Visualizing Complex Functions
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Visualizing Complex Functions
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Re: Visualizing Complex Functions
#2If I had a dollar for every explanation of complex numbers that is basically just "A complex number is a real number plus an imaginary component, where i = sqrt(1)" I would almost have enough money to go back to uni and study math. It's far enough through the curriculum that most people get through the class by symbolic pattern matching and algorithmic question-answering rather than actual understanding (I studied EE), and I am pretty certain that even most (note, most) professors only really understand them as a quirky trick of our mathematical system. Ask them to explain it in English, not just as a mathematical definition, they all seem to come unstuck.
This old but very good lecture series really helped me - I can at least accept that complex numbers are not some ficticious hack now - but I confess I still dont have an intuitive grasp of complex numbers.
Re: Visualizing Complex Functions
#3If I had a dollar for every explanation of complex numbers that is basically just "A complex number is a real number plus an imaginary component, where i = sqrt(1)" I would almost have enough money to go back to uni and study math. It's far enough through the curriculum that most people get through the class by symbolic pattern matching and algorithmic question-answering rather than actual understanding (I studied EE…
a complex number is a point in the plane (in the same way as a real number is a point in a straight line)
Re: Visualizing Complex Functions
#4The images are missing a way to identify zeroes and poles. Here are my takes on this subject with this shader "Complex Maps" https://www.shadertoy.com/view/Ms2Bz3
And this video from a few years ago "Obama deformed by holomorphic complex functions (conformal map)" https://www.youtube.com/watch?v=CMMrEDIFPZY
Re: Visualizing Complex Functions
#5If I had a dollar for every explanation of complex numbers that is basically just "A complex number is a real number plus an imaginary component, where i = sqrt(1)" I would almost have enough money to go back to uni and study math. It's far enough through the curriculum that most people get through the class by symbolic pattern matching and algorithmic question-answering rather than actual understanding (I studied EE…
My definition of complex numbers : 2D points with multiplication such as you multiply their polar radius and add their polar angle (https://en.wikipedia.org/wiki/Polar_coordinate_system) and normal vector addition. (You can find back i=sqrt(-1) and all complex numbers properties / theorems after that of course)
For more visual explanations of complex numbers I recommend Tristan Needham's "Visual Complex Analysis" : http://pipad.org/tmp/Needham.visual-complex-analysis.pdf
Re: Visualizing Complex Functions
#6If I had a dollar for every explanation of complex numbers that is basically just "A complex number is a real number plus an imaginary component, where i = sqrt(1)" I would almost have enough money to go back to uni and study math. It's far enough through the curriculum that most people get through the class by symbolic pattern matching and algorithmic question-answering rather than actual understanding (I studied EE…
a complex number is a point in the plane (in the same way as a real number is a point in a straight line)
What does it mean to multiply two points in the plane? You still need an additional rule for that.
Re: Visualizing Complex Functions
#7Earlier quoted context omitted.
a complex number is a point in the plane (in the same way as a real number is a point in a straight line)
What does it mean to multiply two points in the plane? You still need an additional rule for that.
> What does it mean to multiply two points in the plane? You still need an additional rule for that.
yes, you do. I was answering the question of what complex numbers are. There are many things that you can do with them, that you need to define separately.
Re: Visualizing Complex Functions
#8Earlier quoted context omitted.
What does it mean to multiply two points in the plane? You still need an additional rule for that.
> What does it mean to multiply two points in the plane? You still need an additional rule for that. yes, you do. I was answering the question of what complex numbers are . There are many things that you can do with them, that you need to define separately.
a 2D point is not enough to define complex numbers
Re: Visualizing Complex Functions
#9Earlier quoted context omitted.
What does it mean to multiply two points in the plane? You still need an additional rule for that.
> What does it mean to multiply two points in the plane? You still need an additional rule for that. yes, you do. I was answering the question of what complex numbers are . There are many things that you can do with them, that you need to define separately.
Hmm, I tend to think that what one can do with some objects is an essential part of what they are.
Two-dimensional vectors can just as well be thought of as points in the plane. What sets complex numbers apart is eg that we define complex multiplication of them.
A more general example: a group is not just some set G, but that set with a binary operation satisfying certain axioms.
Re: Visualizing Complex Functions
#10He has subreddit for himself, that's weird.