Motivation for geometrical representation of Complex numbers

[Vinay080]

I am seeing in "slow motion" the development of vectorial system. I am reading the book "A History of Vector Analysis" (by Michael J.Crowe); it seems from my acquaintance that the vector concept came from the quaternions concept; and the quaternions concept came from the act of search for geometrical representation of complex numbers.

And these complex numbers drops down to the concept of √(negative numbers); if complex numbers were true numbers, there is no need for the search of another geometrical representation, as we already have cartesian coordinate system. But, of course, they don't seem to be any usual negative or positive numbers. I thought these numbers to be non-sense (one which doesn't make sense). It seems Gauss, Argand, Buee, Mourey. Warren and Hamilton (?) tried to find a geometrical representation for complex numbers.

According to the angle in which my analysis is going, I don't understand what motivated these all to find a geometrical representation (gr) for Complex numbers. In no thinking route of mine suggest me to find a gr. So, the question is what motivated all these folks to search for a gr of complex numbers? Is gr (of complex numbers) an invention or discovery?

I will be happy if important books and papers are suggested in this regard.

 

[jedishrfu]

With respect to the creation of vector systems that's true. Many scientists rebelled at using quaternions as unnecessarily complicated and so Gibbs borrowed the I, j, k notion and discarded the real part of the quaternion system to create Vector Analysis.

More recently though quaternions have been revived as they handle rotations more naturally.

Some of the history of these developments can found here

https://en.m.wikipedia.org/wiki/Quaternions

 

[Svein]

Is gr an invention or discovery?

As is most of mathematics - the graphical representation is a way of looking at things. It is not the thing itself. But once you accept the definition of multiplication used to define complex numbers, the representation is "sort of" obvious.

That said, the idea of vectors and complex numbers were presented more or less simultaneously way before Gauss - by a Norwegian working in Denmark (see https://en.wikipedia.org/wiki/Caspar_Wessel).

 

[Vinay080]

As is most of mathematics - the graphical representation is a way of looking at things. It is not the thing itself.

Insighful! What is strange (either because of my lack of knowledge or because of its real strage-ness) is that, as said before, if complex numbers are true numbers, there is no need of finding gr (as there already exists cartesian coordinate system); and if they are not true numbers, there is no need then (until there is a reason to give gr), for they are not numbers at all; even then why was gr given?

Though, "geometrical" representation (gr) is not the thing which describes stuff itself, as it is just a way of looking things (as you said), giving gr for non-sense numbers seems to make no sense. We could have kept quite if one guy had tried to give gr, no, but many (Wessel, Gauss, Warren, Buee, Argand, Hamilton) have contributed to it. I don't think they have tried to give gr, without having most probably giving an answer to the question "why am I doing this?", we may say either one person had tried to give gr, and the coming-fellows followed him; but even to follow, it requires them to satisfy their mind to satisfy on why they are following him/her to give gr.

I knew Wessel, I have got his paper "On the Analytical Representation of Direction"; I haven't completed reading. And I don't know technical stuff like dfinition of multiplication used, etc; as I am still starting now, my system will get upgraded within few hours, I hope. Hope you got my question. Thank you

 

[Vinay080]

Some of the history of these developments can found here

https://en.m.wikipedia.org/wiki/Quaternions

I read it; though, I didn't read technical stuff, for I need more muscles. These are the sentences from that link, which seems to be of help (extracted under CC):

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space.

My previous say to Svein stands here. It seems that Hamilton gave gr of complex numbers, simply because he interpreted it and it gave ecstasy for doing that; which seems to be more synthetic than the creation of gr for real numbers, for the measure of space (which I think came from the need). Is this is the only ecstasy interpretation which led everyone to find gr for complex numbers? Is there no better reason for this? If answer for first question is yes, then, we are going to see many many interpretation theories.

 

[Mark44]

Insighful! What is strange (either because of my lack of knowledge or because of its real strage-ness) is that, as said before, if complex numbers are true numbers, there is no need of finding gr (as there already exists cartesian coordinate system)

Complex numbers are numbers -- they just don't happen to be real numbers. In mathematics we don't talk about "true numbers" -- there are integers, rational numbers, real numbers, transcendental numbers (all of which are subsets of the real numbers). Then there are pure imaginary numbers, complex numbers, quaternions, and several more sets of numbers that are not found on the real number line.

; and if they are not true numbers, there is no need then (until there is a reason to give gr), for they are not numbers at all; even then why was gr given?

A graphical representation is an aid to understanding complex numbers. The fact that they aren't "real" is irrelevant, as there are real applications of them in, for example, electronics involving alternating current circuits.

From your quote about Hamilton:

Hamilton knew that the complex numbers could be interpreted as points in a plane...

And as points in a plane, complex numbers behave in ways that are similar to two-dimensdional vectors. When you add two complex numbers, the sum is the same as if you had added two vectors whose magnitudes and directions are the same as the complex numbers. Having a graphical representation makes this much easier to understand.

Furthermore, if you multiply two complex numbers (an operation that vectors don't share), the product is a complex number whose magnitude is the magnitude of the vectors being multiplied, and whose direction is the sum of the angles associated with the two complex numbers. To try to comprehend this without a graphical representation is a lot more difficult than when you have a graphical representation to help you.

 

[Vinay080]

In mathematics we don't talk about "true numbers"

My stand: Langauge involves words; a sense-ible sentence is formed only when words are arranged in correct form, otherwise it forms non-sense sentence. Similarly, a math sentence involves symbols, it forms meaning only when it is arranged in correct form, otherwise it forms non-sense sentence; I see √-1 (and all other square-root of negative numbers) to be such a non-sense (one which doesn't make sense) math statement, which doesn't form true number; and it is just sentence formed formed from worng arragnement of symbols.

A language contains many words, but all the different arrangement of words need not be a meaningful sentence, as it is the case here I think, where a wrong sentence is formed because of wrong arrangement.

Then there are pure imaginary numbers, complex numbers, quaternions, and several more sets of numbers that are not found on the real number line.

All these numbers are from only one wrong pattern √-1. So, there is no "more" sets, there is only one set to be said.

A graphical representation is an aid to understanding complex numbers. The fact that they aren't "real" is irrelevant, as there are real applications of them in, for example, electronics involving alternating current circuits.

That is really about what I am amazed. I look forward to unveil that mystery.

And as points in a plane, complex numbers behave in ways that are similar to two-dimensdional vectors. When you add two complex numbers, the sum is the same as if you had added two vectors whose magnitudes and directions are the same as the complex numbers. Having a graphical representation makes this much easier to understand.

Furthermore, if you multiply two complex numbers (an operation that vectors don't share), the product is a complex number whose magnitude is the magnitude of the vectors being multiplied, and whose direction is the sum of the angles associated with the two complex numbers. To try to comprehend this without a graphical representation is a lot more difficult than when you have a graphical representation to help you.

Okay, I understand, I look forward to learn more on this. I hope this was the main motivation.

 

[Mark44]

In mathematics we don't talk about "true numbers"

My stand: Langauge involves words; a sense-ible sentence is formed only when words are arranged in correct form, otherwise it forms non-sense sentence. Similarly, a math sentence involves symbols, it forms meaning only when it is arranged in correct form, otherwise it forms non-sense sentence; I see √-1 (and all other square-root of negative numbers) to be such a non-sense (one which doesn't make sense) math statement, which doesn't form true number; and it is just sentence formed formed from worng arragnement of symbols.

Again, mathematics does not have the concept of "true" numbers. We have real numbers, the numbers that appear on the real number line, and we have nonreal numbers, including the complex numbers, that are in the complex plane.

A language contains many words, but all the different arrangement of words need not be a meaningful sentence, as it is the case here I think, where a wrong sentence is formed because of wrong arrangement.

In mathematics we can form sentences (equations or inequalities) that are

• Identities (always true) -- for example, ##x^2 - 4 = (x - 2)(x + 2)##
• Conditional statements (sometimes true) -- for example, ##x^2 - 4 = 0##
• Contradictions (never true) -- for example, ##x^2 - 2 = x^2 - 1##

You apparently are under the impression that the imaginary ##i = \sqrt{-1}## is nonsense. The number i is defined to be the number for which ##i^2 = -1##. While it is true that there is no real number whose square is -1, it is not "nonsense" as you put it, because it has applications in the real world. Can something be nonsensical, but still be applied to real concepts?

Here is an example taken from a study guide on AC circuits.

What is the impedance of a network consisting of a 100-ohm-reactance inductor in series with a 100-ohm resistor?

Answer: The impedence is a complex number Z = 100 + j100 ohms. (In Electronics and Engineering, they call the imaginary unit "j", which is just a different name for ##\sqrt{-1}##.)

Every textbook on electronics will have many such examples.

Then there are pure imaginary numbers, complex numbers, quaternions, and several more sets of numbers that are not found on the real number line.

All these numbers are from only one wrong pattern √-1. So, there is no "more" sets, there is only one set to be said.

To sum up for this "wrong pattern" -

• complex numbers, which have the form a + bi, with ##i = \sqrt{-1}## are used to represent the impedance (inductive reactance and capacitive reactance) in AC circuits.
• quaternions, which have the form a + bi + cj + dk, where ##i = j = k = \sqrt{-1}## are used in electrodynamics and computer graphics (see https://en.wikipedia.org/wiki/Quaternion, in the section titled "Quaternions and the geometry of R³".
• octonions, which I haven't mentioned before (see https://en.wikipedia.org/wiki/Octonion) have application in string theory, special relativity, and quantum logic.

For such a "wrong pattern" it turns out to be surprisingly useful.