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, […]

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 […]

[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 […]