 Mathematics for science and technology

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# 1 Units of measurement

In SI, there are seven ‘base units’, which are listed in Box 1. Surprising as it may initially seem, every unit for every other kind of quantity (speed, acceleration, pressure, energy, voltage, heat, magnetic field, properties of radioactive materials, indeed whatever you care to name) can be made up from combinations of just these seven base units. For instance, speed is measured in metres per second. In this course you will work mainly with the familiar base units of length, mass, time and temperature, and some of their combinations, but it is worth knowing that the other base units exist as you may meet them elsewhere.

## Box 1 The SI base units

Physical quantityName of unitSymbol for unit
lengthmetrem
timeseconds
masskilogramkg
temperaturekelvinK
amount of substancemolemol
electric currentampereA
luminous intensitycandelacd

Most of these base units relate to physical descriptions that apply universally. The SI base unit of time, the second, is defined as the period over which the waves emitted by caesium atoms under specific conditions cycle exactly 9 192 631 770 times. Then the SI base unit of length, the metre, is defined by stating that the speed of light in a vacuum is exactly 299 792 458 metres per second.

The SI base unit of mass, the kilogram, is the only fundamental unit that is defined in terms of a specific object. The metal cylinder which constitutes the world's 'standard kilogram' is kept in France. Note that the kilogram is the standard unit of mass, not of weight. In scientific language, the weight of an object is the downward pull on that object due to gravity, whereas its mass is determined by the amount of matter in it.

The SI base unit of temperature is the kelvin, which is related to the everyday unit of temperature, the degree Celsius:

(temperature in kelvin) = (temperature in degrees Celsius) + 273.15

The amount of a pure substance is expressed in the SI base unit of the mole. Whatever the smallest particle of a given substance is, one mole of that substance will contain 6.022 1408 57(74) × 1023 (known as Avogadro’s number) of those particles. A mole of graphite contains Avogadro’s number of carbon atoms. A mole of carbon dioxide (one carbon atom joined to two oxygen atoms), contains Avogadro’s number of these molecules.

It is important to realise that, although in everyday usage it is common to say that you 'weigh so many kilos', there are two things wrong with this usage from the scientific point of view. First, as noted in Box 1, the kilogram is not a unit of weight, but a unit of mass. Secondly, in scientific language, 'kilo' is never used as an abbreviation for kilogram. In science, kilo is always used as a prefix, denoting a thousand: one kilometre is a thousand metres, one kilogram is a thousand grams.

Another familiar prefix is 'milli', denoting a thousandth. One millimetre is one-thousandth of a metre; or put the other way round, a thousand millimetres make up a metre. There are many other prefixes in use with SI units, all of which may be applied to any quantity. Like kilo and milli, the standard prefixes are based on multiples of 1000 (i.e. 103). The most commonly used prefixes are listed in Box 2.

Although scientific notation, SI units and the prefixes in Box 2 are universal shorthand for all scientists, there are a few instances in which other conventions and units are adopted by particular groups of scientists for reasons of convenience. For example, you have seen that the age of the Earth is about 4.6 × 109 years. One way to write this would be 4.6 'giga years' but geologists find millions of years a much more convenient standard measure. They even have a special symbol for a million years: Ma. So, in Earth science texts you will commonly find the age of the Earth written as 4600 Ma.

A few metric units from the pre-SI era also remain in use. In chemistry courses, you may come across the ångström (symbol Å), equal to 10−10 metres. This was commonly used for the measurement of distances between atoms in chemical structures, although these distances are now often expressed in either nanometres or picometres. Other metric but non-SI units with which you are probably familiar are the litre (symbol l) and the ‘degree Celsius’ (symbol °C).

## Box 2 Prefixes used with SI units

 prefix symbol multiplying factor tera T 1012 = 1000 000 000 000 giga G 109 = 1000 000 000 mega M 106 = 1000 000 kilo k 103 = 1000 - - 100 = 1 milli m 10−3 = 0.001 micro μ * 10−6 = 0.000 001 nano n 10−9 = 0.000 000 001 pico p 10−12 = 0.000 000 000 001 femto f 10−15 = 0.000 000 000 000 001
* The Greek letter μ is pronounced ‘mew’.

The following data may help to illustrate the size implications of some of the prefixes:

• the distance between Pluto (the furthest planet in the Solar System) and the Sun is about 6 Tm,
• a century is about 3 Gs,
• eleven and a half days contain about 1 Ms,
• the length of a typical virus is about 10 nm,
• the mass of a typical bacterial cell is about 1 pg.

There are also some prefixes in common use, which don’t appear in Box 2 because they don’t conform to the ‘multiples of 1000’ rule, but never the less produce convenient measures. One is centi (hundredth): rulers show centimetres (hundredths of a metre) as well as millimetres, and standard wine bottles are marked as holding 75 cl. One less commonly seen is deci (tenth), but that is routinely used by chemists in measuring concentrations of chemicals dissolved in water, or other solvents. Later this week you will also come across the decibel, which is used to measure the loudness of sounds.

Look at this worked example, and then try converting between units in the following activity.

## Worked example 1

Diamond is a crystalline form of carbon in which the distance between adjacent carbon atoms is 0.154 nm. What is this interatomic distance expressed in picometres?

Now try this in the following activity.

## Activity 1 Converting between units and scientific notation

Using scientific notation, express:

• a.3476 km (the diameter of the Moon) in metres,
• b.8.0 μm (the diameter of a capillary carrying blood in the body) in nm,
• c.0.8 s (a typical time between human heartbeats) in ms.

• a.A kilometre is 10 3 times bigger than a metre, so

• b.A micrometre is 103times bigger than a nanometre, so

8.0 μm = 8.0 × 103 nm

• c.A second is 103 times bigger than a millisecond, so

8.0 s = 0.8 × 103 ms

To express this in scientific notation, we need to multiply and divide the right-hand side by 10:

When looking at the size of things sometimes all that you need to know is approximately how big, or small it is. You will explore this idea in the next section.

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