Units & Dimensional Analysis

The one trick that makes every unit conversion foolproof: multiply by a cleverly-disguised 1.

High schoolIntro Gen ChemUni Year 1
⏱️ About 18 min

In 1999 a Mars spacecraft was lost because one team worked in pounds and another in newtons — nobody converted. Units aren't decoration; they carry the meaning of a number. The good news: there's a single method that converts any unit into any other without guesswork, and once it clicks you'll never flip a conversion the wrong way again.

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The big idea: Every measurement is a number times a unit. To convert units you multiply by a fraction that equals 1 — a fraction whose top and bottom are the same real quantity written in different units. Multiplying by 1 never changes the amount, only its outfit, and the unwanted units cancel like numbers in a fraction.
🎯 By the end, you'll be able to
  • Name the base SI units for length, mass, time, and amount of substance
  • Build a conversion factor that equals 1 from any equivalence
  • Convert units with the factor-label method so units cancel correctly
  • Chain several conversion factors together in one calculation
📎 Helpful to know first

A number is only half of a measurement

Write down '5' and you've said nothing useful. Five what? Grams, kilometres, seconds? The unit is what turns a bare number into a measurement. In science we lean on the SI system (the International System of Units) so that everyone means the same thing.

A handful of base units cover the quantities chemistry uses most:

  • Length → metre (m)
  • Mass → kilogram (kg)
  • Time → second (s)
  • Amount of substance → mole (mol)
  • Temperature → kelvin (K)
🔑 Metric prefixes scale by powers of ten
Prefixes stretch or shrink a unit: kilo- (k) = 1000×, centi- (c) = 1/100, milli- (m) = 1/1000, micro- (µ) = 1/1 000 000. So 1 km = 1000 m and 1 mL = 0.001 L. Every metric conversion is just sliding a decimal point.

The core trick: multiplying by 1

Here is the whole idea. Since 1 km is 1000 m, the fraction \( \frac{1000\ \text{m}}{1\ \text{km}} \) equals exactly 1 — the top and bottom are the same real distance. Multiplying anything by 1 leaves its value untouched, so you can multiply your measurement by that fraction freely. Pick the version of '1' that puts the unit you want on top and the unit you want to cancel on the bottom.

\[ 1\ \text{km} = 1000\ \text{m} \quad\Longrightarrow\quad \frac{1000\ \text{m}}{1\ \text{km}} = 1 \quad\text{and}\quad \frac{1\ \text{km}}{1000\ \text{m}} = 1 \]
Any equivalence gives you two conversion factors — both equal to 1. Choose the one that cancels the unit you want to get rid of.
✨ Let the units cancel — they'll tell you if you're right
Treat units like algebra: if the same unit appears on a top and a bottom, it cancels. Set up your factors so everything cancels except the unit you want. If the leftover unit is wrong, you flipped a factor — no need to memorise which way to divide. This is the factor-label method.
📝 Worked example: Convert 2.0 days into seconds.
  1. Start with what you have: 2.0 days. Chain factors that each equal 1, cancelling one unit at a time: days → hours → minutes → seconds.
  2. \( 2.0\ \text{days} \times \dfrac{24\ \text{h}}{1\ \text{day}} \times \dfrac{60\ \text{min}}{1\ \text{h}} \times \dfrac{60\ \text{s}}{1\ \text{min}} \)
  3. Cancel units: 'days', 'h', and 'min' each appear top and bottom and disappear, leaving seconds.
  4. Multiply the numbers: 2.0 × 24 × 60 × 60 = 172 800.
✓ 172 800 s, or about 1.7 × 10⁵ s.
📝 Worked example: Convert 90 km/h to metres per second (m/s).
  1. km/h means kilometres divided by hours, so convert the top (km → m) and the bottom (h → s).
  2. \( \dfrac{90\ \text{km}}{1\ \text{h}} \times \dfrac{1000\ \text{m}}{1\ \text{km}} \times \dfrac{1\ \text{h}}{3600\ \text{s}} \)
  3. 'km' and 'h' cancel, leaving m/s. Numbers: (90 × 1000) ÷ 3600 = 90 000 ÷ 3600 = 25.
✓ 25 m/s (a handy rule of thumb: divide km/h by 3.6 to get m/s).
✏️ Practice: Convert 90 km/h to metres per second (m/s).
m/s
Solution
  1. \( \dfrac{90\ \text{km}}{1\ \text{h}} \times \dfrac{1000\ \text{m}}{1\ \text{km}} \times \dfrac{1\ \text{h}}{3600\ \text{s}} \).
  2. km and h cancel, leaving m/s.
  3. (90 × 1000) ÷ 3600 = 25 m/s.
✏️ Practice: How many centimetres are in 3.0 feet? Use 1 inch = 2.54 cm and 1 foot = 12 inches.
cm
Solution
  1. Chain two factors, each equal to 1: feet → inches → centimetres.
  2. \( 3.0\ \text{ft} \times \dfrac{12\ \text{in}}{1\ \text{ft}} \times \dfrac{2.54\ \text{cm}}{1\ \text{in}} \).
  3. 'ft' and 'in' cancel: 3.0 × 12 × 2.54 = 91.44 cm.

Check your understanding

1. Why is it valid to multiply a measurement by a fraction like (1000 m)/(1 km)?
1000 m and 1 km are the same distance, so the fraction equals exactly 1. Multiplying by 1 changes the units without changing the actual amount — that's the whole point of the method.
2. You want to convert 250 cm to metres. Which factor should you multiply by?
You need 'cm' on the bottom to cancel the cm you have, leaving metres on top: 250 cm × (1 m / 100 cm) = 2.5 m. If units don't cancel to what you want, flip the factor.
3. After setting up a conversion, your units cancel down to 'kg·s' but you expected 'm'. What does that tell you?
The leftover units are a built-in error check. If they aren't the units you want, a factor is upside down or one is missing — fix the setup before touching the numbers.
🔎 Related lessons
Next: Significant Figures, Made Simple Density The Mole & Molar Mass
✅ Key takeaways
  • A measurement is a number and a unit; the SI base units include m, kg, s, mol, and K.
  • Any equivalence (like 1 km = 1000 m) gives two conversion factors, each equal to 1.
  • Converting units = multiplying by a form of 1, which changes units but not the amount.
  • Set factors so unwanted units cancel; the leftover unit is your error check.
  • Chain several factors in one line to make multi-step conversions foolproof.
➡️ Your conversions are now airtight — but how many digits should the answer actually keep? Report too many and you're faking precision you don't have. Significant figures are next.
Want to test yourself on this? Try the Chemistry practice test →