From the story. Homer Hickam's account of the Rocket Boys gives heights in feet and miles: their best rocket, flown in May 1960, rose nearly four miles. A NASA engineer would want that in meters, and wouldn't trust a digit more than the measurement could give. (More in October Sky.)
Every measurement is a number and a unit. To change units, multiply by a fraction equal to 1, such as $\dfrac{0.3048\ \text{m}}{1\ \text{ft}}$, written so the old unit cancels. You can chain as many of these as you need: miles to feet to meters.
Key idea
Conversion factors: $1\ \text{ft} = 0.3048\ \text{m}$ and $1\ \text{in} = 2.54\ \text{cm}$ (both exact), $1\ \text{mi} = 5{,}280\ \text{ft}$. Put the unit you want to remove on the bottom.
Significant figures: count from the first nonzero digit; trailing zeros count only with a decimal point. A product or quotient keeps as many significant figures as its least precise factor.
Significant figures: count from the first nonzero digit; trailing zeros count only with a decimal point. A product or quotient keeps as many significant figures as its least precise factor.
Worked example: Four miles high
Express $4\ \text{mi}$ in meters.
- $4\ \text{mi} \times \dfrac{5{,}280\ \text{ft}}{1\ \text{mi}} = 21{,}120\ \text{ft}$.
- $21{,}120\ \text{ft} \times \dfrac{0.3048\ \text{m}}{1\ \text{ft}} \approx 6{,}437\ \text{m}$, about $6.4\ \text{km}$.
Answer: $\approx 6{,}440\ \text{m}$
Worked example: How many digits?
A baseline is paced off as $150\ \text{m}$ (2 significant figures) and the tangent of the angle is $1.428$. Report the height.
- $150 \times 1.428 = 214.2$.
- The baseline has only 2 significant figures, so the height is $210\ \text{m}$. Writing 214.2 m claims a precision the pacing never had.
Answer: $210\ \text{m}$
Common mistake
Turning a factor upside down: $4\ \text{mi} \times \dfrac{1\ \text{mi}}{5{,}280\ \text{ft}}$ leaves miles squared per foot. Check that the units cancel before you multiply.