Complex Numbers
1. Irrational numbers
Irrational numbers - surds |
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Construct root 2Using a compass and straight edge
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Construct root 3Using a compass and straight edge
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2. Complex numbers introduction
Equation with imaginary roots |
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Quadratic equation with imaginary rootsUsing the quadratic formula
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Addition and multiplication of complex numbers |
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3. Division and equity of complex numbers
Dividing by a complex number - conjugateExpressing a fraction with a real denominator
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Complex number equations"Real = real, imaginary = imaginary"
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4. Argand diagram - modulus
Plot on an Argand diagram
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5. Transformations of complex numbers
Understanding multiplication |
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Multiplication involves scaling and rotating
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Transformations involving multiplication |
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6. Conjugate roots theorem
Quadratic equation - conjugate roots |
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Showing that a conjugate is a root |
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Solving a cubic equation with two imaginary roots |
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7. Polar form of a complex number
Converting from polar to rectangular form |
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Express in polar formFinding the modulus and argument
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8. Products and quotients in polar form
Multiplying and dividing in polar formMultiplication: Multiply the module and add the arguments
Division: Divide the module and subtract the arguments
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9. De Moivre's Theorem
Raising a number to a power - expanding |
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Express in polar form and expand |
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10. Applications of de Moivre's Theorem
Raising a number to a power |
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Proving trigonometric identities using de Moivre's Theorem |
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Finding roots |
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12 Revision Questions |
Complex Number Video
Powers of i
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Introduction to Complex Numbers: Addition, subtraction, multiplication, powers of i.
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Complex division in rectangular form
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Solving Complex Equations:
Real parts = Real parts Imaginary parts = Imaginary parts |
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Graphing complex numbers on an Argand diagram and finding the modulus of a complex number.
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Transformations in the Complex Plane
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Complex conjugate roots
Solving quadratic and cubic equations with imaginary roots. |
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Writing a complex number in Polar Form
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Multiplying and dividing in Polar Form and using deMoivre's Theorem to expand a complex number to a power.
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Using deMoivre's Theorem to find roots of a Complex Equation.
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Using deMoivre's Theorem to prove Trigonometric identities. This question also requires use of the Binomial Theorem.
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