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Complex numbers, grade 12 (5 units)

The equation x2=−1\displaystyle {x^2}={-1} has no solution in the real numbers.Subjects

In this course you add a new number, i\displaystyle i, whose square is −1\displaystyle -1, and you get the complex numbers: in them every quadratic equation has a solution, and every number has roots of every order. The course builds on what you learnt in grades 10 and 11 (5 units): trigonometric functions in radians, the sum and difference formulas of angles, quadratic equations, the geometric sequence and proof by induction. The topics, in order:

  • the algebraic form: the number i\displaystyle i, z=a+bi\displaystyle {z}={a+bi}, equality, the four operations, the conjugate;
  • the Gauss plane: a number as a point and as a vector, adding and subtracting, the modulus and distance, equations with zˉ\displaystyle \bar{z} and ∣z∣\displaystyle |z|;
  • equations: the square root of a complex number, quadratic equations, equations of higher degree with real coefficients;
  • the polar form: z=r(cos⁡θ+isin⁡θ)\displaystyle {z}={r(\cos\theta+i\sin\theta)}, multiplying and dividing – rotating and stretching, De Moivre’s theorem;
  • roots: the roots of unity, roots of order n\displaystyle n and the vertices of a regular polygon;
  • geometry: loci in the Gauss plane, translating, rotating and stretching, geometric problems.

Angles in this course are in radians, as on the exam’s formula sheet.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few exercises to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The number i and the complex numbersLearningThe sets of numbers and how they grew, the number $i$ with $i^2=-1$, the complex number $a+bi$, the real part and the imaginary part, powers of $i$ and equality of complex numbers.
  2. The four operationsLearningAdding, subtracting and multiplying complex numbers, the conjugate, dividing with the denominator's conjugate, the laws of arithmetic, and why the complex numbers have no order.
  3. The Gauss planeLearningA complex number as a point and as a vector in the Gauss plane, the real axis and the imaginary axis, adding and subtracting as vectors, the conjugate as a reflection, and the opposite.
  4. The modulus and the conjugateLearningThe modulus as the distance from the origin, $|z_1-z_2|$ as the distance between two points, the properties of the conjugate and the modulus, the triangle inequality, and equations with $\bar{z}$ and $|z|$.
  5. Practice: the algebraic form and the Gauss planePracticePowers of $i$, the four operations, dividing by the conjugate, equality of complex numbers, the modulus and distance, and equations with $\bar{z}$ and $|z|$.
  6. Square roots and quadratic equationsLearningThe square roots of a complex number with $z=a+bi$, the quadratic formula when the discriminant is negative or complex, conjugate solutions, and building an equation from its solutions.
  7. Equations of higher degreeLearningThe biquadratic equation, factoring, a solution found by guessing and dividing polynomials; in an equation with real coefficients the solutions that are not real come in conjugate pairs.
  8. Practice: equations in complex numbersPracticeSquare roots, quadratic equations with real and complex coefficients, equations of the third and fourth degree, equations with $\bar{z}$, and building an equation from its solutions.
  9. The polar formLearningThe modulus and the angle of a complex number, $z=r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta$, from the algebraic form to the polar by the quadrant, and from the polar form to the algebraic.
  10. Multiplying and dividing in polar formLearningIn a product the moduli multiply and the angles add; in a quotient they divide and the angles subtract; multiplying by $r\operatorname{cis}\theta$ is a rotation by $\theta$ and a stretch by $r$.
  11. De Moivre's theoremLearningA power of a complex number in polar form, the proof of the theorem by induction, high powers without opening brackets, and formulas for $\cos2\theta$ and $\cos3\theta$ from the theorem.
  12. Practice: the polar form and De Moivre's theoremPracticeMoving between the forms, multiplying and dividing in polar form, rotating in the Gauss plane, powers by De Moivre's theorem, and the exact value of an angle by comparing forms.
  13. The roots of unityLearningThe solutions of $z^n=1$, $z_k=\operatorname{cis}\frac{2k\pi}{n}$, as the vertices of a regular polygon on the unit circle; they form a geometric sequence, their sum is 0 and their product is $\pm1$.
  14. Roots of order nLearningThe solutions of $z^n=R(\cos\varphi+i\sin\varphi)$ by the formula sheet, the vertices of a regular polygon centred at the origin, and roots found from one known root.
  15. Loci in the Gauss planeLearning$|z-z_0|=R$ – a circle, $|z-z_1|=|z-z_2|$ – a perpendicular bisector, $|z-z_1|=m|z-z_2|$ – Apollonius' circle, lines by $\operatorname{Re}$ and $\operatorname{Im}$, a disc as an inequality; turning them into equations in $x$ and $y$.
  16. Geometric problems with complex numbersLearningTranslating, stretching, reflecting and rotating as operations on complex numbers, the midpoint of a segment and the meeting point of the medians, the parallelogram law, and a sum of angles by a product.
  17. Practice: roots, loci and geometryPracticeRoots of unity and roots of order $n$, a regular polygon in the Gauss plane, loci as equations in $x$ and $y$, and rotations.
  18. Review: complex numbersPracticeQuestions from the whole course: dividing and powers of $i$, a quadratic equation and a triangle in the Gauss plane, the polar form and roots of order 3, loci and where they meet, and a proof with the conjugate.
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