5 Units of measurement

If you were told that the length of a piece of string was 37, you would be rather baffled. 37 what? Is it 37 metres, 37 centimetres, 37 feet, or even 37 miles? Similarly, if somebody says that a friend weighs 100, what does this mean? These examples highlight the importance of having defined units with which to make measurements and the importance of quoting the units when you want to communicate what you have measured.

In science, the units used are known as SI units, which is an abbreviation for ‘Système Internationale d’Unités’ (International System of Units). In 1960, an international conference formally approved this set of metric units as standard, so replacing the many different national systems of measurement that had been used in science up to that time. The advantage of having a standard set of units is that everyone uses them, and there is no need to convert laboriously from one system to another to compare results in different countries. So although in everyday life in the UK people may still buy milk and beer in pints, and measure distances between towns in miles, in the scientific community SI units are used almost exclusively.

So what are these units? At this stage, we shall just introduce the SI units of length, time and mass.

• The basic SI unit of length is the metre, which is abbreviated to m.
• The basic SI unit of time is the second, which is abbreviated to s.
• The basic SI unit of mass is the kilogram, which is abbreviated to kg.

Although a metre is a conveniently sized unit for measuring the height of a person or the width of a room, it is the wrong sort of size to use for quoting the distance between London and Edinburgh, or the breadth of a pinhead. It is therefore conventional and convenient to use larger and smaller multiples of the metre when appropriate; note that these are also SI units. So, for example, large distances can be measured in kilometres (km), and small distances or lengths can be measured in millimetres (mm). The prefix kilo means ‘one thousand’, so a kilometre is one thousand metres. The prefix milli means ‘one-thousandth’, so a millimetre is one-thousandth of a metre. Put another way, one metre is one thousand millimetres. Therefore, 1 km = 1 000 m, 1 mm = m and 1 m = 1 000 mm.

Another common prefix that you may have met is centi (as in centimetre, abbreviated to cm), which means ‘one-hundredth’. So 1 cm = m, and this means that:

100 cm = 100 m = m = 1 m, i.e. 1 m = 100 cm.

If you didn’t know how many millimetres there were in a centimetre, you could use the definitions to work this out. From the definitions,

1 m = 100 cm = 1 000 mm

So if we divide each of these equal lengths by 100, then

m = cm = mm

which gives

m = 1 cm = 10 mm

This means that there are 10 millimetres in 1 centimetre. Alternatively, if you want to know how many centimetres are equivalent to 1 millimetre, you can start from the equation 1 cm = 10 mm and divide these equal lengths by ten, so cm = 1 mm, or 1 mm = cm.

In general, we use the abbreviations for units in all calculations, like the ones above. Within the main text, we sometimes use the full word and sometimes the abbreviation, though as the module progresses we shall use the abbreviations more frequently. Most importantly, you can also see from these examples that the abbreviations for units are both singular and plural, so ‘m’ means metre or metres.

The relationships between the four units of length that have been introduced are summarised in Table 5.1.

The basic SI unit of time, the second (s), will be familiar from everyday life. Longer time intervals may be measured in minutes, hours, days or years, but these are not SI units. Shorter times are measured in smaller multiples (or sub-multiples) of the second. Thus, a millisecond (ms) is one-thousandth of a second, just as a millimetre is one-thousandth of a metre.

You may have been surprised that we said that the kilogram is the SI unit for mass, rather than the unit for weight. After all, in everyday usage we talk about somebody’s weight being so many kilograms. However, in scientific use, the term weight means the downward pull on an object due to gravity, for example, the downward pull that makes an apple fall to the ground. This means that your weight would decrease if you went to the Moon, where gravity is only about 1/6 as strong as on Earth. Your mass, however, is determined by the amount of matter in your body, and, since this doesn’t depend on gravity, your mass is the same wherever you are in the Universe. Weighing scales are always marked in units of mass, for example, kilograms, so to be scientifically correct you should say that somebody has a mass of 55 kilograms, rather than saying that their weight is 55 kilograms.

Now just as the kilometre is equal to one thousand metres, so the kilogram is equal to one thousand grams, 1 kilogram = 1 000 grams. You will have seen the number of grams quoted on all kinds of packaged food. Very small quantities are measured in milligrams, where one milligram (mg) is one-thousandth of a gram, 1 milligram = gram. So, using abbreviations, 1 kg = 1 000 g, and 1 mg = g.

Before we leave the discussion of units, there is one other important point to remember about the relationship between units. Suppose you had to add together 100 cm and 2 m, what would be the first step you would need to take? You would convert 100 cm to 1 m, so the calculation becomes 1 m + 2 m = 3 m. Alternatively, you would convert 2 m to 200 cm, so 100 cm + 200 cm = 300 cm. The same is true for all units, not just the units of length.