Physics › 📐 Foundations

Units & Dimensional Analysis

A number without a unit is just a guess — here's how physicists make measurements bulletproof.

High school
Units & Dimensional Analysis — illustration
Illustrative hero image.
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The big idea: Every physical quantity is a number tied to a unit, and every unit in physics can be traced back to just seven fundamental building blocks. Once you can convert between units and check an equation's dimensions, you have a fast, powerful way to catch mistakes — often before doing any actual math.
🎯 By the end, you'll be able to
  • Name the seven SI base units and identify what physical quantity each one measures.
  • Convert a measurement between metric prefixes and between different unit systems using conversion factors.
  • Use dimensional analysis to check whether an equation is even structurally possible before solving it.
  • Explain why you can only add, subtract, or equate quantities that share the same dimension.
📎 You should already know
  • Basic algebra (rearranging and solving equations)
  • Scientific notation and powers of ten

The Number Alone Means Nothing

Say someone tells you "the distance is 90." Ninety what — meters? Miles? Light-years? Until you attach a unit, the number is meaningless. Physics takes this seriously: NASA's Mars Climate Orbiter was lost in 1999 partly because one team's software delivered thrust data in pound-seconds while the navigation team assumed newton-seconds — a unit mismatch sent a very expensive spacecraft into the Martian atmosphere instead of a stable orbit. Units aren't decoration on a measurement. They're part of its meaning.

Good news: once you're comfortable with the small set of SI base units, the prefixes that scale them, and a simple trick called dimensional analysis, you can convert between almost any units and sanity-check almost any formula — often before plugging in a single number.

🔑 Seven Base Units, Everything Else Is Built From Them

The International System (SI) defines just seven base units. Every other unit in physics — speed, force, energy, pressure — is some combination of these seven, multiplied or divided together.

  • Length — meter (m)
  • Mass — kilogram (kg)
  • Time — second (s)
  • Electric current — ampere (A)
  • Temperature — kelvin (K)
  • Amount of substance — mole (mol)
  • Luminous intensity — candela (cd)

A speed is meters per second (m/s). A force is kilograms times meters per second squared (kg·m/s²) — a combination we just give a shorter name, "newton," for convenience. Learn the seven, and you can reconstruct almost any derived unit from scratch.

\[ [v] = \dfrac{L}{T}, \qquad [F] = \dfrac{M L}{T^{2}} \]
Square brackets mean "the dimension of." Velocity is always length/time; force is always mass·length/time² — regardless of whether you measure in meters and seconds or feet and hours.

Prefixes: The Same Unit, Just Scaled

Rather than invent a brand-new unit every time a quantity gets very large or very small, SI multiplies the base unit by a power of ten and gives that power a prefix. A kilometer is 1000 meters; a millimeter is 1/1000 of a meter. Learn a handful of these and you can read almost any measurement in physics, chemistry, or engineering.

  • giga- (G) = ×10⁹
  • mega- (M) = ×10⁶
  • kilo- (k) = ×10³
  • centi- (c) = ×10⁻²
  • milli- (m) = ×10⁻³
  • micro- (µ) = ×10⁻⁶
  • nano- (n) = ×10⁻⁹

Converting between prefixes is just moving a decimal point. Converting between entirely different unit systems — like kilometers per hour to meters per second — takes one more idea: the conversion factor.

\[ 1 = \dfrac{1000\ \text{m}}{1\ \text{km}} = \dfrac{1\ \text{h}}{3600\ \text{s}} \]
A conversion factor is really just the number 1, dressed up in different units. Multiplying by it never changes the quantity — only the units it's expressed in.

Dimensional Analysis: A Free Error-Checker

Here's the trick that makes units genuinely powerful, not just bookkeeping. Before you trust an equation — one you derived, half-remembered, or found in a textbook — you can check whether the dimensions on both sides match. If they don't, the equation is definitely wrong. If they do match, the equation is at least dimensionally plausible (it could still be off by a pure number like 2 or π, but the underlying physics is structurally sound).

This works because you can only add, subtract, or set equal quantities that share the same dimension. You can add two lengths together. You cannot add a length to a time and get anything with a physical meaning.

\[ T = 2\pi\sqrt{\dfrac{L}{g}} \]
The period of a simple pendulum. Let's check whether this formula's dimensions actually work out to a time.
📝 Worked example: A cyclist rides at 90 km/h. Convert this speed to meters per second.
  1. Write the speed as a fraction so the units are explicit: \(90\ \dfrac{\text{km}}{\text{h}}\).
  2. Multiply by conversion factors that each equal 1, arranged so the unwanted units cancel: \(90\ \dfrac{\text{km}}{\text{h}} \times \dfrac{1000\ \text{m}}{1\ \text{km}} \times \dfrac{1\ \text{h}}{3600\ \text{s}}\).
  3. Cancel km with km, and h with h, leaving only meters and seconds: \(\dfrac{90 \times 1000}{3600}\ \dfrac{\text{m}}{\text{s}}\).
  4. Compute the number: \(90 \times 1000 = 90{,}000\), and \(90{,}000 \div 3600 = 25\).
✓ 25 m/s
📝 Worked example: Check whether the pendulum period formula \(T = 2\pi\sqrt{L/g}\) is dimensionally consistent, given that \(L\) is a length and \(g\) (acceleration due to gravity) has dimension \(L/T^{2}\).
  1. Find the dimension of the quantity inside the square root: \(\left[\dfrac{L}{g}\right] = \dfrac{L}{L/T^{2}}\).
  2. Simplify by multiplying by the reciprocal: \(\dfrac{L}{L/T^{2}} = L \times \dfrac{T^{2}}{L} = T^{2}\).
  3. Take the square root, since the formula applies a square root to this quantity: \(\sqrt{T^{2}} = T\).
  4. The factor \(2\pi\) is dimensionless (a pure ratio), so it doesn't change the dimension. The right-hand side reduces to a single power of time, \(T\), matching the left-hand side.
✓ The dimensions match on both sides (both reduce to a time), so the formula is dimensionally consistent — though this check alone can't confirm the coefficient \(2\pi\) is exactly right.
⚠️ What Dimensional Analysis Can't Tell You

Matching dimensions is a necessary condition, not a certainty. The check can't catch a mistaken pure number — an equation with a stray factor of 2, or \(\pi\) instead of \(2\pi\), passes just as easily as the correct one, because pure numbers carry no dimension at all.

The other classic trap is adding quantities that look similar but aren't — mixing up a radius with a diameter, say, or adding a velocity to an acceleration. These mistakes can "look" fine at a glance, so the real danger is skipping the dimension check because a formula feels familiar. Get in the habit of checking anyway, especially the first time you use a new formula.

Check your understanding

1. Which of these is one of the seven SI base units?
Kelvin (temperature) is one of the seven base units. Newton, joule, and watt are all derived units built from combinations of the base units.
2. A runner's pace is 5 minutes per kilometer. What is this speed in meters per second (to two decimal places)?
5 min/km = 300 s per 1000 m, so speed = 1000 m ÷ 300 s ≈ 3.33 m/s.
3. Someone proposes the equation F = m/v (force equals mass divided by velocity). What does dimensional analysis tell you?
Force has dimension ML/T². Mass divided by velocity has dimension M/(L/T) = MT/L, which is different from ML/T² — so the equation is dimensionally impossible, no matter what unit system you use.
4. Which pair correctly matches a metric prefix to its power of ten?
Kilo- means ×10³ (one thousand times the base unit). Milli- is ×10⁻³, micro- is ×10⁻⁶, and centi- is ×10⁻².
✅ Key takeaways
  • A measurement is a number and a unit together — dropping the unit strips it of meaning, and in real engineering that has caused very expensive mistakes.
  • Every unit in SI can be built from just seven base units: meter, kilogram, second, ampere, kelvin, mole, and candela.
  • Converting between units is just multiplying by cleverly chosen forms of 1; converting between prefixes is just moving a decimal point.
  • Dimensional analysis lets you check whether an equation's units match on both sides — a fast, free way to catch mistakes before you ever calculate a number.