Manipulate real and complex numbers and solve equations

AS 91577

Worksheet 1

QuadraticsGeneral formula:

General solution:

Example 1

Equation cannot be factorised.

Using quadratic formula

We use the substitution

A complex number

The equation has 2 complex solutions

Real Imaginary

Equation has 2 complex solutions.

Example 2

Example 2

Example 2

Adding complex numbers

Subtracting complex numbers

Example

Example

(x + yi)(u + vi) = (xu – yv) + (xv + yu)i.

Multiplying Complex Numbers

Example

Example

Example 2

Conjugate

If

The conjugate of z is

If

The conjugate of z is

Dividing Complex Numbers

Example

Example

Example

Solving by matching terms

Match real and imaginary

Real

Imaginary

Solving polynomials

Quadratics: 2 solutions

2 real roots 2 complex roots

If coefficients are all real, imaginary roots are in conjugate pairs

If coefficients are all real, imaginary roots are in conjugate pairs

Cubic

Cubics: 3 solutions

3 real roots 1 real and 2 complex roots

QuarticQuartic: 4 solutions

4 real roots

2 real and 2 imaginary roots

4 imaginary roots

Solving a cubic

This cubic must have at least 1 real solutions

Form the quadratic.

Solve the quadratic for the other solutionsx = 1, -1 - i, 1 + i

Finding other solutions when you are given one solution.

Because coefficients are real, roots come in conjugate pairs so

Form the quadratic i.e.

Form the cubic:

Argand Diagram

Just mark the spot with a cross

Plot z = 3 + i

z

z =1

z = i

z = -1

z = -i

Multiplying a complex number by a real number.

(x + yi) u = xu + yu i.

Multiplying a complex number by i.

z i = (x + yi) i = –y + xi.

Reciprocal of z

Conjugate

Rectangular to polar form

Using Pythagoras

Modulus is the length

Argument is the angle

Check the quadrant of the complex number

Modulus is the length

Example 1

Polar form

Rectangular form

Example 2

Example 3

Converting from polar to rectangular

Multiplying numbers in polar form

Example 1

Multiplying numbers in polar form

Example 2

Take out multiples of

Remove all multiples of

De Moivre’s Theorem

Example 1

De Moivre’s Theorem

Example 2Take out

multiples of

Solving equations using De Moivre’s Theorem

1. Put into polar form

2. Add in multiples of

3. Fourth root4th root 81

Divide angle by 44. Generate solutions

Letting n = 0, 1, 2, 3

Take note:

Useful websites

Good general levelhttp://www.clarku.edu/~djoyce/complex/

Advanced levelhttp://mathworld.wolfram.com/ComplexNumber.html

Good general levelhttp://www.purplemath.com/modules/complex.htm

Good general level- Also gives proofshttp://www.sosmath.com/complex/complex.html

Problems at 3 levelshttp://www.ping.be/~ping1339/Pcomplex.htm#READ-THIS-FIRST