LEGACY Complex Numbers

Definition

A complex number is a number that is written as: where is the imaginary unit defined as and . The set of all complex numbers is written as

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A-Level Further Core 1

The Complex Conjugate

For , is always real as the imaginary parts cancel so is also always real as it’s a difference of two squares

Basic operations

Addition and Subtraction

For addition and subtraction you combine real and imaginary terms separately

Multiplication

For multiplication just expand brackets ex. since this is equivalent to

Division

For division multiply the numerator and denominator by the complex conjugate of the numerator.

Roots of Polynomials

The possible cases of roots of polynomials are as follows:

Quadratic:

Cubic:

Quartic:

Modulus Argument Form

For any complex number it can be written as: where and .

Modulus

The modulus of a complex number is written as and it is is defined as where . is geometrically represented as the distance from the origin to on the complex plane.

Argument

The argument written as is the angle such that - it is given in radians typically given in the range (This is called the principal argument) This angle represents the angle rotation from the -axis

Basic Operations

Multiplication

Division

You can also prove that:

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A-Level Further Core 2

Exponential Form of Complex Numbers

Derivation

By using the Taylor expansion of and plugging in , you can split the series to obtain

Then multiply by to get the general form of where and

Multiplication and Division

De Moivre’s Theorem

It can be used to find trigonometric identities by applying De Moivre’s Theorem and the binomial expansion of to express in terms of powers of and same for You can also use the following identities

to find other identities by binomially expanding the function in terms of and exponentiating the trig function. These identities themselves can be proved using De Moivre’s Theorem.

Sum of Geometric Series

Nth Root of Complex Numbers

, To Solve

• has the form so by De Moivre’s Theorem
• Compare the modulus to solve for
• Then plug in and solve for

Roots of Unity

The roots of unity are the solutions to , If is a positive integer there is an th root of unity s.t are the roots of unity The roots of unity sum to 0 and form the vertices of a regular -gon If is one root of the equation then the roots of are given by

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