B Is imaginary "i" a purely aesthetic matter?

  • Thread starter fbs7
  • Start date
321
32
Say that I define a set of pairs called ℂ, such that

[a,b] ∈ ℂ iff
a ∈ ℝ, b ∈ ℝ,
[a,b]+[c,d] = [a+c,b+d]
[a,b]*[c,d] = [ac-bd, ad+bc]

Then this has exactly the same properties of a+bi, does it not? You can write any equation that uses i exactly the same way with those pairs, so all interesting thing properties that the normal ℂ has this guy will also have.

Now, this pair notation lacks the wow factor that i has, like "Whaaaaaat!!! Square root of -1!!! Get your eyes away from my daughter you crazy mathematician!!!". Or "Huh? What do you mean an Imaginary number????"

But, maybe more crucial, the equations would look less mysterious and attractive without i all over the place; for example this doesn't look enchanting at all

##e^{[0,1]*[\pi,0]} = [-1,0]##

So, is it possible that mathematicians like i just because it leads to compact, attractive and mysterious equations -- that is, the choice of using it all the time is just basically aesthetic?
 

Drakkith

Staff Emeritus
Science Advisor
2018 Award
20,348
4,060
So, is it possible that mathematicians like i just because it leads compact, attractive and mysterious -- that is, the choice of using it all the time is just basically aesthetic?
I'm betting they like it because it links the real numbers to a more abstract number system and gives useful answers when used in many problems. It's one thing to simply invent a set of numbers that obey these properties, but it's another thing to discover that all those square roots of negative numbers suddenly make a lot of sense and you no longer have to throw them away as invalid answers.
 

WWGD

Science Advisor
Gold Member
3,931
1,650
One way/reason for defining i as it has been done is that it allows for finding roots of polynomial of the type x^2+b . Formally, this is a field extension R[X]/(x^2+1), from which it follows that x^2+1 =0 , so that x^2 = -1. This gels together when having x=i so that x^2 =-1. So the construction/definition of i is not, in this sense, just formal, aesthetic.
 
321
32
Hmm... I see... so the incentive there is to find solutions for polynomials, I see.

Now, in the same way that I defined a particular multiplication rule [a,b]*[c,d] for vectors with 2 elements ( which also allows me to solve X^2 + 1 = 0 ), say that I extend it for bigger vectors as

A = [a1,a2,...an]

and I define

A * B = C

where

cn = ∑ tr(n,i,j) * a(i) * b(j)

where tr(n,i,j) is some transformation; for the 2-element vectors in the beginning we have

c1 = 1 * a1 b1 + 0 * a1b2 + 0*a2b1 + (-1) * a2 b2
c2 = 0 * a1 b1 + 1 * a1b2 + 1*a2b1 + (-1) * a2 b2

So, if ℂ is defined in terms of solving x2+1 = 0, then it stops there, but if we define ℂ as vectors with some particular multiplication rule, then it can not only solve X2+1 = 0 but also we can extend ℂ to multiple dimensions.

Or, asking another way, what is it special that i has, that a 2-dimensional vector doesn't have? Meanwhile 3-dimension, 4-dimension, etc... vectors probably have characteristics that i doesn't have (that is, I'm guessing that, I'm not really a mathematician, just a curious old dude).
 

Drakkith

Staff Emeritus
Science Advisor
2018 Award
20,348
4,060
Or, asking another way, what is it special that i has, that a 2-dimensional vector doesn't have? Meanwhile 3-dimension, 4-dimension, etc... vectors probably have characteristics that i doesn't have (that is, I'm guessing that, I'm not really a mathematician, just a curious old dude).
You might be interested in quaternions, octonions, etc. They are extensions of complex numbers just like complex numbers are extensions of the real numbers.
 

WWGD

Science Advisor
Gold Member
3,931
1,650
One thing to consider is that there is no one way of looking at this. Complex numbers have different types of structure, being a field is just one such type.
Think of a person. They may be a man/woman, parent, lawyer, etc. so there are many types of structure associated with Complex numbers.
 

fresh_42

Mentor
Insights Author
2018 Award
9,492
6,502
[a,b] ∈ ℂ iff
a ∈ ℝ, b ∈ ℝ,
[a,b]+[c,d] = [a+c,b+d]
[a,b]*[c,d] = [ac-bd, ad+bc]
You did just this:
One way/reason for defining i as it has been done is that it allows for finding roots of polynomial of the type x^2+b . Formally, this is a field extension R[X]/(x^2+1), from which it follows that x^2+1 =0 , so that x^2 = -1.
Aesthetic is a relative term. The method above can be found in textbooks about abstract algebra, e.g. van der Waerden. The main purpose is, however, not doing algebra on ##\mathbb{R}[x]/\langle x^2+1\rangle## but doing calculus on ##\mathbb{C}##. For this purpose, it is better to define ##\mathbb{C}## via Cauchy sequences, i.e. a topological approach.

Historically it was a mixture of both: The complex numbers solve analytic equations!
 
32,372
8,350
There are many ways to represent complex numbers. You just rewrote a+bi as [a,b]. There is also a 2x2 matrix representation where you can keep the usual matrix addition and multiplication: [[a,-b],[b,a]].

Writing complex numbers as a+bi is compact and useful, but not the only option.
 
321
32
There are many ways to represent complex numbers. You just rewrote a+bi as [a,b]. There is also a 2x2 matrix representation where you can keep the usual matrix addition and multiplication: [[a,-b],[b,a]].

Writing complex numbers as a+bi is compact and useful, but not the only option.
Oh, wow! That's interesting! So a complex number and a 2x2 matrix of real numbers can be represented the same way!!!

But then I'm confused. That 2x2 matrix is just a particular type of matrix; there are other types of matrices too, for example I once heard of a Hermitian matrix (although I forgot what that is), which supposedly could be written as a one-line sequence of numbers with the help of some special symbols, but people don't get very animated about calling them a number system or extension to ℝ

Maybe the popularity of ℂ is more due to, as you said, compacticity and utility. I for one loved when I found I could write electrical voltage and currents in terms of R+iX instead of the crazy hubbabboo with sines and cosines - that is so practical!!! But then I never thought that "i" meant anything other than an extremely useful shorthand for sines and cosines.
 
32,108
3,993
There are many ways to represent complex numbers. You just rewrote a+bi as [a,b]. There is also a 2x2 matrix representation where you can keep the usual matrix addition and multiplication: [[a,-b],[b,a]].
Interesting! I'd never run into this before.

Just to test this, I tried it on examples of addition and multiplication, using 2 + 3i and 5 - 2i
Addition
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} + \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}7 & -1 \\ 1 & 7 \end{bmatrix}##
The last matrix represents 7 + 1i, which is the sum of 2 + 3i and 5 - 2i
Multiplication
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}16 & -11 \\ 11 & 16 \end{bmatrix}##
Same result that you get by multiplying (2 + 3i) and (5 - 2i).
 

PeroK

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
8,787
3,150
Interesting! I'd never run into this before.

Just to test this, I tried it on examples of addition and multiplication, using 2 + 3i and 5 - 2i
Addition
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} + \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}7 & -1 \\ 1 & 7 \end{bmatrix}##
The last matrix represents 7 + 1i, which is the sum of 2 + 3i and 5 - 2i
Multiplication
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}16 & -11 \\ 11 & 16 \end{bmatrix}##
Same result that you get by multiplying (2 + 3i) and (5 - 2i).
I got into a debate a while back on a non-maths forum where there was an obsession that maths didn't need complex numbers. These matrices were given as virtual proof that complex numbers don't exist! I.e. complex numbers are not "numbers", but really 2x2 matrices in disguise!
 

fresh_42

Mentor
Insights Author
2018 Award
9,492
6,502
I got into a debate a while back on a non-maths forum where there was an obsession that maths didn't need complex numbers. These matrices were given as virtual proof that complex numbers don't exist! I.e. complex numbers are not "numbers", but really 2x2 matrices in disguise!
The same people would probably argue that this is a redundant representation and we could simply write ##a+ib## if it was the other way around. I guess physicists face similar problems: everything that isn't a spatial dimension appears manmade. Guess we're still a young species.
 
1,321
314
On the philosophical level, - if one asserts that math is a language, and asks "do imaginary numbers exist"? We can view the cases above as different ways to say the same thing.

The mathematical power of Eulers identity alone, to effectively analyze and communicate details of the world around us, one could almost argue that "i" is a necessity - or a fundamental "word" portraying meaning in a concise manner.

To me no different then the number 5, in what ways does the number 5 "exist"? We can have 5 things, we can multiply by 5,,, etc... but this is a mathematical representation of what we observe.
 

WWGD

Science Advisor
Gold Member
3,931
1,650
On the philosophical level, - if one asserts that math is a language, and asks "do imaginary numbers exist"? We can view the cases above as different ways to say the same thing.

The mathematical power of Eulers identity alone, to effectively analyze and communicate details of the world around us, one could almost argue that "i" is a necessity - or a fundamental "word" portraying meaning in a concise manner.

To me no different then the number 5, in what ways does the number 5 "exist"? We can have 5 things, we can multiply by 5,,, etc... but this is a mathematical representation of what we observe.
I always disagreed with the claim that Math is a language. Math is a system and it has its own language I think is more accurate.
 
1,321
314
Kinda splitting hairs - yes mathematical concepts are "true" independent of the Language of Mathematics.... But a Real Number is a concept, an Imaginary Number is a concept.... If people want to debate if imaginary numbers "exist" trying to settle or define difference between a language and the concepts it represents and communicates is probably too much.
 

WWGD

Science Advisor
Gold Member
3,931
1,650
Kinda splitting hairs - yes mathematical concepts are "true" independent of the Language of Mathematics.... But a Real Number is a concept, an Imaginary Number is a concept.... If people want to debate if imaginary numbers "exist" trying to settle or define difference between a language and the concepts it represents and communicates is probably too much.
I mean, I would include as part of the conditions that something is a language that it does not have an intrinsic subject matter; language is a tool for communicating as I see it. Math does have an intrinsic subject matter and it is not just used as a form of communicating. But you may have a different layout for Mathematics, language, etc.
 

fresh_42

Mentor
Insights Author
2018 Award
9,492
6,502
I always disagreed with the claim that Math is a language.
Well, it has an alphabet, a syntax, production rules, a grammar and everything a language needs. It's even partly context sensitive. But I also consider music as a language.
 
321
32
Let me ask the other way around.

We see "i" as an useful extension to ℝ because it solves x2+1=0, therefore it makes a group (if my language is right) out of algebraic operators +/-/*// (my language is most probably not right). To what I know, we don't need anything else to solve these polynomial expressions, so, fair is far, that's a nice extension.

Is it possible that are there other extensions to ℝ that provide solutions to other types of non-polynomial problems? For example, if I were to define a number j such that sin(j) = 2, everybody would most likely find that pretty useless (otherwise someone else would have thought of that already), but is it possible that there are other extensions beyond ℂ that would be useful for something, or do the mathematicians think that there's nothing beyond ℂ, and that's really the final frontier?

Pardon again to my imprecise language, maybe a mathematician might help in stating the question in a more... errr... mathematical way.
 

WWGD

Science Advisor
Gold Member
3,931
1,650
Well, it has an alphabet, a syntax, production rules, a grammar and everything a language needs. It's even partly context sensitive. But I also consider music as a language.
Yes, but it has intrinsic content. As I understand it, language is a tool for communicating ideas and has no intrinsic content. Mathematics does have intrinsic content and , IMO uses a special language to deal with a specific albeit somewhat-fuzzily-defined subject matter.
 

fresh_42

Mentor
Insights Author
2018 Award
9,492
6,502
I prefer Chomsky over intrinsic content.
 

Want to reply to this thread?

"Is imaginary "i" a purely aesthetic matter?" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads

Top