Manipulate%20real%20and%20complex%20numbers%20and%20solve%20equations

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Manipulate real and complex numbers and solve
equations

• AS 91577

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Worksheet 1
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Quadratics
General formula
General solution
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Example 1
Equation cannot be factorised.
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Using quadratic formula
We use the substitution
A complex number
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The equation has 2 complex solutions
Imaginary
Real
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Equation has 2 complex solutions.
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Example 2
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Example 2
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Example 2
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Adding complex numbers
Subtracting complex numbers
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Example
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Example
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(x  yi)(u  vi) (xu  yv)  (xv  yu)i.
Multiplying Complex Numbers
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Example
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Example
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Example 2
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Conjugate
If
The conjugate of z is
If
The conjugate of z is
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Dividing Complex Numbers
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Example
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Example
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Example
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Solving by matching terms
Match real and imaginary Real Imaginary
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Solving polynomials
Quadratics 2 solutions
2 real roots
2 complex roots
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If coefficients are all real, imaginary roots are
in conjugate pairs
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If coefficients are all real, imaginary roots are
in conjugate pairs
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Cubic
Cubics 3 solutions
3 real roots
1 real and 2 complex roots
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Quartic
Quartic 4 solutions
2 real and 2 imaginary roots
4 real roots
4 imaginary roots
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Solving a cubic
This cubic must have at least 1 real solutions
Form the quadratic.
Solve the quadratic for the other solutions
x 1, -1 - i, 1 i
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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
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Argand Diagram
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Just mark the spot with a cross
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Plot z 3 i
z
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z i
z 1
z -1
z -i
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Multiplying a complex number by a real
number.(x  yi) u xu  yu i.
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Multiplying a complex number by i.z i
(x  yi) i y  xi.
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Reciprocal of z
Conjugate
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Rectangular to polar form
Using Pythagoras Modulus is the length
Argument is the angle
Check the quadrant of the complex number
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Modulus is the length
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Example 1
Rectangular form
Polar form
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Example 2
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Example 3
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Converting from polar to rectangular
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Multiplying numbers in polar form
Example 1
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Multiplying numbers in polar form
Example 2
Take out multiples of
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Remove all multiples of
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De Moivres Theorem
Example 1
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De Moivres Theorem
Example 2
Take out multiples of
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Solving equations using De Moivres Theorem
1. Put into polar form
2. Add in multiples of
4th root 81
3. Fourth root
4. Generate solutions Letting n 0, 1, 2, 3
Divide angle by 4
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Take note
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Useful websites
Good general level http//www.clarku.edu/djoyce/c
omplex/ Advanced level http//mathworld.wolfram.c
om/ComplexNumber.html Good general
level http//www.purplemath.com/modules/complex.ht
m Good general level- Also gives
proofs http//www.sosmath.com/complex/complex.html
Problems at 3 levels http//www.ping.be/ping133
9/Pcomplex.htmREAD-THIS-FIRST