SI units and unit conversions

The same table measures 120 cm with one ruler and 1,2 m with another. Why are the numbers different? How can we make sure our choice of ruler does not change a calculation's result?

Imagine laying centimetre lengths along the table, then measuring it in metre lengths. The table fits 120 of the first unit, or 1,2 of the second. Measuring means comparing a quantity with a unit: the length is the number of units multiplied by the size of one unit. A larger unit therefore gives a smaller number. Both 120 cm and 1,2 m describe the same length; 120 on its own does not tell us a length.

Suppose we measure distance in metres and time in minutes. Dividing gives metres per minute; expecting metres per second would be a mistake. Compatible units avoid this confusion. The International System of Units, SI, uses the metre, kilogram and second for length, mass and time. Units for other mechanical quantities are built from these using their defining equations. A number may look plausible yet describe a different result if we have not checked its units.

A ruler can measure a table's length l, a shelf's height h, or the distance S along a drawn line. These ask different questions about an object, but each compares a distance with the metre. A letter alone does not define a physical quantity: the problem gives it meaning. Doubling a route doubles the distance travelled without changing the shelf's height. Before substituting values, connect each symbol with a particular measurement.

Now imagine covering the table with paper. Its length is not enough; we also need its width. Area S measures the size of a surface. Its SI unit is m², the area of a square with sides of 1 m. For a rectangle, S=ab: multiply the number of unit squares in a row by the number of rows. Doubling one side doubles the area; doubling both makes it four times as large. In a motion problem, the same letter S may mean distance travelled, so check its meaning and unit.

How much space does a box occupy? We need its length, width and height. A rectangular box has volume V equal to abc: multiply its base area by the number of layers of unit thickness. A cubic metre is the volume of a cube with sides of 1 m. If every dimension of the box doubles, twice as many small cubes fit in each direction, giving eight times as many altogether. These three independent dimensions explain why the unit is cubed.

A load of mass m=500 g can also be written as 0,5 kg, and a duration t=2 min as 120 s. The mass has not decreased and the process has not lasted longer; only the unit has changed. SI expresses mass in kilograms and time in seconds. To check a conversion, work back to the original unit. Half a kilogram contains 500 grams; two minutes contain two groups of 60 seconds.

Later topics introduce other symbols, but we will read quantities in the same way. Electric current I describes the charge passing through a cross-section per unit time; its unit is the ampere. Wavelength \lambda, for example the distance between neighbouring crests of a periodic wave, is measured in metres. For now, distinguish a quantity's meaning from its scale: 'length' suggests metres, but does not tell us which particular distance to measure.

Divide a metre strip into 100 equal parts: each is a centimetre. Divide it into 1000 parts and each is a millimetre. Thus 1\,\text{см}=10^{-2}\,\text{м} and 1\,\text{мм}=10^{-3}\,\text{м}. To convert, multiply the number of parts by the size of each part: 8 cm is 8\times0{,}01=0{,}08 m. If your numerical answer in metres exceeds the number of centimetres, you have converted in the wrong direction.

Imagine dividing a kilogram load into a thousand equal masses. Each has mass 1 g, so 1\,\text{г}=10^{-3}\,\text{кг}. For 250 g we get 250\times10^{-3}=0{,}25 kg, a quarter of the original load. To convert back, divide by 10^{-3}, or multiply by a thousand. This is the same comparison with a unit, applied in opposite directions.

Can we convert 1 cm² using the same factor as for 1 cm? Draw a square: both sides are 0,01 m long. Its area is 1\,\text{см}^2=(10^{-2}\,\text{м})^2=10^{-4}\,\text{м}^2. A metre square fits 100 of these small squares across and 100 along, making 10 000 altogether. The factor appears twice because we convert two dimensions, not because area needs an unrelated rule.

Try the same reasoning with millimetre graph paper. Each side of a square metre fits 1000 millimetre lengths, so the whole square contains a million cells of 1 mm². Therefore 1\,\text{мм}^2=(10^{-3}\,\text{м})^2=10^{-6}\,\text{м}^2. Using only 10^{-3} would convert one side of the rectangle while leaving the other in millimetres; the units would still be inconsistent.

Fill a metre cube with cubes of side 1 cm. Each layer contains 100\times100 small cubes, and there are 100 layers. Thus 1\,\text{см}^3=(10^{-2}\,\text{м})^3=10^{-6}\,\text{м}^3: a small cube occupies one millionth of the large cube. If we only wanted the base area, we would use two factors. The third accounts for height and distinguishes volume from area.

Picture a litre container as a cube with sides of 10 cm, or 1 dm. Then 1\,\text{л}=1\,\text{дм}^3=(0{,}1\,\text{м})^3=10^{-3}\,\text{м}^3. Ten such cubes fit along each edge of a metre cube, giving 10^3=1000 in total. So 2 L is 0,002 m³. A litre measures volume; to find the mass of a litre of liquid, we also need its density.

Time uses the same reasoning but a different set of units: an hour contains 60 minutes, each containing 60 seconds. Therefore 1\,\text{ч}=3600\,\text{с}. Half an hour is 1800 s, and zero duration remains zero in every unit. We cannot borrow the centimetre conversion factor: first establish how many new units make up one old unit, then calculate.

A plate has an area of 30 cm². Express it in m².

Area has two dimensions, so square the length conversion factor.

1 см = 0,01 м, so 1 см² = 0,0001 м².

30 × 0,0001 = 0,003. The result is in m².

Check: 30 cm² is much smaller than a square metre, so the number in m² must be less than one.

30 см² = 0,003 м². This is 30 out of the 10 000 square centimetres in a square metre, a sensible size for a small plate.

Changing the unit changes the number, not the object being measured. Always read a result together with its unit.

Squared and cubed units come from measuring two or three dimensions. The conversion factor must therefore also be squared or cubed.

Check a conversion by reversing it and estimating the size: a small plate should not suddenly cover several square metres.

Next, we will learn to recognise conversion factors from unit prefixes.