How does the imaginary number determine the nature of complex analysis and related geometry?
複分析
- How does complex analysis ground circle folding?
- How does complex analysis ground geometric thinking with Clifford algebras?
读物
- Paul Zeitz mentions enthusiastically the book Visual Complex Analysis by Tristan Needham. University of Cincinnati professor David Herron has put it online.
- Ahlfors. Complex Analysis. Textbook.
- In factoring {$a^2+b^2=(a+ib)(a-ib)$} the only difference between {$ib$} and {$-ib$} is that they are different from each other - they are conjugates. But otherwise they are not distinguishable. This is interesting in taking the square root of the sum of squares as Dirac did.
- {$i$} and {$j$} are examples of two things that have all the same properties and yet are different.
Difference between complex numbers and real numbers
- Quantum possibilities vs. actualities
- Cauchy's integral theorem: for complexes, derivative and integral are mirrors, but not for reals
- There is a sense in which the reals give the magnitude and the imaginaries give the rotation. The function 1/x sends x+iy to x-iy divided by x2+y2. It sends r to 1/r (across the boundary of the unit circle) and it sends theta to -theta.
- Real numbers are used for independent x, y. Imaginary number i denotes a link between two otherwise indepedent variables so that y = ix links indepedent axes by a 90 degree rotation.
- Similarly the polar decomposition of a matrix distinguishes (as for a number) the change in magnitude (scaling) and the rotation. It separates them.
- Complex numbers have two natural coordinate systems that correspond to addition (x,y) and multiplication (r,theta).
- Circle folding relates to "reflection" of the complex conjugate across an x-axis. Thinking of inverse rotation as this reflection.
- The number "i" is highly misleading in that it actually has no priority over "-i". Both are square roots of -1. Thus often (or always?) they should both be referenced - they are a "coupled" pair of numbers, not a single number. They should be referenced by a single Number "I" which is understood to have two meanings.
- The purpose of complex numbers is to define two unmarked opposites (we know them, unfortunately, as "i" and "-i", where one is marked with regard to the other, but in truth they should be both unmarked). The purpose of the real numbers is to provide that context for this unmarkedness. (Is there a simpler way to create it?)
- The quantum world is based on the two unmarked opposites ("i" and "j") as with spin 1/2 particles, "up" and "down". Symmetry breaking - the breaking of the symmetry between "i" and "-i" enforced by complex conjugation - occurs (and is defined to be) when there is a measurement, so that we collapse to the reals, where this symmetry is broken.
- The truth of the heart does not mark the opposites. The truth of the world marks one opposite with regard to the other.
- Ar teisinga? Skaičius turėtų rašyti: xr + yi pabrėžti jog tai skiritingi matai. Bet r tampa 1. Vienas matas gali būti "default" ir užtat išbrauktas. Jisai tada tampa "identity". Every answer is an amount and a unit - šis dėsnis paneigtas.
- Complex numbers: local = global. (Identity theorem). Real numbers: local != global.
- Galois group of C/R