Extension Complex Numbers – Introduction and uses

This lesson comprises six (6) master classes focusing on:

• Imaginary numbers
• Complex numbers
• Arithmetic with complex numbers
• Representations of complex numbers
• Complex plane
• Euler’s formula

Content:

MEX-N1.1

• Use the complex number system
  • develop an understanding of the classification of numbers and their associated properties, symbols and representations
  • define the number \( i \), as a root of the equation \( x^2=−1 \)
  • use the symbol \( i \) to solve quadratic equations that do not have real roots
• Represent and use complex numbers in Cartesian form
  • use complex numbers in the form \( z=a+ib \), where \( a \) and \( b \) are real numbers and \( a \) is the real part \( \mathrm{Re}(z) \) and \( b \) is the imaginary part \( \mathrm{Im}(z) \) of the complex number
  • identify the condition for \( z_1=a+ib \) and \( z_2=c+id \) to be equal
  • define and perform complex number addition, subtraction and multiplication
  • define, find and use complex conjugates, and denote the complex conjugate of \( z \) as \( \bar{z} \)
  • divide one complex number by another complex number and give the result in the form \( a+ib \)
  • find the reciprocal and two square roots of complex numbers in the form \( z=a+ib \)

MEX-N1.2

• Represent and use complex numbers in the complex plane
  • use the fact that there exists a one-to-one correspondence between the complex number \( z=a+ib \) and the ordered pair \( (a,b) \)
  • plot the point corresponding to \( z=a+ib \)
• Represent and use complex numbers in polar or modulus-argument form, \( z=r( \cos \theta +i \sin \theta) \), where \( r \) is the modulus of \( z \) and \( \theta \) is the argument of \( z \)
  • define and calculate the modulus of a complex number \( z=a+ib \) as \( |z|=\sqrt{a^2+b^2} \)
  • define and calculate the argument of a non-zero complex number \( z=a+ib \) as \( \mathrm{arg}(z)=\theta \), where \( \tan \theta=\frac{b}{a} \)
  • define, calculate and use the principal argument \( \mathrm{Arg}(z) \) of a non-zero complex number \( z \) as the unique value of the argument in the interval \( (−\pi , \pi ] \)
• Prove and use the basic identities involving modulus and argument
  • \( |z_1z_2|=|z_1||z_2| \) and \( \mathrm{arg}(z_1z_2)=\mathrm{arg}(z_1)+\mathrm{arg}(z_2) \)
  • \( |\frac{z_1}{z_2}|=|z_1||z_2| \) and \( \mathrm{arg}(z_1z_2)=\mathrm{arg}(z_1)−\mathrm{arg}(z_2) \), \(z_2 \ne 0 \)
  • \( |z^n|=|z|^n \) and \( \mathrm{arg}(z^n)=n \mathrm{arg}(z) \)
  • \( |\frac{1}{z^n}|=\frac{1}{|z|^n} \) and \( \mathrm{arg}(\frac{1}{z^n})=−n \mathrm{arg}(z) \), \( z \ne 0 \)
  • \( \overline{z_1} + \overline{z_2}=\overline{z_1+z_2} \)
  • \( \overline{z_1} \times \overline{z_2}=\overline{z_1z_2} \)
  • \( z \overline{z}=|z|^2 \)
  • \(z+ \overline{z}=2 \mathrm{Re}(z) \)
  • \(z− \overline{z}=2i \mathrm{Im}(z) \)

MEX-N1.3

• Understand Euler’s formula, \( e^{ix}= \cos x+i \sin x \), for real \( x \)
• Represent and use complex numbers in exponential form, \( z=re^{i \theta} \), where \( r \) is the modulus of \( z \) and \( \theta \) is the argument of \( z \)
• Use Euler’s formula to link polar form and exponential form
• Convert between Cartesian, polar and exponential forms of complex numbers
• Find powers of complex numbers using exponential form
• Use multiplication, division and powers of complex numbers in polar form and interpret these geometrically
• Solve problems involving complex numbers in a variety of forms