A recent video from Howie Hua showed how if you split a collection of numbers into equal-sized groups, then find the mean of each group, then find the mean of those means, it turns out this final answer is the same as the mean of the original collection. He was careful to say it usually does not work if the groups were different sizes. Which got me to wondering: just how much of an effect on the final mean-of-means can you have by splitting a collection of numbers into different-sized groups?
Category: My maths
Descriptions of my own new maths that I have created.
-
Where the complex points are: i-arrows
Once upon a time in 2016, I created the idea of iplanes, which I consider to be one of my biggest maths ideas of all time. It was a way of me visualising where the complex points are on the graph of a real function while still being able to see the original graph. But there was a problem with it: the thing I want, which is to see where the complex points are (or at least look like they are) is several steps away from locating them.
However, in my original series of blog posts, I actually already created a solution to this problem! I can draw a complex number as an arrow on the real line, which starts at the real part and extends in the length and direction of the imaginary part. Anyway, combining this arrow model of a complex number from an x-coordinate and a y-coordinate produces an arrow in the plane. The point (p+si,q+ti) is an arrow based at the point (p,q) and extending along the journey (s,t) from there.
This is the representation I need. I have decided to call them i-arrows.
The titles of the eight posts in the series are:
- Where the complex points are: i-arrows
- The complex points on a line using i-arrows
- Further updates on the complex points on an unreal line using i-arrows
- The complex points on a line in finite geometry using i-arrows
- The complex points on a parabola using i-arrows
- The complex points on real circles using i-arrows
- The complex points on unreal circles using i-arrows
- The line joining two complex points using i-arrows
UPDATE: There was a later blog post in 2024 further investigating the line joining two complex points.
-
Where the complex points are: on a real circle
In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph. Ever since I had this idea, I have wondered on and off about the complex points on a circle. It’s time to write about what I’ve found.
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
-
Where the complex points are: on a complex line (again)
It’s been four years since I came up with the idea of iplanes as a way to organise the complex points on a graph, and in the intervening time I have thought about them on and off. For some reason right now I am thinking about them a lot, and I thought I would write down some of what I am thinking.
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
-
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!
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.