The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight in the world of maths.

## Complex numbers

This teacher package brings together all *Plus* articles on complex numbers. In addition to the *Plus* articles, the *try it yourself* section provides links to related problems on our sister site NRICH.

- Introducing and using complex numbers
- Complex numbers, chaos and fractals
- Beyond complex numbers
- Try it your self with NRICH

### Introducing and using complex numbers

Curious quaternions — This article gives a comprehensive introduction to complex numbers, complete with historical perspective. It then goes on to explore quaternions, a generalisation of complex numbers.

A tale of two curricula: Euler's algebra text book — Leonhard Euler was the most prolific mathematician of all time and he wrote an algebra text book whose clarity is hard to beat. This article explores, amongst other things, how Euler introduced complex numbers in this book. In contrast to most modern teachers, Euler introduced them early on, before even discussing quadratic equations, and this nonchalant treatment takes away much of their dread.

Read Euler, read Euler, he is the master of us all. — Robin Wilson takes a look at Euler's life and work.

101 uses of a quadratic equation: Part II — A quick introduction to complex numbers, a bit of history, and interesting applications including the mobile phone.

One L of a discovery — The first third degree transcendental L-function was revealed recently at the American Institute of Mathematics.

The origins of proof III: Proof and puzzles through the ages — For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. Jon Walthoe explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled these problems.

Maths goes to the movies — This article explores the maths used in computer generated movies, and describes how complex numbers and their higher-dimensional cousins, the quaternions, help rotate things.

Pandora's 3D box — Will we ever be able to make computers that think and feel? If not, why not? And what has all this got to do with tiles?

Pandora's 3D box — A new 3D version of the Mandelbrot set has been created, revealing fractal worlds of amazing complexity.

### Complex numbers, chaos and fractals

Chaos and fractals can arise from the simplest of mathematical objects: the humble quadratic equation and the mild-mannered Möbius transformation. To get the full benefit of their beautifully intricate behaviour, you have to look at them as complex functions. These articles tell you how.

Unveiling the Mandelbrot set — A hands-on introduction to the Mandelbrot set and its cousins, the Julia sets, with great images and links to interactive applets for zooming in and out. A short introduction to complex numbers is included.

Computing the Mandelbrot set — A recipe for creating your own computer images of the Mandelbrot set.

A fat chance of chaos — A news item reporting on a break-through in the field of complex dynamics: a proof that Julia sets can be fat.

Pandora's 3D box — A new 3D version of the Mandelbrot set.

Maths in Space — Solving two unrelated problems at once.

Non-Euclidean geometry and Indra's pearls — An intuitive and largely equation-free introduction to hyperbolic geometry and the fractals created by tilings of hyperbolic space. Great movies and images. The article briefly discusses the role of complex numbers and Möbius transformations — a good starting point for a student project.

### Beyond complex numbers

Ubiquitous octonions — Complex numbers were the first step, then there were the above-mentioned quaternions, and next up were the octonions. This article gives an introduction, wonders if there's anything beyond octonions, and looks at possible applications, for example in string theory.

### Try it yourself with NRICH

What are complex numbers? and An introduction to complex numbers — These articles from our sister site NRICH give basic introductions to complex numbers, explore their geometric interpretation and make the link to vectors and trigonometric functions.

Two and four dimensional numbers — This problem from our sister site NRICH investigates matrix representations of complex numbers and quaternions. Requires addition and multiplication of two by two matrices.

Twizzles venture forth, Twizzle twists and Twizzles wind up — These problems from our sister site NRICH use interactive animations to explore the geometric interpretation of complex numbers. Basic level.

Root tracker, Conjugate tracker and Cubic tracker — More animations from NRICH exploring roots of quadratic and cubic polynomials.

Foot prints — This NRICH problem looks at the isometries of the plane and how they can be represented by complex numbers.

Napoleon's theorem, Thebault's theorem and Thousand words — Some geometric NRICH problems involving complex numbers.

Complex sine, Target six, Cube roots and Roots and coefficients — These NRICH problems require you to find complex roots of equations.

## Comments

## Roots of Complex Numbers

To find the root of the complex number a+ib, you find the arcotangent below 90 degrees of the complex ratio a/b, and then divide it by the root required. The cotangent of the result gives one of the say 3 roots of the complex ratio. For the other two roots you add on to the arcotangent 360 degrees and then 720 degrees. This procedure applies for all integer roots of complex numbers, so that there are always n nth roots of a complex number.

## Roots of Imaginary Numbers and Unity.

Further to my comments of 20 May 2011, the n nth roots of +1, -1, i and -i can be obtained by converting a+ib into an imaginary number by making a equal 0 and substituting Cotes's format cos90+isin90 which equals 0+i. It will be observed that cos180+isin180 equals -1, and cos360+isin360 equals +1 which is i to the power of 4. This system also works for division, cos45+isin45 equals the square root of i.