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Visualizing Complex Functions

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Re: Visualizing Complex Functions

#11

Post by user2994cb »

Jedi72 wrote: post #2

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…

You can construct the complex numbers quite nicely with Geometric Algebra: https://www.youtube.com/watch?v=PNlgMPzj-7Q

Re: Visualizing Complex Functions

#12

Post by enriquto »

ttoinou wrote: post #8
enriquto wrote: post #7

Earlier quoted context omitted.

> 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

> 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.

Re: Visualizing Complex Functions

#13

Post by navane »

enriquto wrote: post #12
ttoinou wrote: post #8

Earlier quoted context omitted.

a 2D point is not enough to define complex numbers

> 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.

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.

Re: Visualizing Complex Functions

#14

Post by dragonwriter »

enriquto wrote: post #3
Jedi72 wrote: post #2

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…

a complex number is a point in the plane (in the same way as a real number is a point in a straight line)

> 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.

Re: Visualizing Complex Functions

#15

Post by ttoinou »

enriquto wrote: post #12
ttoinou wrote: post #8

Earlier quoted context omitted.

a 2D point is not enough to define complex numbers

> 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.

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)

Re: Visualizing Complex Functions

#16

Post by mitchtbaum »

Jedi72 wrote: post #2

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…

Steven Wittens made good progress toward an intuitive footing in https://acko.net/blog/how-to-fold-a-julia-fractal/

Re: Visualizing Complex Functions

#17

Post by enriquto »

ttoinou wrote: post #15
enriquto wrote: post #12

Earlier quoted context omitted.

> 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.

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)

> 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.

Re: Visualizing Complex Functions

#19

Post by ttoinou »

enriquto wrote: post #17
ttoinou wrote: post #15

Earlier quoted context omitted.

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)

> 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.

Yes

Re: Visualizing Complex Functions

#20

Post by Jedi72 »

user2994cb wrote: post #11
Jedi72 wrote: post #2

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…

You can construct the complex numbers quite nicely with Geometric Algebra: https://www.youtube.com/watch?v=PNlgMPzj-7Q

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.

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