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Complex Numbers (Metric)

Learn, or revise, what complex numbers are, how to do calculations with them, how to use them to solve trig problems, how to define functions on them and how to represent shapes in the plane.

Arithmetic

Addition, subtraction and multiplication; the complex conjugate and its use in division; the Argand diagram. This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.

What is the square root of \(-1\)?

Why there's only one way to express a given complex number as a real number plus an imaginary one.

Simple arithmetic with complex numbers.

The complex conjugate, \(\bar{z}=x+i\,y\), and how we use it to do division.

Complex numbers as points in a plane.

Modulus and Argument

Complex numbers as vectors with magnitude and direction, and how to do calculations with them. This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.

Why the non-real roots of real polynomials always come in conjugate pairs, and how to use that fact.

Complex numbers as vectors with magnitude and direction.

The modulus of the product is the product of the moduli; the argument of the product is the sum of the arguments. Similar results for division.

The result \(e^{\i\,\theta}=\cos \theta+i\,\sin\theta\).

De Moivre's Theorem

Raising complex numbers to powers. Using complex numbers to solve trig problems. \(n\)th roots of complex numbers. This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.

Raising complex numbers to powers.

Using De Moivre's theorem about complex powers to derive trigonometrical identities.

Finding all \(n\) \(n\)th roots of a complex number.

Complex Functions

Functions of complex \(z\). This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.

For complex \(z\), calculating \(e^z\) and \(\ln z\).

Loci in the Argand Diagram

Shapes in the plane given by conditions on \(z\). This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.