LEC2

Complex Conjugate

$$\begin{array}{r}z=x+yj\\ \bar{z}=x-yj\end{array}$$

We literally just flip the sign of $y$.

Division

$$\begin{array}{r}\frac{w}{z}=w\cdot \frac{1}{z}\\ \\ \frac{1}{z}=\frac{1}{z}\cdot \frac{\bar{z}}{\bar{z}}\\ \\ =\frac{1}{x+yj}\cdot \frac{x-yj}{x-yj}\\ \\ =\frac{x-yj}{x^2+y^2}\\ \\ \\ ⟹\frac{w}{z}=w\cdot \frac{(x-yj)}{x^2+y^2}\end{array}$$

Exponents

Very tedius (I think we will learn about this again with polar representations.)

$$z^k=z(z^{k-1});k\ge 0$$

Modulus

$$|z|=\sqrt{x^2+y^2}$$

• For real numbers modulus is equal to the absolute value.
• Absolute values is kinda weird in complex numbers.
• Modulus represents the magnitude for complex numbers (think like in vectors).
• We can’t compare complex numbers directly, but we can compare their magnitude.
  • The result of comparing magnitudes do not apply to the complex numbers themselves.

Complex Plane

• Thinking more in vectors.

• $(x,y)$ is a coordinate on the Cartesian Plane.

• $x+yj$ too represents a point on the Complex Plane.

• Instead of x, y axes, we have the real and imaginary axes.

• $x,y$ represent magnitude on their respective axis and $|z|$ gives us the magnitude.

• Now operations like addition etc are very similar to vector operations using diagrams.