Units, Quantities and Accuracy

A number alone answers nothing

“The trip costs $37.52$ dollars” is an answer. “The trip costs $37.52$ gallons” is a mistake that the number alone cannot reveal.

Units carry the meaning. Carrying them through every step of a calculation does three jobs at once: it sets the problem up, it checks the setup, and it names the answer.

Letting the units steer

A car gets $32$ miles per gallon. Gas costs $3.45$ dollars per gallon. What does a $348$-mile trip cost in fuel?

Start from what is given and write every unit.

$$348 \text{ mi} \times \frac{1 \text{ gal}}{32 \text{ mi}} \times \frac{3.45 \text{ dollars}}{1 \text{ gal}}$$

Miles cancel against miles, gallons cancel against gallons, and dollars are left standing.

$$\frac{348 \times 3.45}{32} = 37.52 \text{ dollars}$$

The setup was chosen so the units would work out, and nothing else was needed to decide whether to multiply or divide by $32$. Dividing by $3.45$ would have produced dollars in the wrong place, and the units would have said so before any arithmetic happened.

Units inside a formula

Every letter in a model has a unit, and reading them makes the model make sense.

A plumber charges a $40$-dollar service fee plus $25$ dollars an hour.

$$C = 25h + 40$$

Part	Unit	What it means
$C$	dollars	total charge
$h$	hours	time on the job
$25$	dollars per hour	the rate
$40$	dollars	the fee, paid before any work

The $25$ must be dollars per hour, because it multiplies hours and the result has to be dollars. The $40$ must be plain dollars, because it is added to dollars.

$$\underbrace{\frac{\text{dollars}}{\text{hour}}}_{25} \times \underbrace{\text{hours}}_{h} = \text{dollars}$$ $$\text{dollars} + \text{dollars} = \text{dollars}$$

Only quantities with matching units can be added. That single rule catches a large share of modeling errors, because a term with the wrong unit cannot belong in the sum.

Choosing the unit for the display

The same quantity can be reported many ways, and some are readable.

Quantity	Poor choice	Better choice
A city budget	$1{,}847{,}000{,}000$ dollars	$1.85$ billion dollars
A school year	$31{,}536{,}000$ seconds	$180$ days
A drive across Texas	$4{,}435{,}000$ feet	$840$ miles
A virus	$0.00000012$ meters	$120$ nanometers

Pick the unit that makes the typical value a number with a few digits. A graph of a year of temperatures uses months on its axis, not hours.

Defining a quantity

Some questions have no number attached until somebody supplies one.

“How walkable is this neighborhood?” is not a quantity. It becomes one after a definition:

• intersections per square mile — more intersections means shorter, denser blocks
• the fraction of errands reachable within a half mile
• sidewalk miles per road mile

Each is a real quantity, each can be measured the same way by two different people, and each captures something different. Defining the quantity is a decision, and the decision belongs in the report.

Vague question	A quantity that answers part of it
How crowded is the school?	students per classroom
How good is the air today?	micrograms of particulate per cubic meter
How busy is this road?	vehicles per hour past a point
How efficient is this car?	miles per gallon

A good quantity has a unit, a rule anyone can follow, and a clear direction — higher means more of the thing you meant.

Why an answer cannot be sharper than its measurements

Measure a room with a tape marked in tenths of a foot and get $12.4$ ft by $10.2$ ft. A calculator gives the area at once.

$$12.4 \times 10.2 = 126.48 \text{ ft}^2$$

Those last digits are fiction, and here is why.

A reading of $12.4$ means the true length is somewhere between $12.35$ and $12.45$. A reading of $10.2$ means the true width is between $10.15$ and $10.25$. So the true area lies between two products:

$$12.35 \times 10.15 = 125.4 \qquad 12.45 \times 10.25 = 127.6$$

The area is somewhere in a range two square feet wide. Reporting $126.48$ claims to know it to a hundredth of a square foot.

$$\text{Area} \approx 126 \text{ ft}^2$$

Three digits is what two three-digit measurements support. A calculator carries digits; it does not create information. The information came from a tape measure, and the tape measure stopped at tenths.

Significant figures

The digits that carry information are the significant figures.

Measurement	Significant figures
$12.4$	$3$
$0.0086$	$2$ — leading zeros only place the decimal point
$5.60$	$3$ — the final zero says the hundredths place was measured
$1200$	ambiguous; $1.2 \times 10^3$ says $2$

The working rules are short.

Multiplying or dividing: keep as many significant figures as the least precise factor.

Adding or subtracting: keep as many decimal places as the least precise term.

Exact numbers do not count. A dozen is exactly $12$, and $5280$ feet per mile is a definition, so neither limits anything.

Worked examples

Common mistakes

Practice problems