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1.1 Please read this
Communications is a numerate subject, that is, it is often concerned with numbers. Very often these numbers come from measurements and calculations. Important numbers in communications are bandwidth, voltage, current, resistance, and frequency. A familiarity with measurements and the way they are expressed is crucial for anyone who intends to specialize in communications. OK, so it's rather dull. The material in this chapter is taught as part of the National Curriculum in all UK schools, and should be familiar to anyone who has been to school in the UK in the last ten years or so. It is presented here for those people who have somehow missed out on this subject at school, or have forgotten it, or went to school in the days when we measured distance in rods, poles and perches. It is often claimed that learning should be fun; however, sometimes learning is a drag. I suggest that you need to memorize the few facts in this chapter. As the conventions described here are used not only by communications engineers, but all professional scientists, engineers and technologists everywhere on Earth, becoming familiar with it will improve your own communications potential enormously.
1.2 Scientific notation
We engage in numerate activities all the time in daily life, such as shopping and paying bills. In the UK at least, our units of currency, the pound and the penny, are of the right size that paying for things does not involve very large numbers or very small numbers. It is very rare to have to deal with a sum of money bigger than 1,000 pounds or smaller than 1 penny. However, in science and engineering we deal with very large and very small numbers all the time. It is very incovenient to have to write (or say) numbers like 1,000,000,000,000,000 or 0.000,000,000,000,001, and it is rather prone to error. So we use a system called scientific notation, where we always give a number between -1 and 1, and a power of ten. So rather than writing 1 million as 1,000,000, we write it as ``10 to the power 6'' (106). Therefore 2 million is 2 x 106. You should know what the common powers of 10 are, e.g., 103=1000, 104=10,000, 105= 100,000, etc.
Small numbers are represented as negative powers of 10, so 0.001 (one thousandth) is 10-3. It is quite common in communications to work with numbers as large as 109, and as small as 10-6. The power of ten is called the exponent.
Note that exponents are not always easy to display as superscripts on computers and calculators, so there is a convention of writing, say, `E4' to mean `times 104'. Small calculators cannot manage even that much, and normally display the power of ten in a small window to the right of the main display.
An alternative to writing a power of ten directly is to use prefixes, as described below.
1.3 Units and prefixes
Almost everything that we measure has units. Distance has units of metres, or miles, or whatever. Current has units of amps. And so on. When you give the result of a measurement or calculation you should always give the units, if there are any. Leaving out the units in a measurement or result is exactly as bad as leaving out the number.
Because the units we use are not always suitable for the size of number being measured, it is often necessary to express them in scientific notation. For example, the unit `amp' is about a thousand times too big for communications work. Most currents encountered will be 0.001 amps. Rather than writing 0.001, we can write `1 x 10-3', as described above, or `1E-3' as most computer programs do. A better scheme is to use a standard prefix than means `x 0.001'. The prefix we use for this is called `milli' and written `m'. So rather than saying 0.001 amps, I would say `1 milliamp'. I would write this as `1 mA'. Note that I have not changed the numbers or done any calculations here. `1 mA' is exactly the same number as 0.001 amps. It is just easier to write, and can be copied with less chance of a mistake.
As well as `m' for `milli', there are other standard prefixes. The ones that are important in communications are Mega (M), meaning `million' and micro (u), meaning `one millionth'. Note that it is only the difference between a capital letter and a lower-case letter that distinguishes `micro' from `Mega'. Get this wrong and your answer will be wrong by a factor of 1,000,000,000,000!
In order to confuse the student further, we often use units that sound very different, but mean exactly the same. For example, 100 cycles per second is exactly the same as 100 hertz.
Some students have difficulty getting to grips with these ideas; however, I mut stress that they are the standard way of representing large and small numbers, used everywhere on Earth. The best way to become familiar with scientific notation is to use it all the time. For professional engineers and scientists it is second nature, and does not require any extra thought.
1.4 Precision
Whenever we measure something, the measurement is associated with a certain precision. The precision is the amount of `detail' revealed by the measurement. It is not the same as accuracy, which is the `correctness' of the measurement. For example, if I use a ruler to measure the length of something, I cannot measure with more detail than to the nearest millimetre or so. If I express such a measurement as `123.456' millimetres, this must be a mistake. A ruler does not have enough precision to measure distances as small as 0.006 millimetres.
What is true for measurements is also true for calculations. For example, suppose I have a length of electrical cable and I want to cut it into three exactly equal pieces (because I'm a nerd and I've got nothing better to do, presumably). If I measure it and it comes to 1.000 metre, how long must each piece be? 1.000 metres divided by 3 is 0.3333333..... metres. However, it would be wrong to give this as a result, as we don't have any way of measuring a length with this precision. An acceptable answer would be 0.333 metres, or better still, 333 millimetres.
You should realize from this that, in terms of measurements at least, `1.000' is a different number from `1'. This usually comes as a shock to people who have not studied physics or engineering. If I give the result of a measurement as `1.000' it implies that I know the result to three decimal places, and each one is zero. The result could not have been, say, 1.010 if I had measured it more carefully. On the other hand, if I give the result as `1', this means that it might be 1.01, or 1.05, or some other number, but I don't know. In summary you should give enough decimal places in your numbers to reflect how confident you are about their precision.
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