Plots for positive integer powers of x in 3D

A Real-Imaginative Guide to Complex Numbers

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

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Monty Hall at his show Let's Make a Deal

The Reason Why the Monty Hall Problem Continues to Perplex Everyone

The Monty Hall problem is an interesting puzzle loosely based on an American TV game show Let’s Make a Deal hosted by Monty Hall. While the puzzle looked simple, it perplexed some of the brightest mathematical minds in the United States, including the great Paul Erdös who was one of the most prolific mathematicians of the 20th century. This continues to be the case today. I looked upon a number of references to find the source of confusion in the Monty Hall problem but failed. All I found was different solutions. Therefore, I built one myself with the usual from

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Solution for 3 coin tosses

The Coin Toss Puzzle and the Simplest Possible Solution

Recently, I wrote an article on why the Monty Hall problem has perplexed so many brilliant minds where I showed that it was a corner case between 1 open and 1 closed door, while the intuitive but wrong answer is close to the probability curve of 1 open door. Now a coin toss puzzle has appeared on Twitter that has gone viral as it goes against our common intuition of probability and random sequences (such as a series of coin tosses). The puzzle goes as follows. The Problem Flip a fair coin 100 times—it gives a sequence of heads (H)

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Maximum velocity in an FMCW radar

FMCW Radar Part 3 – Design Guidelines

The Bloom’s Taxonomy describes the levels of mastery one attains in a field. Its last two stages are Synthesis and Evaluation. This is where the masters can be differentiated from the experts. In a job interview, for example, a good technique to judge a candidate’s ability is to ask them where the system in question breaks. A little learning is a dangerous thing Drink deep, or taste not the Pierian spring There shallow draughts intoxicate the brain And drinking largely sobers us again While the first two parts of the FMCW radar series addressed the lower levels, Part 3 is

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Multiple objects at different speeds for an FMCW radar

FMCW Radar Part 2 – Velocity, Angle and Radar Data Cube

In Part 1 of FMCW radar series, we described how a radar estimates the range of one or more stationary targets. In Part 2, we talk about estimating the velocities of several moving targets and their directions through forming a structure known as the radar cube. Part 3 presents system design guidelines for an FMCW radar. In a wonderful 1991 paper "Wireless Digital Communication: A View Based on Three Lessons Learned", Andrew Viterbi summarizes the Shannon theory for digital communications in the form of 3 lessons, the first of which was the following. "Never discard information prematurely that may be

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Bat echolocation principle

FMCW Radar Part 1 – Ranging

This is Part 1 of a 3-Part series in which we describe how an FMCW radar finds the range of multiple stationary targets. In Part 2, we talk about estimating the velocities of several moving targets and their directions through forming a structure known as the radar cube. Part 3 presents system design guidelines for an FMCW radar. In his book Multirate Signal Processing, Fred Harris mentions a great problem solving technique: "When faced with an unsolvable problem, change it into one you can solve, and solve that one instead." We will see in this article how an FMCW radar

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