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When Imaginary gets Real

One of the many interesting aspects of imaginary numbers is that we can use them to find out “real” facts (facts about real numbers). Perhaps the most used examples are the derivation of the trigonometric identities for and *. This post offers something more exciting and less-known: the radical forms of and  Consider (frac{frac{1}{2}+frac{sqrt{3}}{2}i}{frac{sqrt{2}}{2}+frac{sqrt{2}}{2}i}) Algebraically, simplify it and write it in the form of (a+bi). [frac{sqrt6+sqrt2}{4}+frac{sqrt6-sqrt2}{4} i] We know the answer. But as a wise man once said, mathematics is all about seeing the same thing from different angles (or something like this).  So let us, calculate the same thing…
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i Cycle

Where is (i^{127})?  If you have read i Cycle, the Sum of Powers, you have already experienced the powers of i, and their hidden cycle. This is a much basic problem and in fact, it is a prerequisite for understanding i Cycle, the Sum of Powers and everything else about complex numbers (okay, it is a bit exaggeration to say so, but you got the point, it is very important). What is the problem here?  Notice that I could ask the “same” problem just using a different power. For example, I could ask: Where is (i^{39})?  You see, the problem…
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i Cycle, the Sum of Powers

[1+i+i^2+i^3+…+i^{57}] Do I need to say what the question is? “Find the sum!” is the immediate question that comes to mind when you start mathematics. When you gain more experience in mathematics, you learn to ask a deeper question: What is the pattern? You know, 57 is irrelevant. The only important thing here is the general question of the finding the sum without actually doing all the additions. However, to get to the point of finding the sum without actually doing all the additions, we should experiment with actual additions first! For example, let us find the sum of the…
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Artin’s Algebra!

This post has nothing to do with Michael Artin Algebra! Artin of this post is my son, and this is the story of him learning algebra. He is 11 years old now, and I guess one of few students on the planet that still has a positive mentality towards mathematics! We are reading together Gelfand’s’ Algebra, page by page, solving its problems one by one. The section “Letters in algebra” includes the following “Magic trick”. Choose the number you wish. Add 3 to it. Multiply the result by 2. Subtract the chosen  number. Subtract 4. Subtract the chosen number once…
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Read Euler, Read Euler!

“Read Euler, read Euler, he is the master of us all” written by Robin Wilson or “Euler: the master of us all” written by William Dunham are to show us how great and multifaceted Euler was as a mathematician. Indeed, he was. In this post, I want to write how great he was as an educator. Today was my first session of a course in number theory. I had to start with “sums of two squares”.  Like most other things in introductory number theory, this one also starts with a very simple observation. Some numbers like 5 ( ( 2^2+1^2…
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Structure, Structure, Structure!

Look at this equality: ( (a + b) + c = a + (b + c) ) , or this one: ( a . (b + c) = a.b + a.c ) .  They are true structurally. In principle, You can just replace one side of the equality with the other side without any extra comment. However, we usually like to describe these in terms of processes. For the first equality, we might say that adding the first two and then the sum to the third is the same as adding the first to the sum of the last two….
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Plan for the Unexpected!

Whatever you teach (or learn), you should look for the links, the more unexpected, the better.  This post is the story of one of them, simple but lovely and strong. I have a class with a group of adult students who never had a positive experience with mathematics (or as they would say, “they hate mathematics”), have very limited knowledge of mathematics, and yet, likely to work in nurseries and primary schools where they have to teach mathematics somehow or other. My aim is to help them not to hate mathematics (if not like it) and have some positive experience of…
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Plan for the Mess!

“Plan for the mess” is one of my favourite teaching ideas. The idea is to bring students to a point where after a heavy messy work (most of the time calculations and symbol pushing) they say “why I didn’t see that earlier!” Here is how recently I used the idea in my linear algebra class. The students had just solved a couple of two equations with two unknowns by whatever tools they had from highschool. Many of them had no idea about any geometric interpretation of such equations (i.e., solution, if there is one, is at the meeting point of…
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Proof-Generated Definitions!

Okay, for a long time, I didn’t know what to blog about. Now, I have decided to write about my teaching ideas that take ages to turn into a piece of research. That is why I have started with this strange title for my first blog. The title comes from the exact same phrase coined by Lakatos in his book Proofs and Refutations.  Here is how I used the idea recently. Last week, I was going to teach the definition of a convergent sequence. Previously, we had played a lot with sequences geometrically: how they are represented on a number line…
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