4 ms·

Visualizing Complex Functions

87 points by vankessel 8y ago

Jedi72 8y ago
If 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. https://youtu.be/BOx8LRyr8mU https://youtu.be/BOx8LRyr8mU
- enriquto 8y ago
  a complex number is a point in the plane (in the same way as a real number is a point in a straight line)
  - empiricus 8y ago
    What does it mean to multiply two points in the plane? You still need an additional rule for that.
    - enriquto 8y ago
      > 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.
      - ttoinou 8y ago
        a 2D point is not enough to define complex numbers
        enriquto 8y ago
        > a 2D point is not enough to define complex numbers alright. Yet, one 2d point is certainly enough to define one complex number, which is what I meant. The data is the same, but they have different methods.
        navane 8y ago
        I struggle with this a lot, too. Maybe "complex numbers" is a misnomer, and we should call them "complex operations", or "complex methods". I have a hunch that for math people complex numbers are tied really close to the operations you perform with them. But to me, the lay man, the numbers are just data. From reading around on HN I infer that there are certain things that are grouped together because they share certain actions. Groups, sets, monads. However, if you were to encounter the number (1, 5) in the wild, you wouldnt know if it was a complex number or just a 2 dimensional vector until it is operated upon.
        ttoinou 8y ago
        The OP was looking for "explanation of complex numbers" You won't explain anything by talking about 2D points only But you can if you specify only how they behave with addition and multiplication (see my others posts)
        enriquto 8y ago
        > The OP was looking for "explanation of complex numbers" > You won't explain anything by talking about 2D points only The OP was complaining about the explanation that complex numbers "have a real and an imaginary part", because it mean nothing to them. It is alright, then, to clarify that this just means that they are points in the plane, and some operations will be defined on these points.
  - dragonwriter 8y ago
    > a complex number is a point in the plane (in the same way as a real number is a point in a straight line) More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line) Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.) But real numbers aren't points on a line, and complex numbers aren't points on a plane.
  - adrian_b 8y ago
    While it is true that 2 real numbers can determine both a complex number and a point in plane and it is also true that 1 real number can determine both a real number and a point in a straight line, these are different mathematical things. The correct view is that the real numbers, also known as scalars, are quotients of 1-dimensional vectors, while the complex numbers are quotients of 2-dimensional vectors. This means that given two 1-dimensional vectors, the second being non-null, there is always a real number by which you can multiply the second vector to obtain the first vector. The same for two 2-dimensional vectors and a complex number. In this case the magnitude of the complex number changes the magnitude of the vector, while the phase rotates the vector. When you understand this fact about complex numbers, than it becomes obvious that the imaginary unit is not imaginary but just a rotation with a right angle and it is trivial that its square, i.e. a rotation with 180 degrees is equivalent with a multiplication by -1. The point are a third, different kind of mathematical entities, distinct from both real & complex numbers and from vectors. In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine spaces) are the primitive objects. From points you can define the vectors as differences of points (i,e, translations). Then from vectors you can define real numbers, complex numbers, quaternions and higher-dimensional matrices as quotients of vectors. (Such definitions define e.g. a 2-dimensional vector as a class of equivalence of pairs of points in plane that are transformed into each other by a translation, and a complex number as a class of equivalence of pairs of 2-dimensional vectors which are transformed into each other by a proportional change in magnitude and a rotation with a fixed angle.) Obviously, my explanation here is very simplified, but it is useful to be aware that e.g. a point in plane, a 2-dimensional vector and a complex number are 3 very different kinds of mathematical objects, even if all 3 are determined by a pair of real numbers. They are very different because the set of operations that can be applied to each is different. For example only the complex numbers are members of a field. You can add 2-dimensional vectors (i.e. compose 2 translations), but you cannot add 2 points from a plane.
    - ttoinou 8y ago
      How do you define a 1D vector then ?
      - 8y ago
- ttoinou 8y ago
  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 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 http://pipad.org/tmp/Needham.visual-complex-analysis.pdf
- user2994cb 8y ago
  You can construct the complex numbers quite nicely with Geometric Algebra: https://www.youtube.com/watch?v=PNlgMPzj-7Q https://www.youtube.com/watch?v=PNlgMPzj-7Q
  - Jedi72 8y ago
    This video is FANTASTIC. Thank you so much. First time I've ever see someone draw what the imaginary number is conceptually! Also seeing a 4 dime sional vector space defined in R2 blew my mind. Thankyou. Students these days are very lucky.
- mitchtbaum 8y ago
  Steven Wittens made good progress toward an intuitive footing in https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
- sannee 8y ago
  I am an EE and our complex analysis class definitely did define complex numbers "rigorously" (in the usual R^2 with multiplication -> wow, it's a field way, none of the fancy algebraic closure stuff). However, later I spoke with some of my classmates and it turned out that this left absolutely zero impression on them and they still believed that "we define i to be sqrt(-1)" high school rubbish...
- vankessel 8y ago
  Thank you for the feedback. That is a very valid complaint and one that is hard to avoid because the complex numbers are basically derived by trying to solve equations like x^2 + 1 = 0. It may be a bit more geometrically intuitive to describe complex numbers as a system of making 2D vectors work with multiplication and division which I sort of touch on later. The explanation is there to provide a brief overview and support the following visualizations, but I will see if I can improve that section.
  - Jedi72 8y ago
    I didnt really intend it as personal feedback for your post, its as much about my failure to understand as it is the explanations. If its useful to you then great, but on the flip side, dont take it too much to heart :)
ttoinou 8y ago
The 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 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 https://www.youtube.com/watch?v=CMMrEDIFPZY
Amendeson530 8y ago
He has subreddit for himself, that's weird.
newprint 8y ago
www.visual.wegert.com is beautiful yearly calendar of visualizations of Complex functions. There is also an entire book devoted to exploring C.A. using visualization.

87 points by vankessel 8y ago

If 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. https://youtu.be/BOx8LRyr8mU https://youtu.be/BOx8LRyr8mU

enriquto 8y ago
a complex number is a point in the plane (in the same way as a real number is a point in a straight line)
- empiricus 8y ago
  What does it mean to multiply two points in the plane? You still need an additional rule for that.
  - enriquto 8y ago
    > 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.
    - ttoinou 8y ago
      a 2D point is not enough to define complex numbers
      - enriquto 8y ago
        > a 2D point is not enough to define complex numbers alright. Yet, one 2d point is certainly enough to define one complex number, which is what I meant. The data is the same, but they have different methods.
        navane 8y ago
        I struggle with this a lot, too. Maybe "complex numbers" is a misnomer, and we should call them "complex operations", or "complex methods". I have a hunch that for math people complex numbers are tied really close to the operations you perform with them. But to me, the lay man, the numbers are just data. From reading around on HN I infer that there are certain things that are grouped together because they share certain actions. Groups, sets, monads. However, if you were to encounter the number (1, 5) in the wild, you wouldnt know if it was a complex number or just a 2 dimensional vector until it is operated upon.
        ttoinou 8y ago
        The OP was looking for "explanation of complex numbers" You won't explain anything by talking about 2D points only But you can if you specify only how they behave with addition and multiplication (see my others posts)
        enriquto 8y ago
        > The OP was looking for "explanation of complex numbers" > You won't explain anything by talking about 2D points only The OP was complaining about the explanation that complex numbers "have a real and an imaginary part", because it mean nothing to them. It is alright, then, to clarify that this just means that they are points in the plane, and some operations will be defined on these points.
- dragonwriter 8y ago
  > a complex number is a point in the plane (in the same way as a real number is a point in a straight line) More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line) Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.) But real numbers aren't points on a line, and complex numbers aren't points on a plane.
- adrian_b 8y ago
  While it is true that 2 real numbers can determine both a complex number and a point in plane and it is also true that 1 real number can determine both a real number and a point in a straight line, these are different mathematical things. The correct view is that the real numbers, also known as scalars, are quotients of 1-dimensional vectors, while the complex numbers are quotients of 2-dimensional vectors. This means that given two 1-dimensional vectors, the second being non-null, there is always a real number by which you can multiply the second vector to obtain the first vector. The same for two 2-dimensional vectors and a complex number. In this case the magnitude of the complex number changes the magnitude of the vector, while the phase rotates the vector. When you understand this fact about complex numbers, than it becomes obvious that the imaginary unit is not imaginary but just a rotation with a right angle and it is trivial that its square, i.e. a rotation with 180 degrees is equivalent with a multiplication by -1. The point are a third, different kind of mathematical entities, distinct from both real & complex numbers and from vectors. In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine spaces) are the primitive objects. From points you can define the vectors as differences of points (i,e, translations). Then from vectors you can define real numbers, complex numbers, quaternions and higher-dimensional matrices as quotients of vectors. (Such definitions define e.g. a 2-dimensional vector as a class of equivalence of pairs of points in plane that are transformed into each other by a translation, and a complex number as a class of equivalence of pairs of 2-dimensional vectors which are transformed into each other by a proportional change in magnitude and a rotation with a fixed angle.) Obviously, my explanation here is very simplified, but it is useful to be aware that e.g. a point in plane, a 2-dimensional vector and a complex number are 3 very different kinds of mathematical objects, even if all 3 are determined by a pair of real numbers. They are very different because the set of operations that can be applied to each is different. For example only the complex numbers are members of a field. You can add 2-dimensional vectors (i.e. compose 2 translations), but you cannot add 2 points from a plane.
  - ttoinou 8y ago
    How do you define a 1D vector then ?
    - 8y ago
ttoinou 8y ago
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 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 http://pipad.org/tmp/Needham.visual-complex-analysis.pdf
user2994cb 8y ago
You can construct the complex numbers quite nicely with Geometric Algebra: https://www.youtube.com/watch?v=PNlgMPzj-7Q https://www.youtube.com/watch?v=PNlgMPzj-7Q
- Jedi72 8y ago
  This video is FANTASTIC. Thank you so much. First time I've ever see someone draw what the imaginary number is conceptually! Also seeing a 4 dime sional vector space defined in R2 blew my mind. Thankyou. Students these days are very lucky.
mitchtbaum 8y ago
Steven Wittens made good progress toward an intuitive footing in https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
sannee 8y ago
I am an EE and our complex analysis class definitely did define complex numbers "rigorously" (in the usual R^2 with multiplication -> wow, it's a field way, none of the fancy algebraic closure stuff). However, later I spoke with some of my classmates and it turned out that this left absolutely zero impression on them and they still believed that "we define i to be sqrt(-1)" high school rubbish...
vankessel 8y ago
Thank you for the feedback. That is a very valid complaint and one that is hard to avoid because the complex numbers are basically derived by trying to solve equations like x^2 + 1 = 0. It may be a bit more geometrically intuitive to describe complex numbers as a system of making 2D vectors work with multiplication and division which I sort of touch on later. The explanation is there to provide a brief overview and support the following visualizations, but I will see if I can improve that section.
- Jedi72 8y ago
  I didnt really intend it as personal feedback for your post, its as much about my failure to understand as it is the explanations. If its useful to you then great, but on the flip side, dont take it too much to heart :)

ttoinou 8y ago

The 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 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 https://www.youtube.com/watch?v=CMMrEDIFPZY

Amendeson530 8y ago

He has subreddit for himself, that's weird.

newprint 8y ago

www.visual.wegert.com is beautiful yearly calendar of visualizations of Complex functions. There is also an entire book devoted to exploring C.A. using visualization.