent b is its order of magnitude. Drag along the ruler — from a proton to a distant quasar spans forty-one powers of ten.
Bench · the cosmic ruler (metres)
order = {{ omExp }}
nearest at this scale
{{ omLabel }}
≈ {{ omValue }} m
{{ omNote }}
§07
Significant figures & error bars
Two measured quantities, each with an uncertainty. Choose an operation and the bench propagates the error the right way — absolute errors add for ± , relative errors add for ×÷ — and rounds the answer to the honest number of figures.
Bench · propagation of errors
A = {{ sfAval }} ± {{ sfAerr }}{{ sfArel }}%
B = {{ sfBval }} ± {{ sfBerr }}{{ sfBrel }}%
{{ sfRule }}
result — {{ sfOpName }}
{{ sfResult }}
± {{ sfResErr }} ({{ sfResRel }}%)
ΔZZ= {{ sfPropExpr }}
{{ sfSfNote }}
✎
Worked example — the period of a pendulum, from dimensions alone
A simple pendulum's period t is believed to depend only on its length ℓ, the bob's mass m, and gravity g, as a product of powers. Find the form of t.
1. Assume t = k ℓa mb gc with k dimensionless. Take dimensions of both sides:
T = La Mb (L T−2)c = Mb La+c T−2c
2. Match exponents of M, L, T on both sides:
b = 0 a + c = 0 −2c = 1 ⇒ c = −½, a = +½
3. The mass drops out entirely, and
t = k√ℓg
The catch: dimensions fix the powers but never the pure number — you must learn from experiment that k = 2π, giving t = 2π√ℓ/g.
✎
Warm-up
quick checks · tap a card to reveal · {{ warmChev }} to fold