MAT187 L02-03 - Complex Numbers

#MAT187 PCE

Imaginary numbers are of the form $b i$ , where $b$ is real and $i = \sqrt{- 1}$ .
e.g. $\sqrt{- 3 i} = 3 i$ , $(2 i)^{2} = - 4$

Complex numbers are numbers of form $x = a + b i$ , where $a$ and $b$ are the real and imaginary parts of $x$ .
Real part of $x$ : $a = Re (x)$
Imaginary part of $x$ : $b = Im (z)$

Complex conjugate of $a + b i$ is $a - b i$ . This is denoted by $\overset{―}{a + b i}$ .
i.e., $\overset{―}{a + b i} = a - b i$

If a quadratic has no solution in the real numbers, it has complex solutions in the form of complex conjugates, $a + b i$ and $a - b i$ .

Operations with complex numbers

Basically just treat complex numbers as functions with an unknown variable (that being $i$ ), and then simplify $i^{2 n}, n ϵ Z$ to $- 1$ in the final result.

Example:
$(3 + 4 i) (2 - 2 i) = 3 \times 2 - 3 \times 2 i + 4 \times 2 i - 4 \times 2 i^{2}$
$= 6 + 2 i - 8 i^{2}$
$= 6 + 2 i - 8$
$= - 2 - 2 i$

Graphing complex numbers

We can graph a complex number by plotting $Re (z)$ on horizontal, and $Im (z)$ on vertical.

Modulus of a complex number, denoted by $| z | = | a + b i |$ , is the length of the vector with $a$ and $b$ as its components.

Argument or phase of a complex number is the angle $ϕ$ between the positive real axis and the line segment created by $a + b i$ .
Often denoted by $a r g (z)$

Important: there are technically infinitely many arguments for a complex number.
Similar to how there are infinitely many $θ$ for any cartesian point, expressed in polar coordinates. There is only one modulus for a complex number, however.

Complex numbers in polar coordinates

Since we can represent complex numbers through moduli and arguments, it is fortuitous to convert them to polar coordinates.

Converting complex numbers from polar to cartesian
If $| z | = r$ , and $a r g (z) = θ$ (as seen in above diagram), then:
$a = r \cos θ$
$b i = r \sin θ i$
$⟹ z = r (\cos θ + \sin θ i)$

Product of arguments
For complex numbers $z_{1}$ and $z_{2}$ :
$a r g (z_{1} z_{2}) = a r g (z_{1}) + a r g (z_{2})$

Product of complex numbers
If $z \neq 0$ (i.e., could be complex or real) and $w$ is a complex number, then $w \times z$ results in:

Rotation of $w$ by angle $a r g (z)$ . If $z$ is not complex, then there is no rotation.
Scaling of $| w |$ by factor of $| z |$ .

Euler's formula to represent complex numbers

Since writing out $z = a + b i$ or $z = r (\cos θ + \sin θ i)$ is a pain to mathematicians apparently, we can use Euler's formula to simplify complex numbers.

Exponential notation of complex numbers
If you have a complex number $z = r (\cos θ + \sin θ i)$ , then the to convert to exponential notation:
$\cos θ + \sin θ i = e^{θ i}$
$⟹ r (\cos θ + \sin θ i) = r e^{i θ}$