Physical Quantities & Measurement

The SI system defines exactly seven base quantities, each with its own base unit. These are: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd). Every other measurable quantity in physics and chemistry can be expressed in terms of these seven. The base units are defined with extreme precision — for example, since 2019 the metre is defined by fixing the speed of light to exactly $c = 299{,}792{,}458 \ \text{m s}^{-1}$, and the second is defined by the frequency of radiation emitted by caesium-133 atoms.

Derived quantities are formed algebraically from base quantities. For example, area is length squared: $A = l^2$, so its SI unit is $\text{m}^2$. Volume has units $\text{m}^3$. Speed is $v = l/t$ with units $\text{m s}^{-1}$. Force is defined by Newton's second law as $F = ma$ (mass × acceleration), so its unit is $\text{kg m s}^{-2}$, which is given the special name 'newton' (N). Energy has units $\text{kg m}^2 \text{s}^{-2}$, called the joule (J). Named derived units are simply shortcuts: $1 \ \text{N} = 1 \ \text{kg m s}^{-2}$ exactly.

Every physical quantity has a dimension — a description of what kind of quantity it is, independent of the unit size. The seven base dimensions are denoted $\mathsf{M}$ (mass), $\mathsf{L}$ (length), $\mathsf{T}$ (time), $\mathsf{I}$ (electric current), $\mathsf{\Theta}$ (temperature), $\mathsf{N}$ (amount of substance), and $\mathsf{J}$ (luminous intensity). A derived quantity's dimension is written as a product of powers of base dimensions. For example, force has dimension $[F] = \mathsf{M L T}^{-2}$, pressure has dimension $[P] = \mathsf{M L}^{-1} \mathsf{T}^{-2}$, and energy has dimension $[E] = \mathsf{M L}^2 \mathsf{T}^{-2}$. The principle of dimensional homogeneity states that both sides of any valid physical equation must have identical dimensions — a powerful tool for checking equations and deriving relationships.

Physical quantities are also classified by whether they have direction. A scalar quantity is fully described by a magnitude and a unit alone — examples include mass, temperature, energy, and speed. A vector quantity requires both a magnitude and a direction — examples include displacement, velocity, acceleration, and force. The distinction matters enormously in calculations: scalar addition is ordinary arithmetic, while vector addition requires geometric or component methods. For instance, walking 3 m east and then 4 m north gives a displacement of magnitude $\sqrt{3^2 + 4^2} = 5 \ \text{m}$ (northeast), not simply 7 m.

No measurement is perfectly exact. Every measured value has an associated uncertainty that reflects the limitations of the measuring instrument and the skill of the experimenter. Significant figures (sig figs) are the digits in a measurement that carry meaning about its precision. When combining measurements mathematically, rules govern how precision propagates: for multiplication and division, the result should have as many sig figs as the least precise input; for addition and subtraction, the result should be rounded to the least precise decimal place. For example, $12.3 \ \text{m} \times 4.56 \ \text{m} = 56.1 \ \text{m}^2$ (3 sig figs), not $56.088 \ \text{m}^2$.

1 \ \text{J} = 1 \ \text{kg m}^2 \text{s}^{-2}

• SI Base Units and Their Modern Definitions — Explores how each of the seven base units is defined using fundamental physical constants, following the 2019 SI revision.
• Dimensional Analysis and Unit Checking — Extends the dimensional formula concept into a systematic problem-solving tool for deriving and verifying equations.
• Significant Figures, Precision, and Experimental Uncertainty — Builds on the idea that all measurements carry uncertainty, with formal methods for quantifying and propagating errors.
• Scalars and Vectors — Develops the distinction between scalar and vector physical quantities introduced here, including vector addition and components.