Why 72 km/h and 20 m/s Describe Exactly the Same Speed

A tailor's "2 metres of cloth" and a chemist's "500 mg tablet" both mean something only because everyone agrees on what a metre and a milligram are. A number without a unit is meaningless.

A number can also mislead by claiming more precision than the instrument had — a school ruler cannot measure $12.3456$ cm.

This part covers SI units, significant figures, rounding in calculations, and combining errors.

The SI system has seven base quantities, each with a base unit, and every other unit is a derived unit formed by multiplying and dividing powers of these.

The seven base units:

- 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)

Derived units.

$$\text{force: }1\text{ N}=1{\text{ kg m s}}^{-2}\text{energy: }1\text{ J}=1\text{ N m}=1{\text{ kg m}}^2{\text{ s}}^{-2}$$

$$\text{power: }1\text{ W}=1{\text{ J s}}^{-1}\text{pressure: }1\text{ Pa}=1{\text{ N m}}^{-2}$$

Worked example 1 — speed.

$$72{\text{ km h}}^{-1}=\frac{72\times 1000\text{ m}}{3600\text{ s}}=20{\text{ m s}}^{-1}$$

Worked example 2 — energy.

$$1\text{ kWh}=1000\text{ W}\times 3600\text{ s}=3.6\times 10^6\text{ J}$$

An everyday example. The "unit" on a household electricity bill is one kilowatt-hour, which is $3.6\times 10^6$ joules.

The substance. The radian and steradian are dimensionless derived units for plane and solid angle, and temperature is written in kelvin, not "degrees kelvin".

Significant figures are the digits known reliably plus the first uncertain digit; they are counted using a fixed set of rules about non-zero digits and zeros.

The rules, with examples:

- All non-zero digits count — $2.64$ has 3
- Zeros between non-zero digits count — $2.005$ has 4
- Leading zeros never count — $0.00340$ has 3 (the digits $3$, $4$ and the final $0$)
- Trailing zeros after a decimal point count — $4.500$ has 4
- Trailing zeros in a whole number without a decimal point are ambiguous — write $4500$ as $4.5\times 10^3$ (2) or $4.500\times 10^3$ (4)

Worked example — changing units.

$$2.30\text{ m}=2.30\times 10^2\text{ cm}=2.30\times 10^3\text{ mm}$$

All three have 3 significant figures; changing the unit does not change the precision.

An everyday example. A kitchen scale showing 250 g and a jeweller's balance showing 2.350 g differ in precision — the jeweller's reading has four significant figures down to a milligram.

The substance. Exact numbers, such as the $2$ in $2\pi r$ or a count of $20$ coins, have unlimited significant figures.

In multiplication and division, keep as many significant figures as the least precise quantity has; in addition and subtraction, keep as many decimal places as the quantity with the fewest decimal places.

Worked example 1 — division. A mass of $4.237$ g has a volume of $2.51$ cm${}^3$.

$$\rho =\frac{4.237}{2.51}=1.688\cdots \to 1.69{\text{ g cm}}^{-3}(3\text{ significant figures})$$

Worked example 2 — multiplication.

$$12.6\text{ cm}\times 4.2\text{ cm}=52.92\to 53{\text{ cm}}^2(2\text{ significant figures})$$

Worked example 3 — addition.

$$436.32\text{ g}+227.2\text{ g}+0.301\text{ g}=663.821\to 663.8\text{ g}(1\text{ decimal place})$$

Rounding off. If the digit dropped is more than $5$, raise the previous digit; if less than $5$, leave it. If it is exactly $5$, leave an even previous digit and raise an odd one: $2.745\to 2.74$ but $2.735\to 2.74$.

An everyday example. **Put a $1.25$ kg packet on a sack weighed as $12$ kg on a rough scale**: the total is $13$ kg, not $13.25$ kg, because the sack's weight is known only to the nearest kilogram.

The substance. Keep extra digits in the middle of a calculation and round only the final answer.

**For sums and differences the absolute errors add; for products and quotients the relative errors add; and a quantity raised to power $p$ contributes $p$ times its relative error.**

$$Z=A\pm B:\Delta Z=\Delta A+\Delta BZ=AB\text{ or }\frac{A}{B}:\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$

$$Z=\frac{A^p{B}^q}{C^r}:\frac{\Delta Z}{Z}=p\frac{\Delta A}{A}+q\frac{\Delta B}{B}+r\frac{\Delta C}{C}$$

Worked example 1 — sum. Resistors of $(100\pm 3)\Omega$ and $(200\pm 4)\Omega$ in series give $(300\pm 7)\Omega$.

Worked example 2 — difference. Temperatures of $(50.0\pm 0.5)$ and $(20.0\pm 0.5)$ in ${}^{\circ }$C differ by $(30.0\pm 1.0){}^{\circ }$C. Errors add even when the quantities subtract.

Worked example 3 — quotient. With $V=(100\pm 5)$ V and $I=(10\pm 0.2)$ A:

$$R=\frac{V}{I}=10\Omega \frac{\Delta R}{R}=\frac{5}{100}+\frac{0.2}{10}=0.07R=(10\pm 0.7)\Omega$$

Worked example 4 — power. A square of side $(2.0\pm 0.1)$ cm has area $4.0$ cm${}^2$ and

$$\frac{\Delta A}{A}=2\times \frac{0.1}{2.0}=0.10A=(4.0\pm 0.4){\text{ cm}}^2$$

An everyday example. Measuring a room's length and width with a tape to order floor tiles: the errors in both measurements feed into the error in the area.

The substance. **Pure numbers such as $\frac{1}{2}$ carry no error**, so they do not add to the relative error.