# Complex Numbers We start by *assuming* that there is a number $i$ such that: $i^2 = -1$ We then state that $i$ lives in a different dimension. Specifically, it lives *perpendicularly* to the real number line:  Now, at first this seems very arbitrary. Why should we state that this number $i$ is in a dimension that is perpendicular to our real number line, and 1 unit above it? To see why this may actually be a reasonable way of defining things, let us recall what it means to rotate a point 90 degrees counter clockwise, via the image below:  We see that a 90 degree rotation moved our point as follows: $(3, 2) \longrightarrow (-2, 3)$ The rule associated with all 90 degree rotations is: $(a,b) \;\;\; \overbrace{\longrightarrow}^{\text{90 rot}} \;\;\; (-b,a)$ Now, *if* we treat $i$ just like any other number, let us investigate the following operation: the multiplication of a complex number by $i$. For example: $i \cdot (3 + 2i)$ If we distribute the $i$ we end up with: $3i + 2i^2 = -2 + 3i$ Where the expression on the right makes use of the fact that we *defined* $i^2 = -1$. Notice that if we write this point via coordinates instead of addition we see that multiplying by $i$ looks like: $(3, 2i) \;\;\; \overbrace{\longrightarrow}^{\text{mult by i}} \;\;\; (-2, 3i)$ So, we can see that: > Multiplying a complex number by $i$ has the same effect as **rotating it by 90 degrees**. To summarize: $i$ has a defining property of it's square being equal to $-1$. When we multiply by $i$ it has the same effect as rotating by 90 degrees. >Hence, maybe it isn't that crazy to *geometrically position* $i$ at a 90 degree angle to the real number line. #### 3 facts about multiplication * $Z \cdot 1 = Z \;\;\;\;$ (geometrically this means that we stretch and rotate the vector $1$ to get $Z$ ) * $Z \cdot i = Rot90(Z) \;\;\;\;$ (rotate $Z$ by 90 degrees) * $Z \cdot (c + di) = c \cdot Z + d \cdot (Zi) \;\;\;\;$ (We can view multiplying by $Z$ as a transformation that keeps gridlines parallel and evenly spaced-a Linear Transformation) See [here](https://photos.google.com/photo/AF1QipN3ZyMBj6sbDI21omb-yfXV4018EY0TeuazUqHj). ### A new take Key distinction for complex numbers vs simply using two dimensions: complex numbers have rotation *built in*; hence they are great for periodic functions. In other words both are represented in 2d plane, but complex numbers have a unique rotational component. --- #### References * [3 blue 1 brown, live stream 1](https://www.youtube.com/watch?v=5PcpBw5Hbwo&t=90s) * [Imagine numbers are real, playlist-incredible resource](https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF) * [Beautiful book on complex analysis](https://complex-analysis.com/content/table_of_contents.html)