Complex Numbers and their Geometry

(Note: I lied. This will be my first “neural dump.” I began writing about Euler’s Formula, but felt what follows was worthy of its own post and a better foundation for what will follow when I tackle Euler.)

Complex numbers arose from the fact that there is no solution for x in the equation x^2 = -1 in \mathbb{R}, the set of real numbers.

Early mathematicians being the devil-may-care mavericks that they were, were all like, “Screw it. Let’s just invent a new number. We’ll just call this number i and say that the solution is x = i.” Or, in other words, this new number i they imagined up (see what I did there) is equal to \sqrt{-1}.

    \[i = \sqrt{-1}\]

I don’t know this part for a fact, but I am guessing the guardians of \mathbb{R} laughed these heretics out of the room when they vouched for i to join the club. You can’t blame them. \mathbb{R} does stand for “real” after all.  Being the rebels they’d already proven themselves to be, the cult of i started their own club, or field as the math geeks like to call them.  To show those \mathbb{R} chumps who was boss they decided to include the real numbers as well as imaginary numbers in this new field, which they dubbed the field of complex numbers, denoted \mathbb{C}.

Now, that i was defined, mathematicians could solve other troublesome problems of the time, like:

x^2 = -16

x = \sqrt{-1 \cdot 16} = 4\sqrt{-1} = 4i

It turns out that any number containing i, like 4i, 10i, etc, is strictly speaking an imaginary number. But as I said,  \mathbb{C} subsumes \mathbb{R}, hence the “Complex” part of the name. What gives? Where are the real numbers?


(x - 1)^2 = -16

The solution is found like so:

x - 1 = \sqrt{-1 \cdot 16} = 4\sqrt{-1} = 4i

x = 1 + 4i

So the solution to the problem (x - 1)^2 = -16 is x = 1 + 4i. Notably,  x = 1 + 4i has both a real part, 1, and imaginary part, -4i. So there you go. A complex number is a number with both a real part and an imaginary part.

complex number = real part + imaginary part

As it turns out the real part can be zero, which explains why all of the imaginary numbers are also complex numbers, or the imaginary part can be zero, meaning all of the real numbers are also complex numbers.

Formally, a complex number z consists of an ordered pair of real numbers (a, b) where a is the real part and b is the imaginary part. Using our earlier notation we can say, z = a + bi.

Complex Number Algebra

Now, it wouldn’t be a field if the operations of addition and multiplication weren’t defined, so all complex numbers observe the following algebraic rules:

(1)   \begin{equation*} z_1 + z_2 = (a_1, b_1) + (a_2, b_2) = (a_1+a_2, b_1+b_2) = a_1 + a_2 + (b_1+b_2)i \end{equation*}

(2)   \begin{equation*} c \cdot z = c \cdot (a,b) = (c\cdot{a}, c\cdot{b}) = ca + cbi \end{equation*}

(3)   \begin{equation*} z_1 \cdot z_2 = (a_1\cdot{a_2} - b_1\cdot{b_2}, a_1\cdot{b_2} + a_2\cdot{b1}) \end{equation*}

(4)   \begin{equation*} \frac{1}{z} =  \end{equation*}

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