Complex Numbers

Students' Learning Outcomes

• Identify complex numbers, complex conjugate, absolute value or modulus of a complex number.
• Apply algebraic properties and perform basic operations on complex numbers.
• Demonstrate additive identity and multiplicative identity for the set of complex numbers.
• Find additive inverse and multiplicative inverse of a complex number $z$
• Demonstrate the following properties of a complex number $z$

  $$|z| = | - z| = |\bar{z} | = | - \bar{z} |\qquad \circ \bar{z} = z,z\bar{z} = |z|^2$$ $$\overline{z_1}+\overline{z_2}=\overline{z_1}+\overline{z_2}\circ \overline{z_1}\overline{z_2}=\overline{z_1}\overline{z_2}:\left(\frac{z_1}{\overline{z_2}}\right)=\frac{\overline{z_1}}{\overline{z_2}},z_2\mathrm{\ne }0$$$\overline{z_1} +\bar{z_2} = \overline{z_1} +\bar{z_2}\qquad \circ \overline{z_1}\overline{z_2} = \overline{z_1}\overline{z_2} :\left(\frac{z_1}{\bar{z_2}}\right) = \frac{\overline{z_1}}{\bar{z_2}},z_2\neq 0$

• Apply the Geometric interpretation of a complex number, modulus of a complex number and algebraic operations of a complex number.
• Find real and imaginary parts of complex numbers of the type $(x + iy)^n$ and
• Solve the simultaneous linear equations with complex coefficients.