Complex Numbers are Planar Numbers for Rotations and Waves
Channel: Math 4 Wisdom
Date: 2023-10-05

John Harland of Palomar College gives this second talk on complex numbers, which he would prefer to call "planar numbers".  They are natural for describing rotations and waves.  

This talk was recorded on September 29, 2023.  Here are John's slides:
https://www.math4wisdom.com/files/JohnHarlandComplexNumbersTalk-9-29-23.pdf
And his prior talk:
https://www.youtube.com/watch?v=clgBV9kwu80

Andrius Kulikauskas gives 20 minutes of preliminary remarks about how the complex numbers are fundamental amongst number systems.  We are investigating how to model cognitive frameworks.

This is a session of the Math 4 Wisdom study group for the mathematics of the divisions of everything.  These cognitive frameworks are presented here:
https://www.math4wisdom.com/wiki/Research/20170929TimeSpaceDecisionMaking
https://www.youtube.com/watch?v=9x69-Xxv6L0

Join us! The first step is to sign up for our Math 4 Wisdom discussion group 
https://www.freelists.org/list/math4wisdom

We have sessions at the introductory (community college) level and the advanced (graduate student) level of math.

1:00 Andrius reviews the twosome, threesome, foursome and notes how they may relate to complex numbers and quaternions.
7:20 More broadly there are division algebras and Clifford algebras.
14:00 Clifford algebras may lack multiplicative inverses.
16:00 Division algebras (such as octonions) may be nonassociative.
19:10 Real Clifford algebras manifest eightfold Bott periodicity and may model the eight-cycle of divisions of everything.
21:40 John reviews his earlier presentation from July 24, 2023. Complex numbers arose in the quadratic formula (where they are frivolous in finding real solutions)  and then the cubic formula (where they are essential in calculating real solutions). Quadratic example of area.  Cubic example of volume.
32:40 Does adding the square root of -1 to your number system goof it up? Complex numbers as ordered pairs.
34:55 These are pairs of numbers... so why call them numbers?  John lists the properties of a field.
38:10 Kirby: We could call a matrix a number!  
38:50 John: These pairs of real numbers satisfy the axioms for a field.  
40:20 John completed his review and continues.  Pairs (a,b) are written a+bi. John shows i^2=-1 by construction and not by caveat.
44:40 We may also define complex numbers as a subset of 2x2 matrices. 
49:40 Unit complex numbers describe rotations in 2D and unit quaternions in 3D.
51:15 Real numbers exist on a line. Complex numbers exist on the plane. Addition and multiplication are geometric in terms of vectors. 
58:00 When you multiply a complex number by itself you go in a circle or in a spiral.
59:40 Multiplying by i relates two different axes. 
1:00:00 What are complex numbers used for? Fundamental theorem of Algebra.
1:02:10 Natural models of periodic phenomena. Consider rotations by 30 degrees.
1:06:45 Model, visualize waves. A helix in time.  Projecting gives a real wave as in trig.
1:09:25 Adding two real waves is hard. Adding complex waves is simply adding amplitudes. 
1:13:00 Go from complex waves to real waves easily.
1:13:40 Multiply by i to rotate.
1:15:25 Complex numbers  may model slack.
1:16:15 Complex numbers reveal the behavior of electronic circuits and finite difference equation systems.
1:17:10 In conclusion, "imaginary" and "complex" are bad terms. John would call real numbers the "linear numbers" and call complex numbers the "planar numbers".
1:21:00 Kirby and John discuss visualization of waves.
1:27:30 A problem with complex numbers is that to visualize them you need 3 or more dimensions. 
1:29:20 Kirby: Animation! Andrius: Virtual reality!
1:30:20 Ryan was struck by the huge gap between Cardano 1545 and Descartes 1637. John: Descartes used "imaginary" ironically. Andrius: Descartes was an idealist.
1:33:00 Ryan liked the spiral and the matrices. John and Andrius want to talk more about rotations in 2D (complex numbers) and 3D (quaternions) and 2x2 matrices.
1:34:25 Andrius: Grateful that we think fresh with different levels of math. 
1:35:45 Ryan: How to calculate the angular rotations? Euler's formula. Multiplication implements rotation geometrically. 
1:41:45 Kirby: Can emphasize the problem with real numbers.  Andrius: geometrically, the square root of 180 degrees is 90 degrees but also 270 degrees. 
1:44:10 John: A large part is familiarity. Math is experiential. They become natural when you start working with them. Calculations are helping SU(2) sink in for him. 
1:45:30 Kirby: Multiplication comes up with the Mandelbrot set, convergence and divergence. Opportunity for schools.
1:46:45 Andrius: John does what Gauss did, lots of calculations to build up intuition. How to make that fun?

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Visit http://www.math4wisdom.com to contact me, Andrius Kulikauskas, and learn more about Math 4 Wisdom!