Units and Measurements

• Every non-zero digit counts.
• Zero between two non-zero digits counts.
• Leading zeros never count.
• Trailing zeros count only if there’s a decimal point.

1. Multiply/divide: keep as many sig figs as the least-precise input.
2. Add/subtract: keep as many decimal places as the least-precise input.

Velocity, average velocity, speed: all dimensionally identical, $[M^0 L T^{-1}]$ — same kind of quantity regardless of value.

4700 m written unambiguously: $4.700 \times 10^3\text{ m}$ — the exponent form fixes the sig-fig count that trailing zeros alone can’t.

1. Dropped digit $< 5$: round down.
2. Dropped digit $> 5$ (or exactly 5 with non-zero digits after): round up.
3. Dropped digit exactly 5, nothing after: round so the preceding digit becomes even.

E.g. $2.735 \to 2.74$ (5 becomes even), $2.745 \to 2.74$ (4 was already even).

Multiplying/dividing measured quantities: percentage uncertainties add.

Sides $16.2 \pm 0.1$ cm ($0.6\%$) and $10.1 \pm 0.1$ cm ($1\%$) → area uncertainty ≈ $1.6\%$.

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

Plane angle (radian) and solid angle (steradian) are dimensionless — both are ratios of lengths/areas.

Every term in a valid equation must carry identical dimensions, on both sides.

Passing this check only proves an equation isn’t obviously wrong — it can never catch a missing dimensionless constant (e.g. the $\tfrac{1}{2}$ in $s = ut + \tfrac12 at^2$).

Found by matching powers of $L$ and $T$ in $a = k\,v^x r^y$; the value of $k$ needs the full derivation.

1. Can never recover a dimensionless constant.
2. Can’t handle a sum of several same-dimensioned terms.
3. Says nothing about relations built from sines, logarithms, or exponentials.