You almost certainly came across the Number Line at school (even if you didn’t pursue mathematics or the history of underground railways to an advanced level). Let me refresh your memory and take you on a few scenic detours.
The Number Line began with the construction of a station at Zero (which at the time was absolutely the coolest place to be in the city). For reasons lost in the mists of time, some people settled in a place east of Zero, called One. After doing whatever it was they did in Zero, they needed to get home quickly for the night, and it wasn’t long before a line segment was established between Zero and One. Actually, it was little more than a dirty underpass, with carts pulled by rabbits, but it did the job.
Over the course of time, as the city grew larger, successive stations were built east of the earlier ones: Two (east of One), Three (east of Two), Four, Five, and so on. (There was a very positive mood in the air during this phase of expansion.) The initial line segment was extended, and extended again (and again), and a timetabled eastbound railway service was eventually established. Phew. Rabbits were still used to propel the carriages, but let’s abstract away from the obvious animal-welfare issues for now:
In the interests of fairness (or more likely so that somebody somewhere could make a profit), any journey between two successive stations was considered to be of equal value to any other. (Somehow, the people in the city hadn’t yet encountered money.) Furthermore, the values of journeys starting from Zero were named after their destinations: a journey to One was said to have value one, a journey to Two was said to have value two, and so on.
Now, some people started making journeys from Two to Five, which really sent the valuers into a spin for a bit. But, given that all the trains stopped at every station – to give the rabbits a rest, even if no one was getting on or off – the valuers soon realised that this was just like going from Two to Four and then from Four to Five. Following a series of tedious steps that I won’t bore you with, they concluded (on the assumption that values accumulate in the way physical things do) that a journey from Two to Five had the same value as a journey from Zero to Three, i.e. three.
As a bonus, the valuers realised that going from Zero to Two and then from Two to Five could be seen as having the value of the latter journey (three) over and above the value of the former (two). But this was just the same as a journey from Zero to Five (with obligatory rabbit stops), which had value five. In other words, three over and above two is just the same as five.
In short, three plus two equals five. (There are many more results of the same kind, but there’s not enough space here – or indeed anywhere else – to list them all.)
Now, the commuters in this city – most people using the Number Line were commuters – found that they got in a bit of a muddle from time to time and jotted down the value of the first part of their journey before the value of the second part of their journey. So instead of working out the total value as three plus two, say, they worked it out as two plus three. But luckily the valuers never noticed. That’s because it didn’t make any difference: two plus three also equals five! Since the commuters got away with doing their calculations backwards, this is known as the commutative law. (Not really.)
Westbound services introduced
The Number Line was a huge success, but for a long time there were only eastbound trains. This was partly because there was only one physical track, and partly because it made things easier for the valuers. (I’m not quite sure why they had so much power. Perhaps they really were making a profit from all of this.) There were problems with the lack of a westbound service, though. Obviously, commuters who needed to go west had to walk, which limited how far they could travel. More seriously, new carriages had to be built on a daily basis as the old carriages travelled ever eastward. And the rabbit population was constantly migrating eastward too (though fortunately new rabbits always seemed to be in plentiful supply for some reason).
So, starting from Zero and working eastward, a new westbound track was gradually constructed, eventually extending as far as the eastbound track (which is as far as anyone has ever cared to go). Once again, in the interests of fairness (or perhaps for more nefarious reasons – we’ll never know), the valuers decided that they were only interested in the start and end points of journeys. Even if you went east from One to Five and then west from Five to Three, the value of your journey would be two, as if you had never gone further east than Three. (Once again, I’m afraid the rabbits’ wellbeing wasn’t taken into consideration.)
But the purpose of the westbound track wasn’t to allow such contorted journeys to be made. No, it was so that commuters could travel west by train (and to reduce wastage of raw materials in constructing carriages). The value of a westbound journey was decreed by the valuers to be the same as that of the corresponding eastbound journey. So a journey from Five to Two had the same value as a journey from Two to Five, i.e. three.
This kept the commuters happy too, because their beloved commutative law still seemed to hold. For instance, a westbound journey from Five to Zero via Two had value three (the Five-to-Two part) plus two (the Two-to-Zero part), but it didn’t matter if you worked it out as two plus three.
Going east from Three to Five and then west from Five to Four (a journey of value one) was a little more complicated. This had a calculated value of two minus one (meaning a journey of value two in one direction followed by a journey of value one in the opposite direction). And since going east from Three to Four and then west from Four to Two had a value of one minus two, and the total journey from Three to Two also had a value of one, it seemed as though the commuters could indeed continue to be carefree about which number they wrote down first, whether they were combining journeys using plus or minus. (That’s not to say that the journeys from Three to Four via Five and Three to Two via Four were the same: of course they had different destinations, but the valuers didn’t care too much about that.)
‘All westbound services terminate at Zero’
I just said that two minus one equals one minus two. But I chose my starting station carefully. If a commuter started at One, went east to Three (a journey of nominal value two) and then west to Two (a journey of nominal value one), they’d have no problem: two minus one equals one (the value of any journey from One to Two). The same commuter could also go east to Two and then west to Zero: one minus two equals one. OK so far. Commutativity rules! (So said the graffiti in the tunnels between Three and Four at one point.)
But what if a commuter set out westbound from One on a journey of nominal value two and then returned eastbound on a journey of nominal value one? Well, it’s not entirely clear what this would mean.
Looking at the diagram of the Number Line, it would seem that the commuter could continue through Zero until they came to Three; continuing eastbound would take them to Four. That doesn’t seem right: the journey that began westbound was really an eastbound journey in disguise, and the commuter would end up having made a journey of value two plus one, i.e. three. This feels uncomfortable.
The valuers decided that they would forbid portions of journeys that passed through Zero. They did this by decreeing that all westbound services officially terminated at Zero (before restarting as eastbound services).
Even so, sometimes commuters fell asleep and accidentally went straight through Zero: their westbound journeys turned into eastbound journeys. It didn’t take long for them to come up with a little hack that restored their beloved commutative law. In this, they were helped by the polarity inspectorate. Anyone entering a station on the Number Line was issued with a pink token. Anyone leaving a station had to surrender a pink token. But anyone staying on a train passing through Zero had their pink token replaced by a blue one (or vice versa).
Although it wasn’t officially sanctioned by the valuers (indeed they viewed the practice somewhat negatively), commuters who found themselves in possession of a blue token would imagine themselves to be somewhere else. Arriving at Three, say, they would say to themselves that they were at ‘Blue Three’ (as opposed to ‘Pink Three’). The value of a journey beginning with a pink token and ending with a blue token (or vice versa) was just the value of a journey from Zero to the destination station over and above the value of a journey from the start station to Zero.
So a journey from Three to Two via Zero had a value not of one but of three plus two, i.e. five. Any overall journey from Three to Two would still have a value of one, though, because going via Zero meant that you either had a blue token at the start (in which case you couldn’t have entered at Three) or at the end (in which case you weren’t allowed to leave at Two).
Values for stations, stations for values
So the commutative law survived this phase of the Number Line’s expansion, but at the cost of some complexity (the creative interpretation of the polarity inspectorate’s tokens) and an arguably draconian restriction on people’s freedom of movement. If only those blue stations weren’t just convenient fictions dreamt up by the commuters.
Remember how the valuers named the values of journeys after the successive stations east of Zero? So for every station east of Zero, there was a unique corresponding value (the value of a journey from Zero to that station). And every (complete) journey’s value had a unique corresponding station too.
For any initial station besides Zero, any other station also had a unique corresponding value with respect to the initial station (the value of a journey between the stations). However, it was not true that every (complete) journey’s value had a unique corresponding station with respect to the initial station: for each value up to and including the value corresponding to the initial station itself, there were two stations reachable from the initial station by a journey with that value: one to the east and one to the west.
It was to the benefit of both commuters and valuers (but sadly of little consequence to the rabbits) to rectify this situation. It so happened that land was beginning to be reclaimed from the great ocean to the west of Zero, enabling a symmetrical expansion of the Number Line (bidirectional along its entire length):
Tokens were no longer issued, and staff of the obsolete polarity inspectorate were mostly reemployed as animal welfare officers. The blue line in the diagram was a nod to the old blue tokens, but most people were ignorant of the history and assumed it was blue because the western extension of the Number Line was under what used to be the sea.
Now there was a one-to-one correspondence between (direction-sensitive) values of journeys from any given station and stations on the Number Line. A journey to the next station to the east from any station had value one (or east one), a journey to the next station to the west had value west one, and so on.
Because of this one-to-one correspondence, the distinction between stations and journeys was often overlooked. It was extremely convenient for commuters to be able to work out that a journey from (East) Three going west seven was a journey to the station corresponding to (east) three west seven, which equals west four, so the destination was West Four.
The city’s infrastructure development didn’t stop there, of course. Multiple-journey tickets were introduced, each named after a different station on the Number Line. A north–south line was opened up from Zero, with carriages hauled by unicorns (though some considered these to be imaginary creatures and supposed it was really an electric line). There was eventually a whole grid of east–west and north–south lines, with a fully commutative ticketing system. It might sound complex, but it was really quite straightforward to use.
And then people discovered that there were some places east, west, north and south of Zero (or any other station) that simply could never be reached by rail: the transfinite stations like Omega. Probably just as well for the rabbits.
I realise this departure into mathematics has taken us into ‘aimless, unprincipled’ territory – but you were warned. I had originally planned to write a post using traditional mathematical notation when the idea popped into my head (it was going to be about the failure of commutativity for the additive inverse), but then I’d have had to pay for a ‘business’ WordPress account just to be able to install a MathJax plugin. Since I couldn’t justify that expense, I had to work around the restrictions, meaning that this might have been a slightly more interesting post than you’d have had otherwise. Maybe. Well, rabbits.