The number 4380 got picked from a random pressing of the numeric keypad. It could be any number. We’re going to discuss the semi-random number, 4380, as a way to get, maybe, a fresh insight into what numbers do.
Numbers for counting
The first thing to say is that 4380, as written (and we’ll write it differently later…) is a ‘natural number’. Natural numbers are the numbers we use to count with. Mathematicians seem to have come up with two lists of natural number, one starting at zero (0, 1, 2, 3, …) and the other starting at one (1, 2, 3, 4, …). For our purposes it might be easier to appropriate the 2nd list (notice how we just used the number ‘2’ to label the list we wanted! Labelling/ordering is an important use of numbers.). Then if we have a pile of paperclips on the desk, and we want to know how many there are, we can pair off the first paperclip with the number 1, the next with the number 2, and so on until there are no more paperclips left. The number of paperclips is the number that you assign to the final paperclip. In the picture below, the final paperclip was paired off with the number 4380, so there are 4380 paperclips. There has been a ‘one-to-one mapping’ of the numbers to the paperclips. This one-to-one mapping is called ‘counting’. You might not have realised you were doing this last time you counted something, but you were! It is, of course, important that you assign the numbers ‘in order’ – you can’t just select random numbers to pair the paperclips off with. But that’s OK, because we all manage to do it fine (none of us count 1, 7, 3, 4380, 32 etc).
Now, your anger may be growing as you consider that your time is being wasted being taught ‘how to count’. I’ll try to win you back by explaining the role of numbers in science, rather than in desk stationery management.
Quantifying physical properties
In science, numbers are not just used for counting. That’s true for everyday life too – people are comfortable saying that the temperature is 21 °C, but nobody is ‘counting the degrees up to 21…’ Numbers quantify a property of something that we are interested in. For example, a device might have an electrical resistance of 4380 Ω (ohms, named after the Swede, Georg Ohm). However, whereas paperclips are discrete entities that ‘come in ones’, 4380 Ω is not the only possible resistance between 4379 Ω and 4381 Ω. That’s OK too, because we are all used to decimal numbers, such as 4380.27 Ω. In providing a value of resistance, numbers are being used to quantify a continuous variable, rather than simply enumerating discrete objects (counting).
In the previous paragraph, the number 4380 took on an added layer of meaning, one which is often not considered. There were exactly 4380 paperclips, but there might not be exactly 4380 ohms. How many ohms there are depends on the number of significant figures (s.f.) to which we are quantifying the resistance. In this case, 4380 Ω might mean ‘a resistance in ohms at least 4375 and less than 4385’, or it might mean ‘a resistance in ohms at least 4379.5 and less than 4380.5’. It isn’t possible to tell from the information given, because we don’t know whether the zero in the fourth column is a product of rounding or not. We should write 4380 Ω (to 3 s.f.) for the first case, and 4380 Ω (to 4 s.f.) for the second case, but many times, we don’t bother.
In praise of standard form
The number 4380 can be expressed as 4.38 x 103. The expression of a number as a power of ten (3 in this case) multiplied by a number between 1 and 10 (4.38 in this case) is called ‘standard form’. The same number can be expressed 4.380 x 103. Now, some people may say that 4.38 x 103 and 4.380 x 103 are the same number. After all, we are taught that adding zeroes on to the end of a number after the decimal point does not change its value. Well, it might not change its value, but it is changing what you are saying about the quality of your knowledge of that value. Standard form can be really useful like this because the number of significant figures can be made unambiguous, unlike in the number 4380, where confusion reigned over where the rounding had taken place. So:
- 4.38 x 103 → corresponds to → 4380 (to 3 s.f.) → corresponds to → ‘at least 4375 and less than 4385’
- 4.380 x 103 → corresponds to → 4380 (to 4 s.f.) → corresponds to → ‘at least 4379.5 and less than 4385.5’
There are even more confusing numbers. Consider a resistance of 50000 Ω. Written out like this it is impossible to tell whether it has been quoted to 1, 2, 3, 4 or 5 significant figures. So it could mean ‘between 45000 Ω and 55000 Ω’ or ‘between 49500 Ω and 50500 Ω’ and so on.
When science, maths, commerce, and law meet (and make no sense)
Here in the UK, we used to use the imperial units pounds (lb) and ounces (oz) to weigh our household goods. So a 1 lb box of gravy powder was labelled ‘1 lb’ – how sensible. From a commercial/legal standpoint the label meant ‘we have put in 1 lb of gravy powder. It might be a bit more or a bit less, but it’s one pound to within the regulations that we are bound by…’ Mathematically/scientifically, the label really meant ‘a weight of at least 0.5 lb and less than 1.5 lb’ because the weight was only quoted to one significant figure, but no-one took this pedantic view. Instead, everyone thought of it in the commercial/legal sense, even if they weren’t aware how far from a pound the manufacturers were allowed to get away with.
Then in the dim and distant past, we decided to/were forced to adopt the metric system and label all our goods in grams (g). But, of course, manufacturers still made gravy powder in the same sized boxes – production techniques are expensive to change, and besides, people wanted the same amount of gravy powder as before. A modern box of gravy powder in shown on the left in the picture below.
A pound of gravy powder and half a pound of coffee
Now, it is true that 1 lb is approximately equal to 454 g, but in fact the relabelling has changed the meaning of the label. The mathematical/scientific meaning of this label is that we know the weight to the nearest gram, a claim that was not made when labelling in pounds. For the same reason, half-pound bags of coffee are labelled 227 g, as in the picture above. Do we really believe they are ‘at least 226.5 g and less than 227.5 g’? Well, I think that’s quite an optimistic view of the packing process (although if gravy powder and coffee packers tell me it’s true then I apologise…).
I think we probably realise these examples of packaging are really saying is ‘a bit less than or a bit more than the stated value, within the regulations I’m bound by rather than to the nearest gram, but I’ve been forced to label it in grams’. But it is definitely open to misinterpretation. Manufacturing in imperial units and then labelling in metric is inherently confusing (although, I admit it might not ruin your day, and we may all have more pressing issues in life to worry about).
Should the manufacturers label the gravy powder 450 g and the coffee 230 g instead? That would seem like a more sensible rounding, giving a leeway of 5 g either way. Well, maybe, but then the ‘target weights’ have shifted by 4 g and 3 g respectively, and that may not be desirable, especially as 230 isn’t half of 450. Also, we can lose information about ‘where the rounding is’. Here’s a can of tomatoes:
At first sight, 400 g seems more sensible than 454 g as a measure – at least it’s not an imperial measure pretending to be a metric measure. However, we are back to the same problem we had when dealing with 4380 – we don’t know whether this weight is quoted to:
- 1 s.f., in which case the weight is 400 g ± 50 g
- 2 s.f., in which case the weight is 400 g ± 5 g
- 3 s.f., in which case the weight is 400 g ± 0.5 g
Maybe we should get food producers to label like this: 400 g ± 5 g.
And in fact, the tomato weight is probably 400 g ± ‘some other value dictated by regulations, rather than a value formed from the rounding’. Let’s imagine the ‘leeway’ is 20 g (tomato canners please get in touch to let us know). Then, maybe we should get food producers to label the tomatoes 400 g ± 20 g, not because we are mathematical pedants, but because we actually care about how different from the ‘target weight’ our goods might be.
This isn’t going to happen any time soon. So the only moral we have really arrived at is don’t work in one system and label in another – it will fail to make sense from whichever angle you look at it… If you are now laughing at the UK from Europe, where the more sensible metric system rules, then fair enough. However, if you are laughing at us from the US, bear in mind that you have your own issues! I suppose at least the US is consistent – in the UK we are labelling pounds in grams but we haven’t got round to changing our road signs to kilometres yet.
So if we aren’t going to manage to revolutionise the world of packaging, maybe we should find a more fruitful area to start a campaign. Here’s a tongue-in-cheek suggestion: let’s lobby for all numbers to be expressed in standard form at all times, except when counting… Maybe even when counting…
Or perhaps we should all just question what numbers mean a little more often wherever we meet them…