This lecture contained just a basic introduction to complex numbers, one which I will not note in detail. This first week’s notes are just the selected bits of information I thought were interesting.
It is possible to write polynomial equations with real coefficients which do not have real solutions. You must extend the Reals into the Complex numbers in order to be able to solve the equations.
If you write polynomial equations with complex coefficients, there is no need to extend the complex numbers, the solutions of said equation will always be complex.
<aside> ➕ $\mathbb{C}$ is the final extension of the natural numbers to be able to solve any polynomial equation.
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<aside> ➕ Definition ($\mathbb{C}$)
$$ \mathbb{C} = \{z=x+iy|x=\Re(z)\in \mathbb{R},y=\Im(z)\in \mathbb{R}\} $$
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Notably:
$$ \mathbb{R}=\{ z \in \mathbb{C}|\Im(z) = 0 \}\subseteq \mathbb{C} $$
Also, $\mathbb{C}$ equipped with addition and multiplication from the reals, satisfies the field axioms:
The following is a formula for the multiplicative inverse of a complex number $z$:
$$ z^{-1}:=\frac{\Re(z)-i\Im(z)}{\Re^2(z)+\Im^2(z)} $$
<aside> ➕ Fundamental Theorem of Algebra
Every non constant polynomial of degree $D$ has $D$ roots in $\mathbb{C}$.
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Polar form of a complex number has the following properties: