Where the complex points are

When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation \(x^2 + 1 = 0\), which if you’re only using real numbers has no solutions, but with complex numbers has the solutions \(x=i\) and \(x=-i\).

As someone who likes to imagine the physical reality of everything, this has always caused me considerable difficulties. The equation \(x^2 + 1 = 0\) can be thought of as the equation that tells you where the parabola with equation \(y = x^2 + 1\) meets the x-axis.

Only the parabola with equation \(y = x^2 +1\) doesn’t meet the x-axis. If our complex number solutions are to be believed, then it meets the x-axis in the points \((i,0)\) and \((-i,0)\), but I certainly can’t see those points on my graph. Where are they?

Presumably there are a whole host of points with complex coordinates, which are points where various things meet that don’t look like they meet. These points must be somewhere, and they must be some place that is somehow related to the graphs I see in the real plane. But where is this place?

Well, about a week ago, I finally found the place where the complex points are!

You can read the rest of this blog post, and all seven of the blog posts in the series, in PDF form here. 

The titles of the seven posts in the series are:

• Where the complex points are
• Where the complex points are on a line
• Where the complex points are on a parabola
• Where the complex points are on the graph of a function
• Where the idea came from for where the complex points are
• Where the complex points are on a complex line (again)
• Where the complex points are on a real circle

UPDATE: There was a later blog post in 2016 where I slightly modified the idea from i-planes to i-arrows, and a later blog post in 2024 further investigating the line joining two complex points using i-arrows.