Monthly Archives: February 2015

Why are vector spaces so good

In the basics we saw that vector spaces (v.s.) are defined over a field. Field is the keyword. Recall that examples of field include real numbers ‘r’, complex numbers ‘c’, rational numbers ‘q’. Let’s take the last example of q. q consists of numbers of the form a/b where a and b are integers like -2, 3, 5, 10 etc.. Examples of numbers that are not rational include square root of 2, pi, e etc.. v.s. of dimension n are sets of n – tuple of numbers where each number in the tuple comes from the underlying field. The question is, why do we need a field, that is, why can’t we just define it over the set of integers ‘z’. Recall also that z shares all properties with q except that not all elements have a multiplicative inverse i.e., there is no number for 21 in z so that the product of both of these numbers equal 1 but if we ask for an element in q, then we have 1/21 * 21 = 1. But that’s ok because nothing is stopping me from defining n – tuples from z. For example let’s look at z^2 which consists of numbers of the form (a,b) where a and b are integers. Well, the world didn’t come to an end anyway, in fact, we can see that if we take two 2- tuples of this form and add them coordinate-wise, we get a number of the same form with different integers. Secondly, if we multiply (coordinate-wise) a 2-tuple with an integer we still end with a different 2-tuple where each coordinate is again an integer, so what’s the problem?

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Digression about…

If you noticed, a significant of aspect this blog is digression all over so I thought it is only natural for me to write about it. It turns out that one of the main reasons why math is ‘hard’ for those who feel that it is hard is that they learn math the hard way which is why it is hard. That doesn’t obviously sound like a thing, but the truth is that it is as good as any other reason. Let’s take an example.

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Continuous time zones, yes please

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.

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Some basics

OK, this post is supposed to be a gentle introduction to some of the words or terminologies used in math. What is a ‘set’? A set is a collection of distinct objects, eg: {apple, orange}, {football, basketball} etc.. As we can see, a set is written with ‘{‘, ‘}’ brackets, the examples are read as ‘a set of (or consisting of) apple and orange’ and so on. Individual entities of a set are called as ‘elements’ simply because they make up the set that we are interested in. As we can see, we can’t do much just with a set, well it’s not true cause lots of interesting objects are almost always defined using a set, but anyway what we really care about is ‘relations’ and ‘mappings’.

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