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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
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