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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, […]
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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 […]
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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 […]
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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’ […]
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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 […]
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Structure, Structure, Structure!
Look at this equality: or this one: 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 […]
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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 […]
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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. […]
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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 […]