Quantities and units

Suppose you measure a table and report that its length is 120. That number leaves something out: 120 of what? Centimetres and metres describe very different lengths.

Writing the length as $120\,\mathrm{cm}$ gives the number a reference. The centimetre is the unit, and 120 is the numerical value in that unit.

A quantity and its numerical value

A physical quantity is a property that can be measured or expressed quantitatively. Length, mass and time are examples. For a length, we can write:

$$\ell = 120\,\mathrm{cm} = 1.20\,\mathrm{m}$$

The table has not changed. We have changed the unit used to express its length, so the numerical value changes with it. A larger unit requires a smaller numerical value for the same quantity.

The physical quantity stays the same. Both expressions describe the same length. This distinction matters whenever we compare measurements or substitute values into an equation.

Converting units

Since one metre equals one hundred centimetres, the ratio below equals one. Multiplying by it changes the expression without changing the length:

$$120\,\mathrm{cm}\times\frac{1\,\mathrm{m}}{100\,\mathrm{cm}} = 1.20\,\mathrm{m}$$

The centimetres cancel, leaving metres. Writing the units throughout the calculation helps us choose the correct conversion factor.

For an area, the conversion factor must also be squared. A square with sides of one metre has sides of one hundred centimetres, so:

$$1\,\mathrm{m}^{2} = 10{,}000\,\mathrm{cm}^{2}$$

Base and derived quantities

A system of units chooses base quantities and defines other quantities through relationships between them. In SI, length, mass and time are among the base quantities. Their units are the metre, kilogram and second.

Area is a derived quantity: multiplying two lengths gives an area, measured in square metres. Density relates mass to volume, giving kilograms per cubic metre. The units reflect how these quantities are defined.

This is an introduction to that distinction. The full set of SI base quantities and units, dimensional reasoning and the limits of a measurement belong to the rest of the module.

Try the conversion

A sheet is $30\,\mathrm{cm}$ long and $20\,\mathrm{cm}$ wide. Find its area in square centimetres, then convert that area to square metres. Check your result by converting the two lengths to metres before multiplying them.

Both routes should describe the same area. If the numerical answers disagree after conversion, check whether you converted the area using a squared factor.