Dimensional analysis

Dimensional analysis is the technique of carrying units alongside numbers through a calculation and treating them as algebraic objects. Units multiply, divide, and cancel; if your final answer is supposed to be a distance and the units work out to seconds, you know you’ve made an error before you even check the numbers.

Worked example. Converting 60 mph to meters per second:

60 mi/hr × (1609.344 m / 1 mi) × (1 hr / 3600 s)
= 60 × 1609.344 / 3600 m/s
= 26.8224 m/s

Notice how the “mi” and “hr” units cancel diagonally. The result has units m/s — which is what was asked for. Any other unit in the answer would indicate a math error.

This is more than a teaching technique. NASA’s 1999 Mars Climate Orbiter loss happened because Lockheed Martin’s ground software output pound-seconds and the spacecraft’s software expected newton-seconds. The conversion factor wasn’t applied; the resulting trajectory commands sent the orbiter into the Martian atmosphere. $125 million plus the mission.

For human calculations, dimensional analysis catches roughly:

• Forgotten conversions (miles to kilometres, hours to seconds).
• Inverted ratios (multiplied when should have divided, or vice versa).
• Conflated quantities (treating mass as weight; treating volume as flow).

The rule of thumb: every step of every calculation should have units attached. The numerical answer is right only if the unit answer is also right.