## Why the Constant e Arises in Complex Plane as a Rotation

In the tutorial on how complex numbers arose, we asked three questions. The first two were answered in the same article while the answer to the third question, repeated below, is explained here. Why is the expression $e^{i \theta}$ a rotation of 1 by $\theta$ radians on a unit circle? Is it possible to make sense out of a number like $2.71828^{\sqrt{-1}\cdot\theta}$? The constant e is a special number discovered by Jacob Bernoulli while studying compound interests. It appears in many other forms as well which are all related to each other but that topic is a complete account in

## I/Q Signals 101: Neither Complex Nor Complicated

There was a recent discussion on GNU Radio mailing list in regards to the simplest possible intuition behind I/Q signals. Why is I/Q sampling required? Question: The original question from Kristoff went like this: “… when you mention `GNU Radio complex numbers’, you also have to mention I/Q signals, which is a topic that is very difficult to explain in 10 seconds to an audience who has never seen anything about I/Q sampling before.” Comment: According to Jeff Long: “This is a great thing to try to figure out. If we can come up with an answer that gives someone

June 18, 2020 On a cold morning in August 2015, I narrowly missed a train to my office in Melbourne city. With nothing else to do in the next 20 minutes, my mind wandered towards an intuitive view of complex numbers, something that has puzzled me since long. In particular, I wanted to seek answers to the following questions. (a) What is the role of the number $\sqrt{-1}$ in mathematics? What sets it apart from other impossible numbers, e.g., a number $k$ such that $|k|=-1$? (The origins of this question might lie in how I cut apple slices for my