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

Read More

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

Read More

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

Read More

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

Read More

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

Read More

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

Read More

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

Read More

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

Read More

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)

Read More

Sign up for the Sum Comfort newsletter and get a free e-book of mathematical quotations.

No spam ever, obviously.

Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

On twitter