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 derivations of the trigonometric identities for and *.This post offers something more exciting and less-known: the radical forms of and Consider Algebraically, simplify it and write it in the form of . 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 in some other way. We […]

### i Cycle

Where is ? If you have read $latex i$ Cycle, the Sum of Powers, you have already experienced the powers of and their hidden cycle. This is a very basic problem, and in fact, it is a prerequisite for understanding Cycle, the Sum of Powers and everything else about complex numbers (okay, it is a bit of an exaggeration to say so, but you get 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 […]

### i Cycle, the Sum of Powers

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 […]

### 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 […]