manipulate real and complex numbers and solve equations
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Manipulate real and complex numbers and solve equations. AS 91577. Worksheet 1. Quadratics. General 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. Imaginary. - PowerPoint PPT PresentationTRANSCRIPT
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Manipulate real and complex numbers and solve equations
AS 91577
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Worksheet 1
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QuadraticsGeneral 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
Real Imaginary
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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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QuarticQuartic: 4 solutions
4 real roots
2 real and 2 imaginary 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 solutionsx = 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 =1
z = i
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
Polar form
Rectangular 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 Moivre’s Theorem
Example 1
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De Moivre’s Theorem
Example 2Take out
multiples of
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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
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Take note:
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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