As any physics student knows (or should know), units are important things. By ‘unit’ I mean a measure of the kind of quantity you are dealing with. So if it’s mass, then a kilogram, a gram, an ounce, etc are all units; if it’s distance, then kilometres, light-years, feet are all units. Units are essential – it’s not very helpful to say that the distance from Hamilton to Auckland is 130. If I had a dollar for every time I’ve had to yell ‘UNITS’ at a student who has missed them out, I’d be … well, maybe not a millionaire, but at least able to affford one more cup of coffee a week during term time.
Units are very useful too. Last week I was trudging through some pretty intense algebra for some of my research work. The potential for mistakes is huge, and it’s difficult to be sure you get the right answer out at the end. (It’s research, which means, amongst other things, you can’t go and look up the right answer in a text book – it’s up to you to work through it, and to know that you have worked through it correctly.) Units help in this process because when you have an equation, the units (more formally, the dimensions) must balance. Here’s a rather trivial example.
The distance a ball falls when in free fall is given by the equation s = g t^2 / 2 (g times t squared, then divided by 2). Here, distance is denoted by ‘s’, time by ‘t’, the acceleration due to gravity by ‘g’. Let’s check the units. On the right-hand side, ‘g’ is an acceleration, so would carry the SI unit of metres per second squared. ‘t’ is a time, so would carry the unit of seconds (in SI). And 2 is just 2. No unit there. So on the right hand side we have
metres per second squared, times seconds squared
which is of course just metres. And that matches the unit of the left hand side, which is a distance. That example isn’t too taxing, but when you get nasty equations that need manipulating and solving it is a very good check that things are reasonable. (Of course it can’t be used to spot all mistakes – if I’d written s = g t^2 / 3, this method wouldn’t have picked up a problem.)