Category Archives: Personal Experiences

My mathematics related experiences

Feb17

Listening Maths

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Earlier I told about new mediums now available to enjoy maths. You can find a list of YouTube channels, to begin with here. And a list of dedicated math podcasts here.

In this post, I want to point out some episodes from the  podcast “In Our Time” which discuss mathematics:

Apart from them, another podcast worth listening to is “The Wizard of Mathematics, Srinivasa Ramanujan” by Prof. Srinivas Kotyada on FM Gold:

Feb10

A topic I wanted to discuss for long time

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If you are an average maths undergraduate student (like me), you might have ended up in a situation of choosing between “just completing the degree by somehow passing the courses without caring about the grades” and “repeating a course/taking fewer courses so as to pass all courses with nice grades only”. Following is a nice discussion from Reddit:

Is it okay to take less credits?
byu/OldPlan inlearnmath

Dec14

Childhood Maths – II

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I found two documents that I was very proud of as a child. Both were the result of trying to understand the kind of things Ramanujan did in free time, a result of the little AMTI books I read as a child. I will share the second document in this post and the other one was in the previous post.

Following document was created in MS Word on my old Windows XP desktop. The calculations were done using some Microsoft advanced calculator:

primes

I was not happy with the result though since the pattern didn’t continue which was supposed to continue according to Ramanujan.

Dec7

Childhood Maths – I

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I found two documents that I was very proud of as a child. Both were the result of trying to understand the kind of things Ramanujan did in free time, a result of the little AMTI books I read as a child. I will share one document in this post and another one in the next since both are not related to each other in substance.

Following is a calculation table generated using some spreadsheet software:

caylay

As you can see, it was motivated by the famous Taxicab number story.

Nov24

How to deal with failure?

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If you are an average student (like me…) and are part of an insane curriculum (like having 5 advanced math courses in one semester) you may fail in few of your courses at college, even after trying your best to pass them.  In such situations you may be able to control your panic by reading this Reddit post:

I think I am going to fail Calc II. What can I do better?
byu/Thaw6663 inmath

I will suggest you to go through the comments of that post.

Aug20

Four Examples

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Following are the four examples of sequences (along with their properties) which can be helpful to gain a better understanding of theorems about sequences (real analysis):

  • \langle n\rangle_{n=1}^{\infty} : unbounded, strictly increasing, diverging
  • \langle \frac{1}{n}\rangle_{n=1}^{\infty} : bounded, strictly decreasing, converging
  • \langle \frac{n}{1+n}\rangle_{n=1}^{\infty} : bounded, strictly increasing, converging
  • \langle (-1)^{n+1}\rangle_{n=1}^{\infty} : bounded, not converging (oscillating)

I was really amazed to found that x_n=\frac{n}{n+1} is a strictly increasing sequence, and in general, the function f(x)=\frac{x}{1+x} defined for all positive real numbers is an increasing function bounded by 1:

 

Thre graph of x/(1+x) for x>0. Plotted using SageMath 7.5.1

The graph of x/(1+x) for x>0, plotted using SageMath 7.5.1

 

Also, just a passing remark, since \log(x)< x+1 for all x>0, and as seen in prime number theorem we get an unbounded increasing function \frac{x}{\log(x)} for x>1

dort

The plot of x/log(x) for x>2. The dashed line is y=x for the comparison of growth rate. Plotted using SageMath 7.5.1

 

Aug13

Magic Cubes

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Last week I attended a talk (by a student) about Magic Squares. I learned a bunch of cool facts about them (like how to devise an algorithm to construct them). Towards the end of the talk, one student from the audience suggested the possibility of Magic Cubes. I got very excited about this idea since it pointed towards the stereotypical mathematical ideology of generalizing the examples in order to see the deeper connections.

I myself don’t know much about Magic Cubes (or even Magic Squares) but would like to quote W. W. Rouse Ball & H. S. M. Coxeter from pp. 217 the book “Mathematical Recreations and Essays” (11th Ed.) :

A Magic Cube of the n^{th} order consists of the consecutive numbers from 1 to n^3, arranged in the form of a cube, so that the sum of the numbers in every row, every column, every file, and in each of the four diagonals (or “diameters “), is the same-namely, \frac{1}{2}(n^3 + 1). This sum occurs in 3n^2 + 4 ways. I do not know of any rule for constructing magic cubes of singly-even order. But such cubes of any odd or doubly-even order can be constructed by a natural extension of the methods already used for squares.

I would like to read about these magic hyper-cubes in future. And if you know something interesting about them, let me know in the comments below.

May13

Intra-mathematical Dependencies

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Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

mat-dependency (1)

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.

Mar20

Teaching Mathematics

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One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with  concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!

Feb25

Division algorithm for reals

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You must have seen long-division method to compute decimal representation for fractions. Astonishingly, I never pondered about how one would divide an irrational number to get decimal representation. Firstly, this representation will be approximate. Secondly, we have been doing this in name of “rationalizing the denominator” stating the reason that division by irrationals is not allowed. But, in fact, this is the same problem as faced while analysing division algorithm for Gaussian integers.

Bottom line: Numbers are just symbols. We tend to assign meaning to them as we grow up. Since the set of real numbers, rational numbers and integers  form an Euclidean domain, we can write a division algorithm for them. For example, we don’t have special set of symbols for 3 divided by π, but 3 divided by 2 is denoted by 1.5 in decimals.