1. What a unit is
A chosen reference quantity against which other quantities of the same kind are measured. A quantity is a (number, unit) pair: the number says "how many of these"; the unit says "of what."
Strip the unit and the number means nothing. "60" — is that miles per hour, metres per second, kilometres per hour, beats per minute, dollars? A measurement carries its unit the way a name carries a person's identity. Drop it and the rest is noise.
Examples and non-examples
- $3 \text{ m}$ — a length
- $72 \text{ kg}$ — a mass
- $60 \text{ mi/h}$ — a speed
- $20 \text{ °C}$ — a temperature
- "$3$" — three what? rocks? metres? years?
- "$72$" — body weight, exam score, age?
- "$60$" — speed, frequency, count?
- "$0.05$" — fraction, dollars, seconds?
In 1999 NASA lost the $125M Mars Climate Orbiter because one team specified thruster output in pound-force seconds and another assumed newton-seconds. The numbers were technically correct; the units were a mismatch. The spacecraft burned up in the Martian atmosphere. The lesson burned in with it: always annotate units throughout a calculation, never just at the end.
2. Two systems: US customary and metric (SI)
Two measurement systems dominate everyday life, and you'll need to move between them constantly. The US customary system (inches, pounds, gallons) is the historical English/Imperial lineage. The metric system, codified as SI (Système International d'Unités), is the worldwide scientific standard — decimal everywhere, seven base units, the rest derived.
US customary — the essentials
| Dimension | Relationships |
|---|---|
| Length | $1 \text{ ft} = 12 \text{ in}$ · $1 \text{ yd} = 3 \text{ ft}$ · $1 \text{ mi} = 5280 \text{ ft}$ |
| Weight | $1 \text{ lb} = 16 \text{ oz}$ · $1 \text{ ton} = 2000 \text{ lb}$ |
| Capacity | $1 \text{ cup} = 8 \text{ fl oz}$ · $1 \text{ pt} = 2 \text{ cups}$ · $1 \text{ qt} = 2 \text{ pt}$ · $1 \text{ gal} = 4 \text{ qt}$ |
| Time | $1 \text{ min} = 60 \text{ s}$ · $1 \text{ h} = 60 \text{ min}$ · $1 \text{ day} = 24 \text{ h}$ · $1 \text{ wk} = 7 \text{ days}$ |
Notice the irregularity: 12, 3, 5280, 16, 2, 2, 4. Each conversion factor is its own number to remember. That's the cost of a system that grew up by historical accretion rather than design.
SI — seven base units
| Quantity | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Every other SI unit is derived from these seven. Force (the newton) is $\text{kg} \cdot \text{m}/\text{s}^2$; energy (the joule) is $\text{kg} \cdot \text{m}^2 / \text{s}^2$; pressure (the pascal) is $\text{kg}/(\text{m} \cdot \text{s}^2)$. The whole edifice of physics is just these seven letters in different combinations.
SI was designed once, on purpose, by mathematicians and physicists. US customary evolved over centuries from medieval English trade. The metric system has internal consistency on its side; the customary system has habit and culture on its side. Both are with us, and you have to speak both.
3. The "multiply by 1" trick
Here's the entire technique, in one sentence:
Multiply by a fraction equal to 1, chosen so the unwanted unit cancels.
Because $1 \text{ ft} = 12 \text{ in}$, the fraction
$$ \frac{12 \text{ in}}{1 \text{ ft}} = 1 $$is genuinely the number 1 — top and bottom name the same physical length. Multiplying by it doesn't change the value of anything; it just relabels.
To convert $66$ inches to feet, you want "inches" to cancel out and "feet" to be what's left. Pick the form of 1 with inches on the bottom:
$$ 66 \text{ in} \times \frac{1 \text{ ft}}{12 \text{ in}} = \frac{66}{12} \text{ ft} = 5.5 \text{ ft} $$The in on the top of $66 \text{ in}$ cancels the in on the bottom of the fraction, leaving feet. The arithmetic — $66 / 12 = 5.5$ — is ordinary division. The cleverness is structural, not numerical.
The single move, visualized
Whenever you write the conversion factor, look at your starting unit. Whichever unit you want to get rid of goes on the bottom of the fraction. If feet are on the top of your starting quantity and you want inches, use $\frac{12 \text{ in}}{1 \text{ ft}}$. Flip the fraction and you'd be converting in the wrong direction.
4. Chaining: multi-step conversions
The trick scales. To go from weeks to seconds, you don't need a "weeks → seconds" conversion factor — you chain the ones you know:
$$ 2 \text{ wk} \times \frac{7 \text{ d}}{1 \text{ wk}} \times \frac{24 \text{ h}}{1 \text{ d}} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{60 \text{ s}}{1 \text{ min}} = 1{,}209{,}600 \text{ s} $$Read across left to right: weeks cancels with weeks, days with days, hours with hours, minutes with minutes. The only unit not cancelled is the one on the top right — seconds — which is exactly what you wanted.
Two things to notice. First, you never had to look up a "weeks to seconds" number; the chain assembled it from things you already know. Second, the units are doing real work — they're acting like algebraic variables that genuinely cancel. If you ended up with weeks-squared per hour, you'd know something went wrong before you reached for a calculator.
5. Metric prefixes — the decimal shortcut
The metric system's superpower is uniformity: every conversion within it is a power of 10. You memorize one set of prefixes and you're done — for length, mass, volume, anything.
| Prefix | Symbol | Factor | Example |
|---|---|---|---|
| tera | T | $10^{12}$ | $1 \text{ TB} = 10^{12} \text{ bytes}$ |
| giga | G | $10^{9}$ | $1 \text{ GHz} = 10^{9} \text{ Hz}$ |
| mega | M | $10^{6}$ | $1 \text{ ML} = 10^{6} \text{ L}$ |
| kilo | k | $10^{3}$ | $1 \text{ km} = 1000 \text{ m}$ |
| — | — | $10^{0}$ | base unit |
| centi | c | $10^{-2}$ | $1 \text{ cm} = 0.01 \text{ m}$ |
| milli | m | $10^{-3}$ | $1 \text{ mm} = 0.001 \text{ m}$ |
| micro | μ | $10^{-6}$ | $1 \text{ μg} = 10^{-6} \text{ g}$ |
| nano | n | $10^{-9}$ | $1 \text{ ns} = 10^{-9} \text{ s}$ |
The bolded three — kilo, centi, milli — cover almost everything you'll run into day to day. Converting within the metric system is then just moving the decimal point:
$$ 3 \text{ km} \;\to\; 3000 \text{ m} \;\to\; 300{,}000 \text{ cm} \;\to\; 3{,}000{,}000 \text{ mm} $$The decimal slides three places per prefix step (because each prefix differs from its neighbour by $10^3$ across the common range). No lookup table required.
A common careless mistake is treating millimetres and metres as if they were neighbours on a decimal staircase like cents and dollars. They are not. A factor-of-1000 misread can turn a 3 mm hairline crack into a 3 m structural fissure on paper. Always confirm which prefix you're reading.
6. US ↔ metric, the conversions worth memorizing
You'll use these often enough that committing them to memory is worth it. The first one is exact by international agreement (since 1959 the inch is defined as $2.54 \text{ cm}$); the rest are approximate to the digits shown.
| Dimension | From | To | Factor |
|---|---|---|---|
| Length | $1 \text{ in}$ | cm | $2.54$ (exact) |
| Length | $1 \text{ mi}$ | km | $\approx 1.609$ |
| Length | $1 \text{ km}$ | mi | $\approx 0.621$ |
| Mass | $1 \text{ lb}$ | kg | $\approx 0.4536$ |
| Mass | $1 \text{ kg}$ | lb | $\approx 2.205$ |
| Capacity | $1 \text{ gal (US)}$ | L | $\approx 3.785$ |
| Capacity | $1 \text{ L}$ | qt | $\approx 1.057$ |
Worked example — 100 yards in metres, using nothing but the inch and the inch-to-cm equivalence:
$$ 100 \text{ yd} \times \frac{3 \text{ ft}}{1 \text{ yd}} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}} = 91.44 \text{ m} $$Four conversion factors, four cancellations, one answer. You didn't need a yard-to-metre constant — you only needed the inch.
7. Compound units and powers
Real-world quantities often live in compound units — units made by combining two or more dimensions. Speed is length divided by time (m/s, km/h, mi/h). Density is mass divided by volume (kg/m³, g/cm³). Acceleration is length divided by time-squared (m/s²). The "multiply by 1" trick handles all of them, you just convert each factor independently.
Speed: 60 mi/h in m/s
$$ 60 \frac{\text{mi}}{\text{h}} \times \frac{1609.34 \text{ m}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{60 \cdot 1609.34}{3600} \approx 26.82 \text{ m/s} $$Two conversions in one expression: miles into metres on the top, hours into seconds on the bottom. The fraction structure is keeping the bookkeeping honest.
Powers of units: area and volume
A subtle, very common trap: when units are raised to a power, the conversion factor is raised to the same power.
$1 \text{ m} = 100 \text{ cm}$. So how many cm² in a m²?
$$ 1 \text{ m}^2 = (1 \text{ m})^2 = (100 \text{ cm})^2 = 10{,}000 \text{ cm}^2 $$Not 100. The factor is squared. The same goes for volume: $1 \text{ m}^3 = (100 \text{ cm})^3 = 10^6 \text{ cm}^3$. Imagine a cube one metre on each side; it contains a million centimetre-cubes. Visualize the geometry once and the rule sticks.
Density: 1 g/cm³ in kg/m³
Water's density is $1 \text{ g/cm}^3$ — a famously round number. In SI base units it looks different:
$$ 1 \frac{\text{g}}{\text{cm}^3} \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{(100 \text{ cm})^3}{1 \text{ m}^3} = \frac{10^6}{1000} \frac{\text{kg}}{\text{m}^3} = 1000 \frac{\text{kg}}{\text{m}^3} $$Same substance, same density, different-looking numbers. Without units to anchor it, "1" versus "1000" would look like a contradiction.
8. Temperature — the affine exception
Every conversion we've done so far has been a pure multiplication: scale by some factor, done. Temperature is the one place that pattern breaks, because the three temperature scales don't share a zero point.
- Celsius (°C): water freezes at $0°$, boils at $100°$.
- Fahrenheit (°F): water freezes at $32°$, boils at $212°$.
- Kelvin (K): absolute zero at $0 \text{ K}$; same degree size as Celsius. Used in science because no negative temperatures.
The conversions are affine — scale and shift:
$$ F = \tfrac{9}{5} C + 32 \qquad C = \tfrac{5}{9}(F - 32) \qquad K = C + 273.15 $$Body temperature $98.6 \text{ °F}$ in Celsius:
$$ C = \tfrac{5}{9}(98.6 - 32) = \tfrac{5}{9}(66.6) = 37 \text{ °C} $$The order matters. Subtract first, then scale — or you'll be scaling a Fahrenheit reading as if it were a Celsius one.
Treating $F = 1.8 C$ as the temperature conversion is a classic error. It works for differences in temperature (a $10°$ rise in C corresponds to an $18°$ rise in F), but not for absolute readings — because the two scales disagree about where zero is. The $+32$ in $F = \tfrac{9}{5} C + 32$ is the shift that accounts for that disagreement.
For quick mental Celsius → Fahrenheit: "double and add 30." So $25 \text{ °C} \approx 80 \text{ °F}$ (actually $77$). It overshoots by a few degrees but gets you in the right ballpark — fine for packing a suitcase, not for medicine.
9. Dimensional analysis as a sanity check
Here is the deepest payoff of treating units as algebraic objects: units catch errors before numbers do. If you write down a formula and the units don't come out right, the formula is wrong — full stop. You don't have to compute anything; the units already disagree.
Suppose someone hands you "$s = \tfrac{1}{2} a t$" as the formula for displacement from constant acceleration $a$ over time $t$. Does it dimensional-check?
$$ [a][t] = \left(\frac{\text{m}}{\text{s}^2}\right)(\text{s}) = \frac{\text{m}}{\text{s}} $$That's a velocity, not a position. The formula is wrong. The correct one is $s = \tfrac{1}{2} a t^2$:
$$ [a][t]^2 = \left(\frac{\text{m}}{\text{s}^2}\right)(\text{s}^2) = \text{m} \;\checkmark $$You can catch this without plugging in a single number. The units have done all the work.
Whenever you derive or remember a formula, run the unit check before the numerical one. It costs nothing and rules out an entire class of mistakes — sign errors aside, this is the cheapest error-detection method in all of applied math.
Dimensional consistency in equations
The rule generalises: both sides of an equation, and every term in a sum, must have the same units. "$5 \text{ m} + 3 \text{ s}$" isn't just unusual — it's meaningless. There's no number that "$5 \text{ m} + 3 \text{ s}$" denotes, the way "$5 + 3 = 8$" denotes the number eight. Mixing dimensions in a sum is a flag that something's gone wrong upstream.
10. Common pitfalls
The single most common conversion error. If you want to go from inches to feet but multiply by $\frac{12 \text{ in}}{1 \text{ ft}}$, you'll get a number 144× too large (and your units won't cancel). Always look at which unit you want to cancel — that's the one that goes on the bottom.
Feet-and-inches aren't decimal. Three inches is $3/12 = 0.25$ feet, so $5 \text{ ft } 3 \text{ in} = 5.25 \text{ ft}$. Many calculators and forms quietly assume decimal, which is fine for metric but disastrous for customary units.
$1 \text{ m}^2 = 10{,}000 \text{ cm}^2$, not $100$. When the unit has an exponent, the numerical factor inherits the same exponent. Volume is even more dramatic — $1 \text{ m}^3 = 10^6 \text{ cm}^3$.
"mm" and "m" differ by a factor of $1000$, not $10$. So do "μg" and "mg". A single dropped letter is a three-orders-of-magnitude error — fatal in medication dosing or engineering tolerance.
$F = 1.8 C$ is wrong. The $+32$ matters. Skip it and "freezing" reads as $0 \text{ °F}$ instead of $32 \text{ °F}$, an error of nearly $20 \text{ °C}$ at room temperature. Temperature is the one conversion that needs both a scale and a shift.
If you carry units only at the start and the end, you've thrown away your error-checker for the entire middle of the calculation. Annotate units on every line. The Mars Climate Orbiter's engineers didn't, and the spacecraft paid for it.
11. Worked examples
Try each one before opening the solution. The goal isn't to match the final number — it's to see whether your setup follows the cancellation pattern.
Example 1 · Convert $48 \text{ in}$ to feet
Step 1. Pick the conversion factor with inches on the bottom so it cancels:
$$ \frac{1 \text{ ft}}{12 \text{ in}} $$Step 2. Multiply:
$$ 48 \text{ in} \times \frac{1 \text{ ft}}{12 \text{ in}} = \frac{48}{12} \text{ ft} = 4 \text{ ft} $$Check. $4 \text{ ft} \times 12 \text{ in/ft} = 48 \text{ in}$ ✓
Example 2 · Convert $72 \text{ km/h}$ to m/s
Step 1. Two cancellations: km → m on top, h → s on the bottom.
$$ 72 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} $$Step 2. Cancel and compute:
$$ = \frac{72 \cdot 1000}{3600} \frac{\text{m}}{\text{s}} = \frac{72{,}000}{3600} \frac{\text{m}}{\text{s}} = 20 \text{ m/s} $$Mnemonic. For km/h → m/s, divide by $3.6$. (Since $1000/3600 = 1/3.6$.) So $72 / 3.6 = 20$. ✓
Example 3 · Convert $98.6 \text{ °F}$ to Celsius
Use $C = \tfrac{5}{9}(F - 32)$.
Step 1. Shift first:
$$ F - 32 = 98.6 - 32 = 66.6 $$Step 2. Scale:
$$ C = \tfrac{5}{9}(66.6) = \tfrac{333}{9} = 37 \text{ °C} $$So normal human body temperature is $37 \text{ °C}$, a number worth knowing by heart.
Example 4 · How many seconds in a year?
Use the chain $1 \text{ yr} = 365 \text{ d} \cdot 24 \text{ h} \cdot 60 \text{ min} \cdot 60 \text{ s}$:
$$ 1 \text{ yr} \times \frac{365 \text{ d}}{1 \text{ yr}} \times \frac{24 \text{ h}}{1 \text{ d}} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{60 \text{ s}}{1 \text{ min}} $$ $$ = 365 \cdot 24 \cdot 60 \cdot 60 = 31{,}536{,}000 \text{ s} \approx 3.15 \times 10^7 \text{ s} $$Mnemonic. $\pi \times 10^7$ seconds is one year to within $0.5\%$ — a number cosmologists use to sketch timescales mentally.
Example 5 · Convert $5 \text{ m}^2$ to cm²
$1 \text{ m} = 100 \text{ cm}$, so $(1 \text{ m})^2 = (100 \text{ cm})^2 = 10{,}000 \text{ cm}^2$. Therefore:
$$ 5 \text{ m}^2 \times \left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^2 = 5 \times 10{,}000 \text{ cm}^2 = 50{,}000 \text{ cm}^2 $$Watch out. Not $500$ cm² — the factor of $100$ is squared because the unit is squared.
Example 6 · Diagnose: "$5 \text{ ft } 8 \text{ in} = 5.8 \text{ ft}$"
Wrong. Feet and inches aren't decimal — $8$ inches is $8/12$ of a foot, not $0.8$ of one.
$$ 5 \text{ ft} + 8 \text{ in} \times \frac{1 \text{ ft}}{12 \text{ in}} = 5 + \tfrac{8}{12} = 5 + 0.\overline{6} \approx 5.67 \text{ ft} $$The student-written "$5.8$" is off by about $0.13 \text{ ft}$, or roughly $1.6$ inches — small but pervasive in any context that demands precision.
Example 7 · Express $1 \text{ atm}$ in kPa
By international definition, the standard atmosphere is exactly $101{,}325 \text{ Pa}$. Converting Pa to kPa is just dividing by $1000$:
$$ 1 \text{ atm} = 101{,}325 \text{ Pa} \times \frac{1 \text{ kPa}}{1000 \text{ Pa}} = 101.325 \text{ kPa} $$Useful sanity-check for blood-pressure devices and meteorology, where pressures are often quoted in kPa or hPa (1 hPa = 1 millibar).