§1A measurement is a number and a unit
“5” is not a length; “5 mm” is. The number counts and the unit carries the physics — drop the unit and the physics goes with it.
This is why unit errors are among the most expensive mistakes in engineering: a quantity handled without its unit can silently change meaning by a factor of a thousand or a million. The discipline of this page — track units through every calculation, and check they balance — is the cheapest error-detection available, and §6 turns it into a formal test.
Contents§2The seven base units
Since 2019 all seven are fixed to constants of nature rather than physical artefacts — the units no longer depend on a lump of metal in a vault.
| Quantity | Unit (symbol) | Fixed by |
|---|---|---|
| Length | metre (m) | the speed of light, c |
| Mass | kilogram (kg) | the Planck constant, h |
| Time | second (s) | the caesium hyperfine frequency |
| Electric current | ampere (A) | the elementary charge, e |
| Temperature | kelvin (K) | the Boltzmann constant, k |
| Amount of substance | mole (mol) | the Avogadro constant, N_A |
| Luminous intensity | candela (cd) | a fixed luminous efficacy |
| For mechanical engineering the working three are the metre, kilogram and second — length, mass and time — from which force, pressure, energy and power all follow. The kelvin shares its size with the degree Celsius, offset by the freezing point of water: K = °C + 273.15. | ||
§3Derived units
Multiply and divide base units and you get every other unit. Many combinations earn a name — but each is only shorthand for a product of base units.
| Quantity | Unit | In base units |
|---|---|---|
| Frequency | hertz (Hz) | s⁻¹ |
| Force | newton (N) | kg·m·s⁻² |
| Pressure, stress | pascal (Pa) | kg·m⁻¹·s⁻² (= N/m²) |
| Energy, work | joule (J) | kg·m²·s⁻² (= N·m) |
| Power | watt (W) | kg·m²·s⁻³ (= J/s) |
| Electric charge | coulomb (C) | A·s |
| Voltage | volt (V) | kg·m²·s⁻³·A⁻¹ (= W/A) |
| Resistance | ohm (Ω) | kg·m²·s⁻³·A⁻² (= V/A) |
| Read the chain on the hero drawing: mass × acceleration builds the newton; the newton over an area is the pascal; the newton through a distance is the joule; the joule per second is the watt. Four of the most-used units in this Library, all one family. | ||
§4Coherence
The SI is a coherent system: its derived units are pure products of base units with no numerical factors. This is the property that makes consistent-unit equations need no conversion constants.
Put F = ma in base units: a mass of 12 kg accelerating at 3 m/s² gives 12 × 3 = 36 in units of kg·m·s⁻² — which is the newton, no factor required, so the force is 36 N directly. The same calculation in mixed units (pounds, feet, minutes) would need a conversion constant to come out right. Work in coherent SI and the arithmetic is the physics.
The single most useful habit that follows: reduce to base SI units (metres, kilograms, seconds, and the coherent derived units built from them) before computing, and the answer emerges in coherent units with nothing to reconcile. Every worked example across this Library is set up that way.
Contents§5The prefix ladder
Prefixes rescale a unit by powers of ten; engineering keeps to the powers divisible by three so the digits stay legible.
| Prefix | Symbol | Factor | Prefix | Symbol | Factor |
|---|---|---|---|---|---|
| tera | T | 10¹² | deci | d | 10⁻¹ |
| giga | G | 10⁹ | centi | c | 10⁻² |
| mega | M | 10⁶ | milli | m | 10⁻³ |
| kilo | k | 10³ | micro | µ | 10⁻⁶ |
| hecto | h | 10² | nano | n | 10⁻⁹ |
| deca | da | 10¹ | pico | p | 10⁻¹² |
| Beyond this range sit peta/exa/zetta/yotta and the 2022 additions ronna and quetta upward, and femto/atto/zepto/yocto with ronto and quecto downward — completing 10³⁰ down to 10⁻³⁰. | |||||
Rules that keep prefixes honest: only one prefix on a unit (nm, never mµm); the prefix binds before any exponent, so 1 mm² = 10⁻⁶ m²; and mass prefixes ride on the gram, not the kilogram (mg, g, Mg). Prefer moving the prefix over writing strings of zeros — 21 µm and 13.8 MPa are legible where their expanded forms are not.
Contents§6Dimensional analysis
Every physically valid equation is dimensionally homogeneous: both sides carry the same combination of mass, length and time. Checking that is a free, powerful sanity test.
Kinetic energy ½mv²: [M] × [L T⁻¹]² = [M L² T⁻²] — exactly the dimensions of energy (the joule). Homogeneous ✓.
Pendulum period T = 2π√(L/g): inside the root, [L] ÷ [L T⁻²] = [T²]; its square root is [T], a time — as a period must be. Homogeneous ✓ (and note the dimensionless 2π is invisible to the check, as constants always are).
The test cuts both ways: an equation that fails is certainly wrong, while one that passes is only possibly right — dimensional analysis catches structural errors, not a missing factor of two. It also builds units: force must be [M L T⁻²], so its unit must be kg·m·s⁻², which is why the newton is defined exactly so.
Contents§7Accepted non-SI units
A handful of non-SI units are sanctioned for use alongside the SI because practice demands them — but they are not coherent, so they must be converted before they enter a calculation.
| Unit | Quantity | In SI |
|---|---|---|
| minute, hour, day | time | 60 s, 3600 s, 86 400 s |
| degree (°) | angle | π/180 rad |
| litre (L) | volume | 10⁻³ m³ |
| tonne (t) | mass | 10³ kg |
| bar | pressure | 10⁵ Pa |
| The radian and steradian are dimensionless SI units — an angle is a ratio of two lengths, so it carries no base dimension, which is why the 2π in the pendulum check was invisible to the dimensions. | ||
§8Quick reference
The working core of the page on one card rack.
Base seven
m · kg · s · A · K · mol · cd
Mechanical family
N = kg·m/s² · Pa = N/m²
J = N·m · W = J/s
Coherence
base-unit inputs → base-unit answer
no conversion constants
Dimensions
both sides must match
fail ⇒ wrong; pass ⇒ maybe
Prefixes
steps of 10³ · one only
mass prefixes on the gram