Is imaginary "i" a purely aesthetic matter?

[fbs7]

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]

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]

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.

 

[fbs7]

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,...a_n]

and I define

A * B = C

where

c_n = ∑ 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]

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]

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]

[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!

 

[mfb]

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.

 

[fbs7]

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.

 

[Mark44]

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]

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]

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.

 

[Windadct]

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]

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.

 

[Windadct]

Kinda splitting hairs - yes mathematical concepts are "true" independent of the Langauge 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]

Kinda splitting hairs - yes mathematical concepts are "true" independent of the Langauge 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]

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.

 

[fbs7]

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]

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]

I prefer Chomsky over intrinsic content.

 

[WWGD]

I prefer Chomsky over intrinsic content.

I am not familiar with Chomsky's layout, I just know of terms such as generative grammar, etc. Is he still doing Linguistics? I only hear of him re politics.

 

[fresh_42]

Don't make me search my book ... It was sooo long ago ... I have problems with 'intrinsic'. Isn't that true for any language which has a meaning?

 

[WWGD]

Don't make me search my book ... It was sooo long ago ... I have problems with 'intrinsic'. Isn't that true for any language which has a meaning?

I don't think so. Does German, e.g., have any subject matter that distinguishes it from other languages? I think no; it is used to communicate ideas, like any other human language. Some of the content of German may be unique to living conditions in Germany, but not otherwise, I believe.

 

[fresh_42]

I don't think so. Does German, e.g., have any subject matter that distinguishes it from other languages? I think no; it is used to communicate ideas, like any other human language. Some of the content of German may be unique to living conditions in Germany, but not otherwise, I believe.

You cannot translate idioms and sayings and you cannot really translate Shakespeare. These things are a distinction and every language has them, often even dialects. Their existence is intrinsic as well.

 

[WWGD]

You cannot translate idioms and sayings and you cannot really translate Shakespeare. These things are a distinction and every language has them, often even dialects. Their existence is intrinsic as well.

Because there are aspects of life in Germany that do not have a parallel in other countries (e.g., English) , so that the abstract grammatical layout is different to account for this. German language is used to communicate ideas experienced by Germans. Maybe t o make my point more clear ( although I may be wrong) you need an interpretation and assign semantics to a (formal)language. Would you say German language is a Syntactic construct or Semantic, Mixed, etc? I say German language is used as a means to describe the German experience.

 

[DaveE]

A 2-D vector field also has the properties required to be an algebraically defined number. I don't recall them all; operators with identities and inverse operators, existence of zero, etc. That is why so many concepts in analysis can be treated as a complex number or as a 2-D vector. In electronics, for example, the amplitude and phase of a wave, voltage, impedance, etc. are all expressed as a complex number in analysis when they could also be understood as a vector. It is really about choosing a notation that facilitates stealing analysis techniques from another perspective in the world of math.

 

[WWGD]

A 2-D vector field also has the properties required to be an algebraically defined number. I don't recall them all; operators with identities and inverse operators, existence of zero, etc. That is why so many concepts in analysis can be treated as a complex number or as a 2-D vector. In electronics, for example, the amplitude and phase of a wave, voltage, impedance, etc. are all expressed as a complex number in analysis when they could also be understood as a vector. It is really about choosing a notation that facilitates stealing analysis techniques from another perspective in the world of math.

Agreed, choosing the right representation in a context is key, and IMO underrated as a means to finding solutions to a problem.

 

[Klystron]

Let me ask the other way around...[snip]...
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?

Howdy,
Having trouble with example sin (j) = 2 at least using trig functions, as arcsin(2) is undefined. Could approximate sin function with a Taylor series but I assume you are only making a point since no units (degrees, radians) specified. My response, then is that functions (and arithmetic operators) have certain input requirements that apply even for examples.

You might be interested in quaternions, octonions, etc. They are extensions of complex numbers just like complex numbers are extensions of the real numbers.

If Drak's reply does not answer your 'extensions' question, there are many forms and fields beyond complex numbers. Set theory and group theory that you mention in previous posts, have marvelous extensions that I am currently learning, along with attempts at learning more abstract algebras and geometries.

JFTR I am another old dude non-mathematician. Cool thing about mathematics is that we never run out of new forms and ideas to learn.

[edit: forgot: my original degree was technically Math before Computer Science was separated.]

 

[fresh_42]

Because there are aspects of life in Germany that do not have a parallel in other countries (e.g., English) , so that the abstract grammatical layout is different to account for this.

What about the words that do have a correspondence in their meaning, but don't exist anyway? E.g. there is no German word for "sophisticated" and no English word for "schweigen", although the situations in which they are used do exist equally. Then by your definition, those words are intrinsic.

 

[WWGD]

What about the words that do have a correspondence in their meaning, but don't exist anyway? E.g. there is no German word for "sophisticated" and no English word for "schweigen", although the situations in which they are used do exist equally. Then by your definition, those words are intrinsic.

I am kindof reaching, but aren't C,C#, Java, etc. all means/tools for dealing with the problem of how to program? Each language is used to address the same content/issue but it is just that, a vehicle for it. But everyday language starts being an obstacle to discuss these issues.

 

[Klystron]

On the Langauge sub-theme:

If one accepts that language derives from culture expressed in an environment -- Inuit language has n words for 'snow' comes to mind from Anthropology -- then the expression "Mathematics is the language of Science" not only describes math as a language but science as its environment (culture).

 

[WWGD]

On the Langauge sub-theme:

If one accepts that language derives from culture expressed in an environment -- Inuit language has n words for 'snow' comes to mind from Anthropology -- then the expression "Mathematics is the language of Science" not only describes math as a language but science as its environment (culture).

But can't the language aspect be studies separately and independently-from the content? We can make a purely syntactic analysis of Mathematical language as a first-order language. These discussions get weird after a while and seemingly drift of, so sorry if I am not making too much sense.

 

[PeroK]

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?

The importance of $\mathbb{C}$ is that it is an algebraically closed field. In terms of numbers, that's part of the reason that further extensions are not normally considered numbers. But, of course, you have complex vectors and complex matrices etc. And, as others have said, there are quaternions etc.

Note that there do exist solutions to $\sin z = 2$ for complex $z$.

Interestingly, from a physics point of view, Quantum Mechanics is built on complex vector spaces, and complex functions of a real variable. In general, although measurable quantities: position, momentum, spin etc. are real numbers, the underlying structure of QM is complex.

 

[fbs7]

Ohhhhhhh... so the keyword is "algebraically closed". Wow, I'm starting to get it! From wikipedia... "F is algebraically closed field if every non-constant polynomial has a root"... which it says it is equivalent to saying that "F has no proper extension".

That means that ℂ cannot be extended just by polynomials, correct? Does that mean that it can't be extended either with any function that can be expressed as infinite series? I mean, an infinite series is a polynomial, but it's infinite, so is it possible that ℂ can be extended through some non-polynomial function that cannot be expressed as a series?

This is fascinating stuff!

 

[wle]

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.

They really became important for expressing the solutions to cubic polynomials (see here). Basically, cubic polynomials always have (at least) one real root and it is possible to say how to compute it, but you sometimes need complex numbers in the intermediate steps. For example, cubic polynomials of the form $$x^{3} + p x + q$$ have a root given by the Cardano formula, $$x = \Biggl(- \frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}\Biggr)^{1/3} + \Biggl(- \frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}\Biggr)^{1/3} \,.$$ That root is real, but depending on the values of $p$ and $q$ it's possible for the term $\frac{q^{2}}{4} + \frac{p^{3}}{27}$ under the square roots to be negative. In that case the Cardano formula can still be meaningful and gives the correct (real) result, but only if you accept it containing intermediate expressions $\Bigl(a \pm b \sqrt{-1}\Bigr)^{1/3}$ with imaginary parts that end up cancelling out.

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).

Or if you're familiar with the properties of Pauli matrices, $$\begin{eqnarray*}
\bigl( a \, \mathbb{I} + i b \, \sigma_{y} \bigr) \bigl( c \, \mathbb{I} + i d \, \sigma_{y} \bigr) &=& ac \, \mathbb{I} + i (ad + bc) \, \sigma_{y} - b d \, {\sigma_{y}}^{2} \\
&=& (ac - bd) \, \mathbb{I} + i (ad + bc) \, \sigma_{y} \,,
\end{eqnarray*}$$ since ${\sigma_{y}}^{2}$ is the identity. (It doesn't matter if you use $\sigma_{y}$ or $-\sigma_{y}$.)