So the solutions to the n'th roots of unity of the form z^n = 1 are n-gons on the unit circle. This means for an arbitrary w in the complex field z^n = w gives solutions on a circle with radius n'th root of modulus w and are 2*pi/n apart so it is also an n-gon.
Pretty fun stuff to plot around using Geogebra to give a geometric intuition how the complex numbers behave.
Complex numbers are awesome for bashing through (some) IMO geometry problems. Points P, Q, R are collinear exactly when (p-q)/(p-r) is real, rotation is multiplication by exp(iϕ), etc.
Nice but I prefer the much powerful version of 3 points being collinear precisely when ((p-q)^2 + (q-r)^2 - (p-r)^2)^2 = 4(p-q)^2(q-r)^2 which works for any field(!) with non-characteristics 2.
If they are non-collinear/degenerate then the difference becomes 16 * area^2 of the triangle PQR, in which case one can easily derive Heron's formula.
I’m curious as to how many high schools around the world teach complex numbers. In England in the late 80s when I did A level maths (age 17-18) it was part of the syllabus, but now it is only available in the harder A-level further maths, which is often not taught in the standard non-selective non-private schools, making entry to university STEM courses more difficult.
In Portugal it is part of the standard curriculum although in my time they were poorly motivativated and besides solving some classes of quadratic equations no other applications were mentioned.
The Finnish high/secondary school (= years 10 to 12) gives a choice of two alternative math curricula, the more comprehensive option giving a reasonable foundation for university-level math and is usually very much recommended to anyone interested in STEM. It includes complex numbers (at least superficially) as well as a lot of other stuff that the narrow version doesn’t, like limits, integration, basic linear algebra, and analytic geometry.
How deep did your high school go on complex numbers?
I recall we covered them very lightly (mostly to explore what happens to roots of negative numbers), but we certainly came nowhere near the group theory stuff or linear algebra aspects mentioned in this post.
So the solutions to the n'th roots of unity of the form z^n = 1 are n-gons on the unit circle. This means for an arbitrary w in the complex field z^n = w gives solutions on a circle with radius n'th root of modulus w and are 2*pi/n apart so it is also an n-gon.
Pretty fun stuff to plot around using Geogebra to give a geometric intuition how the complex numbers behave.
I built these two interactive visualizations to gain some intuition for this. [1][2]
1: https://static.laszlokorte.de/complex-roots/ 2: https://static.laszlokorte.de/complex-plane/
Complex numbers are awesome for bashing through (some) IMO geometry problems. Points P, Q, R are collinear exactly when (p-q)/(p-r) is real, rotation is multiplication by exp(iϕ), etc.
Nice but I prefer the much powerful version of 3 points being collinear precisely when ((p-q)^2 + (q-r)^2 - (p-r)^2)^2 = 4(p-q)^2(q-r)^2 which works for any field(!) with non-characteristics 2.
If they are non-collinear/degenerate then the difference becomes 16 * area^2 of the triangle PQR, in which case one can easily derive Heron's formula.
Reading this made me feel like I am in highschool again
I’m curious as to how many high schools around the world teach complex numbers. In England in the late 80s when I did A level maths (age 17-18) it was part of the syllabus, but now it is only available in the harder A-level further maths, which is often not taught in the standard non-selective non-private schools, making entry to university STEM courses more difficult.
In Portugal it is part of the standard curriculum although in my time they were poorly motivativated and besides solving some classes of quadratic equations no other applications were mentioned.
The Finnish high/secondary school (= years 10 to 12) gives a choice of two alternative math curricula, the more comprehensive option giving a reasonable foundation for university-level math and is usually very much recommended to anyone interested in STEM. It includes complex numbers (at least superficially) as well as a lot of other stuff that the narrow version doesn’t, like limits, integration, basic linear algebra, and analytic geometry.
How deep did your high school go on complex numbers?
I recall we covered them very lightly (mostly to explore what happens to roots of negative numbers), but we certainly came nowhere near the group theory stuff or linear algebra aspects mentioned in this post.
Reading the title, I thought this was going to be about the origins of the Unity game engine.