# 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.

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.

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.

## SAQ 4

Try to think of another quantity that is strictly discrete, in that it can only have a finite number of values.

### Answer

How about the number of people who will come to my party tonight? Or the number of bricks in a house?

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.

## Exercise 2

Which items in the following list are fundamentally analogue and which fundamentally discrete?

The price of petrol

The amount of heat from a fire

The speed of a car

The energy of a star

The size of the audience at a play

The pressure of the atmosphere.

### Discussion

I would say that items 2, 3, 4 and 6 were definitely analogue, although we might choose to measure them with discrete instruments. Items 1 and 5 are discrete.