"When we see it for the first time, it looks so abstract that it seems impossible something like this could have any real-world applications" - Edward Frenkel
As the under-graduation curtains close, confronting and comprehending the ideas of abstract algebra seem barbaric to my mind. Right from high school, one can view the negligence or rather the amount of incompetence people exhibit while teaching this area. As a consequence, leaving students, like myself, blindsided to the essence and worth of the field. This, and the following few post are meant to encapsulate my thoughts on this area of the subject and it's significance in human civilization.
Much of abstract algebra involves properties of integers and sets. In this post, let's specifically look at the concept of modular arithmetic. It is an application of the well known division algorithm, it is basically an abstraction of a method of counting. For instance, if it is now September, what month will it be 25 months from now? of course, the answer is October, but the interesting fact is that you didn’t arrive at the answer by starting with September and counting off 25 months. Instead, without even thinking about it, you simply observers that 25 = 2.12 + 1, and you added one month to September. Similarly, if it is now Wednesday, you know that in 23 days it will be Friday. This time, you arrived at your answer by noting that 23 = 7.3 +2, so you added two days to Wednesday instead of counting off 23 days. If your electricit’s is off for 26 hours, you must advance your clock 2 hours, Since 26=2.12 + 2. Surprisingl, this simple idea has numerous important applications in mathematics and computer science.
In general, when a = qn + r, where a is the quotient and r is the remainder upon dividing a by n, we get, a mod n = r. As A consequence, 6 mod 2 = 0, since 6= 3.2 + 0.
If a and b are integers and n is a positive integer, then, a mod n = b mod n, if n divided a-b. This very modular arithmetic is often used in assigning an extra digit to identification numbers for the purpose of detecting forgery or errors. In the next post, we‘ll discuss a few examples and delve deeper into abstract algebra in general.
Image credits : Institute for Advanced study