# A-level Mathematics/AQA/MFP2

## Contents

- 1 Roots of polynomials
- 2 Complex numbers
- 2.1 Square root of minus one
- 2.2 Square root of any negative real number
- 2.3 General form of a complex number
- 2.4 Modulus of a complex number
- 2.5 Argument of a complex number
- 2.6 Polar form of a complex number
- 2.7 Addition, subtraction and multiplication of complex numbers of the form x + iy
- 2.8 Complex conjugates
- 2.9 Division of complex numbers of the form x + iy
- 2.10 Products and quotients of complex numbers in their polar form
- 2.11 Equating real and imaginary parts
- 2.12 Coordinate geometry on Argand diagrams
- 2.13 Loci on Argand diagrams

- 3 De Moivre's theorem and its applications
- 4 Hyperbolic functions
- 5 Arc length and area of surface of revolution
- 6 Further reading

## Roots of polynomials[edit]

The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficents of the polynomials are real.

## Complex numbers[edit]

### Square root of minus one[edit]

### Square root of any negative real number[edit]

### General form of a complex number[edit]

where and are real numbers

### Modulus of a complex number[edit]

### Argument of a complex number[edit]

The argument of is the angle between the positive x-axis and a line drawn between the origin and the point in the complex plane (see [1])

### Polar form of a complex number[edit]

### Addition, subtraction and multiplication of complex numbers of the form x + iy[edit]

In general, if and ,

### Complex conjugates[edit]

### Division of complex numbers of the form x + iy[edit]

### Products and quotients of complex numbers in their polar form[edit]

If and then , with the proviso that may have to be added to, or subtracted from, if is outside the permitted range for .

If and then , with the same proviso regarding the size of the angle .

### Equating real and imaginary parts[edit]

### Coordinate geometry on Argand diagrams[edit]

If the complex number is represented by the point , and the complex number is represented by the point in an Argand diagram, then , and is the angle between and the positive direction of the *x*-axis.

### Loci on Argand diagrams[edit]

represents a circle with centre and radius

represents a circle with centre and radius

represents a straight line — the perpendicular bisector of the line joining the points and

represents the *half* line through inclined at an angle to the positive direction of

represents the *half* line through *the point* inclined at an angle to the positive direction of

## De Moivre's theorem and its applications[edit]

### De Moivre's theorem[edit]

### De Moivre's theorem for integral *n*[edit]

### Exponential form of a complex number[edit]

### The cube roots of unity[edit]

The cube roots of unity are , and , where

and the non-real roots are

### The n^{th} roots of unity[edit]

The equation has roots

### The roots of z^{n} = α where α is a non-real number[edit]

The equation , where , has roots

## Hyperbolic functions[edit]

### Definitions of hyperbolic functions[edit]

### Hyperbolic identities[edit]

### Addition formulae[edit]

### Double angle formulae[edit]

### Osborne's rule[edit]

Osborne's rule states that:

- to change a trigonometric function into its corresponding hyperbolic function, where a product of two sines appears, change the sign of the corresponding hyperbolic form

Note that Osborne's rule is an *aide mémoire*, not a proof.

### Differentiation of hyperbolic functions[edit]

### Integration of hyperbolic functions[edit]

### Inverse hyperbolic functions[edit]

#### Logarithmic form of inverse hyperbolic functions[edit]

#### Derivatives of inverse hyperbolic functions[edit]

#### Integrals which integrate to inverse hyperbolic functions[edit]

## Arc length and area of surface of revolution[edit]

### Calculation of the arc length of a curve and the area of a surface using Cartesian or parametric coordinates[edit]

## Further reading[edit]

The AQA's free textbook [2]