3.4 Discrete things

In contrast to analogue quantities, which change continuously, discrete quantities change in a series of clear steps.

Again, an example should make this clear. In Subsection 3.2, I considered the case of an analogue volume control. A discrete volume control is shown in Figure 6.

Figure 6 A discrete volume control

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A discrete volume control which increases volume in steps not incrementally.

Figure 6 A discrete volume control

Turning a volume control like this is rather a different sensation from the smooth feeling of the one illustrated in Figure 4. For a start, the movement progresses through a series of definite clicks. And if you listen to the sound as the control moves clockwise from click to click, you can hear the volume increasing not smoothly, but in a series of steps, each one sharply and distinctly higher than the one before. This is just what we mean by a discrete quantity.

Think back again to my example of a thermometer. I claimed that, for any two temperatures there is an infinite number of possible temperature differences between them. This cannot be true of a discrete quantity, which only has a fixed number of possible values between any two points on its scale.

You may now have a question in mind. Didn't he say earlier that volume is an analogue quantity? Now he's suggesting that it is a discrete quantity. Which is true? Well, both can be true in a way. Temperature and volume are fundamentally analogue quantities in that they are infinitely variable. But we may choose to treat them as if they were discrete for our own convenience. Take the example of a thermometer again. Perhaps you own a clinical thermometer that looks something like the one shown in Figure 7.

Figure 7 A discrete thermometer

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A drawing of a digital thermometer

Figure 7 A discrete thermometer

The thermometer in Figure 7 has no column of mercury and no scale – just a window in which we can read out a temperature value to two decimal places. Now, if I warm the bulb of this sort of thermometer and watch what happens in the window, I will see the temperature rise in a series of distinct steps: 97.18, 97.19, 97.20, … . The values in-between these steps are simply ignored. I could, of course, design a thermometer with a much wider window, so I could record temperatures like 97.1843750927341. But why would I want to? For my purposes, the difference between 101.8374923 and 101.8374924 is of no interest: I still feel ill. And no matter how wide the window is, it can't record all possible temperatures, because an infinite number of infinitely small temperature differences are possible. So I can treat temperature as if it were a discrete quantity because it suits my purpose. This happens whenever we measure something.

However, you should note that many quantities are fundamentally discrete, in that there is truly only a fixed range of values they can take.

In fact anything that we can count is likely to be a discrete quantity. The weight of a pile of sand is an analogue quantity, but the number of grains of sand in the pile is discrete.