Here we will see how the phenomenon of time zones be made very ‘easy’, that is, for non-frequent travelers it becomes a pain every time they have to figure out how long the travel duration is even though most airlines provide the information in the ticket. I personally have tried computing the duration of actual travel in my mind just using the local time of departure and local time of arrival. Why would someone do this? well, why not? it’s just a good mental exercise, the calculations are often interesting when there are more than 1 or 2 stops or worse if we miss the connecting flights. In the latter case we have no other option but to compute ourselves or well ask someone to compute it for us. So, what makes the timezones computations kinda hard or interesting or complex? From the mathematical point, it is because of the fact that the time zones are not continuous. Let me try to motivate what I mean. One reason why timezones can be confusing is because that the places where times change are often arbitrary in the sense that they are chosen by a specific set of people, so not most people know the reason why India just has 1 timezone or why US has 6 timezones. Well the question here is not why we need multiple timezones but why those specific numbers. One might try to make a crude relation that the number of timezones for a country increase with the size of the country in terms of the area of the land. Well this might be a very bad assumption because the area even if we assume countries to be rectangle is length times breadth (product of difference between extreme points of latitudes and longitudes ) but only breadth (longitudes) contributes to the timezones. A smarter option would be to just use longitudes, well that might be bad too, because of daylight savings (see the timezones of US when the daylight savings are on, it’s ridiculous!). Anyway, a mathematician would easily see that the way timezones work is completely arbitrary with the current setup. How do we fix this? Here we go, I’ll explain the process since I figured that, that is the best way to explain it a clean way.
We first choose a point in the east, preferably the places in Japan where the sun apparently rises first, let’s call the point 1. We need only one coordinate since the latitudes are not useful (perhaps there’s a fancier way to make timezones using latitudes but we will not go there). Similarly we choose the point in the west where the sun sets the last, let’s call the point 0. This means that each point (or place on the earth) corresponds to a particular number between 0 and 1. Of course lots of points share the same number. Now let’s say that we have a function or a mapping defined as f that takes any number in the interval 0, 1 and outputs a number or time in our case. What we will now do is impose some conditions on f so that it becomes mathematically nice. One nice thing to have is a continuous f and perhaps smooth too. Note that we don’t have continuity in the current world since there are uneven jumps at random points on the longitudes. So the next thing we might want is f be a nice polynomial. This is actually not required but let’s say we need this just because it helps explanation easier. That’s it!
What have we done? We replaced the arbitrary timezones with nice mathematical surrogates. We will now see the immediate consequences of this. Let me first warn you that this is not looking good. The above constructions say that whenever we move in along longitudes we need to use the polynomial to figure out the time, which means every time you travel east or west even if it’s just two blocks, time changes include your travel time plus the calculation given by the polynomial. Nightmare! Well, not so much because these numbers will be fairly insignificant. But when you actually travel 200 or 300 miles these will be significant which is nice. We will also have calculators (and apps) to do this for us so it’s all good. Another thing is that once everyone gets used to this setup, it will be easy for everyone. What does this achieve overall? First, one will be better at one of the fundamental ideas in applied math and some areas in computer science, engineering in a vague sense (not that one should be, but why say no to something free and good). Second, the world will be better off with continuous functions than otherwise (I’ll leave it for you to think about)!