# Browsing category complex numbers

## Complex transformations (and incorrect wording?)

You know how I often bang on about how 'impossible' exams are really nothing of the sort? Well, just for a change, I'm going to bang on about how sometimes exam boards get it wrong. I'm looking at the 2014 Edexcel FP2 paper (the normal one, not the (R) one

## Ask Uncle Colin: Powers and polar form

Dear Uncle Colin, I've been given $u = (2\sqrt{3} - 2\i)^6$ and been told to express it in polar form. I've got as far as $u=54 -2\i^6$, but don't know where to take it from there! - Not A Problem I'm Expecting to Resolve Hello, NAPIER, and thanks for your

## Ask Uncle Colin: A Complex Conundrum

Dear Uncle Colin, I'm told that $z=i$ is a solution to the complex quadratic $z^2 + wz + (1+i)=0$, and need to find $w$. I've tried the quadratic formula and completing the square, but neither of those seem to work! How do I solve it? - Don't Even Start Contemplating

## A curious identity

There's something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It's one of those that just begs, "prove me!" So, here goes! I'd start with the

## Ask Uncle Colin: A Complex Battle

Dear Uncle Colin, I'm supposed to solve $(1+i)^N = 16$ for $N$, and I don't know where to start! -- Don't Even Mention Other Imaginary Variations -- Reality's Enough Hello, DEMOIVRE, there are a couple of ways to attack this. The simplest way (I think) is to convert the problem

## Ask Uncle Colin: A hellish trigonometric identity

Dear Uncle Colin, @CmonMattTHINK unearthed the challenge to prove that: $\tan\left( \frac 3{11}\pi \right) + 4 \sin\left( \frac 2{11}\pi \right) = \sqrt {11}$. Wolfram Alpha says it's true, but I can barely get started on the proof and I'm worried no-one will like me. Grr, Really Obnoxious Trigonometry Has Evidently

## Ask Uncle Colin: An imaginary curve?

Dear Uncle Colin, I was playing with parametric equations and stumbled on something Wolfram Alpha wouldn't plot: $x=t^i;\, y = t^{-i}$. Does this curve really exist? Or am I imagining it? -- A Real Graph? A Non-existant Drawing? Hi, ARGAND -- what you're trying to plot certainly exists; whether or

## A numerical curiosity

A numerical curiosity today, all to do with $\i$th powers. Euler noticed, some centuries ago, that $13({2^\i + 2^{-\i}})$ is almost exactly $20$. As you would, of course. But why? And more to the point, how do you work out an $\i$th power? It's all to do with the exponential

## Complex mappings

Just for a change, an FP3 topic. I've been struggling to tutor complex mappings properly (mainly because I've been too lazy to look them up), but have finally seen - I think - how to solve them with minimal headache. A typical question gives you a mapping from the (complex)