# Axis labels – accurate but not at all obvious

I had a conversation with a class this morning regarding the labelling of axes on graphs.In particular, how we should indicate the units. Most quantities we deal with in physics carry units. A speed might be 35 km/h, a distance might be 16.8 mm, a pressure could be 28 kPa. Saying that a speed is 35 is wrong and meaningless. 35 km/h is rather different to 35 m/s or 35 inches/day.

If we are plotting two quantities, for example to find a relationship between them, we need to indicate what units the axes are in. A typical way to do that would be to include the unit in brackets after the label. So if we are plotting a distance, we might label our axis ‘distance (m)’, with the ‘(m)’ indicating that the numbers on the plot are in metres. But let’s say we are dealing with small distances, such as the spacing between atoms in a crystal. Then nanometres (one billionth of a metre) would be more appropriate as units. So we could label our axis as ‘distance (nm)’, indicating the numbers on the plot are in nanometres.

That’s the easy, pragmatic approach an engineer might take. For the physicist, however, it is not rigorous enough. So the physicist takes the same plot, crosses out ‘distance (nm)’ and instead writes ’10^{9} distance / m’.

How do we decipher that? (One of my students this morning suggested this was like a secret code physicists use to make sure their work is unreadable to anyone outside their area.) To a physicist or mathematician it’s perfectly logical. Let’s suppose we want to indicate on the plot a distance of 5.6 nanometres (that is, 5.6 x 10^{-9} metres). Then ’10^{9} distance / m’ is just:

10^{9} distance /m = 10^{9} (5.6 x 10^{-9} m) / m = 10^{9} x 10^{-9} x 5.6 x m/m = 5.6

We are left with our ‘5.6’ to show on the plot. Note how the 10^{9} at the front cancels the ‘nano’ within the nanometre, and the ‘/m’ cancels the ‘metre’ within the ‘nanometre’. It’s easy and logical once you’ve got it, but if you haven’t, it looks a unweildy and pointless mess.

Really, it comes down to a physicist recognizing a unit for what it actually is – something that the quantity is multiplied by – rather than a tag-on descriptor. I’m currently revising my presentation on preparing for the NZQA Scholarship Physics exam, which I’m giving again on Saturday (19th Sept). There’s a deeply annoying question from last year’s exam which says

By considering energy conservation, show that at the lowest point of the jump,

mgh= 1/2k(23-L)^{2}, where h is the change in height of Emma’s centre of mass.

Why is that wrong? Yes, I do mean wrong, rather than just clumsy. *L *here is a length (the length of a bungy cord, as it happens). So what is 23-*L*? The 23 is dimensionless – something dimensionless minus a length makes no sense (unless we go into geometric algebra where it is quite reasonable but I really don’t think that the examiners are wanting that…) It should say ’23 m’, or ’23 metres’. “Ah”, you retort, “but here we are using S.I. units, so *m *is in kg, *g* in m/s^{2}, *h* in metres, *k* in N/m and *L* in metres. When we understand the units, there is no issue. 23 – *L *is 23 minus the number of metres that *L *is”. That would be true, if the question had said this. But it actually says that ‘*h*‘ is the change in height of Emma’s centre of mass – i.e. it implies that *h* is dimensioned, it isn’t just the number of metres that the height changes by. A similar statement tells us that *L *also is a length, and is therefore a number times a unit.

You might think that is over pedantic, but it does betray some sloppy thinking regarding the nature of quantities on behalf of the examiners.

So what should it say? “… *mgh *= 1/2 *k* (23 m – *L*)^{2} …” would be the way to write this.

## 4 Responses to “Axis labels – accurate but not at all obvious”

In this particular case, isn’t there some ambiguity in using the letter “m” in two different ways within the same equation?

Yes. But note the italics for mass, and the non-italics for metres. I think I got it right. Actually, mistaking a unit for a quantity in a formula is something that I’ve seen students do as well.

I did notice that, but when it comes to written work (as opposed to typed) it’s much harder to distinguish between italics and non-italics. Or is there some standard way of doing this when writing that I’m just not aware of?

How about “distance / nm”? Still gives you 5.6 on the axis scale but avoids the awkward 10^9 inversion of nano.

Students not labelling axes get referred to xkcd.com/1578