Views On Complex Numbers

[fresh_42]

Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted to gather the many different views that can be found elsewhere - Euler's and Gauß's perspectives, i.e. various historical views in the light of the traditionally parallel development of mathematics and physics, e.g. the use of complex coordinates in kinematics, the analytical or topological views, e.g. the Radish or the mysterious Liouville's theorem about bounded entire functions that are already constant, or the algebraic view that led to the many non-algebraic proofs of the fundamental theorem of algebra. The complex numbers have so many faces and appear in so many contexts that I could as well have written a list of bookmarks. All of that is true to some extent. The real reason is, that I want to break the automatism of the association of complex numbers with, and the factual reduction to points in the Gaußian plane
$$
\mathbb{C}=\{a+i b\,|\,(a,b)\in \mathbb{R}^2\}\neq \mathbb{R}^2.
$$
We need two dimensions to visualize complex numbers but that doesn't make them two-dimensional. They are a one-dimensional field in the first place, i.e. a single set of certain elements that obey the same axiomatic arithmetic rules as the rational numbers do. They are one set that is not just a plane! The reason they exist and bar us from visual access is finally a tiny positive distance we can see.

Continue reading...

 

[Hornbein]

Complex numbers contain two independent quantities so they are two-dimensional. An Excel spreadsheet with a hundred columns can be one-hundred dimensional. And so forth.

 

[fresh_42]

Complex numbers contain two independent quantities so they are two-dimensional. An Excel spreadsheet with a hundred columns can be one-hundred dimensional. And so forth.

Sorry, but this is wrong, and why I wrote that article. Complex numbers are scalars (one quantity) and as such one-dimensional. Every field is a one-dimensional vector space over itself. Please read the article and its purpose before you spread misinformation. They can be viewed as a two-dimensional real vector space, but this is a reduction. A reduction that doesn't allow complex analysis. Moreover, the complex numbers are an infinite-dimensional rational vector space. So claiming they are two-dimensional is as right and wrong as it is to claim they are infinite-dimensional. Both perspectives are insufficient to perform analysis and physics.

A spreadsheet in Excel is two-dimensional, no matter how many columns it has. You can make an Excel file three-dimensional by adding more sheets to the same file, but that's it.

 

[martinbn]

Some people call certain objects "comex curves", other people call them "Riemann surfaces".

 

[MidgetDwarf]

Complex numbers contain two independent quantities so they are two-dimensional. An Excel spreadsheet with a hundred columns can be one-hundred dimensional. And so forth.

To further fresh_42 response.

It is essentially as he says. Every field is a one dimensional vector space over itself.

If you want to understand why. Then it requires first learning about field extensions. Send me a pm if you are interested in source.

I do not want to derail this insight.

 

[fresh_42]

It makes a difference whether you first associate ##z## or ##a+i b## with a complex number. And it makes a difference if your view guides you or you guide your views. I prefer the latter, but this may mean cutting old braids. If people are used to all possible ways to look at complex numbers then there is nothing to learn here, except perhaps a nice proof of the FTA. The perspective of a two-dimensional real vector space is narrowing the situation. It hides that complex numbers are a field, an algebra, a ring. All those perspectives are lost if we concentrate on ##\mathbb{R}^2=\{(a,b)\}## instead of ##\mathbb{C}=\{z\}.##

It is written for students who aren't trained to accept only one view. It was written to keep in mind that ##\mathbb{C}## is a field in the first place, not a vector space and even less a plane. Cauchy's work is the real achievement in calculus, neither Newton's nor Leibniz's. And Cauchy's theorems are not linear algebra.

 

[martinbn]

One has to keep in mind that the set of complex numbers is not just a field. It is also a topological space and as such is homeomorphic to the plane and in this context dimension two is the correct one.

 

[Haborix]

Perhaps the next insight article should be about the not-so-trivial dimensions of mathematical objects.

 

[fresh_42]

One has to keep in mind that the set of complex numbers is not just a field. It is also a topological space and as such is homeomorphic to the plane and in this context dimension two is the correct one.

I haven't said that this is an incorrect view, except if it is reduced to ##\mathbb{R}^2##. I think, it just shouldn't be the first view. But, hey, let's consider it as vector space over the rationals.

 

[fresh_42]

Perhaps the next insight article should be about the not-so-trivial dimensions of mathematical objects.

Do you have some specific objects in mind? I'd prefer a topic that isn't in the category of "religion" like arithmetic rules or complex numbers. Those topics are of the kind that literally everybody has to say something about, everybody has a different view and complains if you don't hit their perspective, everybody knows it better, but none of those critics wrote it down before. It is tedious.

How about history? I have a rather thick book about the history of mathematics between 1700 and 1900. It is a bit biased towards French mathematicians since it is by a French author, but not as much as the Britannica is biased towards British people.

 

[martinbn]

Perhaps the next insight article should be about the not-so-trivial dimensions of mathematical objects.

The Krull dimenssion of any field is zero. From that point of view the field of complex numbers is a point.

 

[Haborix]

Do you have some specific objects in mind? I'd prefer a topic that isn't in the category of "religion" like arithmetic rules or complex numbers. Those topics are of the kind that literally everybody has to say something about, everybody has a different view and complains if you don't hit their perspective, everybody knows it better, but none of those critics wrote it down before. It is tedious.

How about history? I have a rather thick book about the history of mathematics between 1700 and 1900. It is a bit biased towards French mathematicians since it is by a French author, but not as much as the Britannica is biased towards British people.

As to your initial question, not really. But inspired by some of the discussion in this thread, it could be interesting to survey the different way dimension is used in mathematics and/or how the same object, e.g., complex numbers, have many different dimensions depending on how you look at them. It doesn't have to be exhaustive or advocate a point of view. The latter is probably what usually invites the most nitpicking (well, more nitpicking than I already expect from mathematicians ).

 

[fresh_42]

As to your initial question, not really. But inspired by some of the discussion in this thread, it could be interesting to survey the different way dimension is used in mathematics and/or how the same object, e.g., complex numbers, have many different dimensions depending on how you look at them. It doesn't have to be exhaustive or advocate a point of view. The latter is probably what usually invites the most nitpicking (well, more nitpicking than I already expect from mathematicians ).

Dimension is indeed an interesting topic! There came at least half a dozen keywords to my mind without doing any research (17 so far). And it is not as trivial as it sounds! Let me take the chance and make a little survey here:

a) Does any of the participants, critics, or readers here want to write that article instead of me?

b) Would you mind giving me some keywords I should not forget to mention?

I have already read Krull, and topology. However, the latter was a bit unprecise and I'm not sure I know what had been meant in post #7, i.e. which objects and topologies had been meant. I mean, since zero wasn't excluded, it sounded like the comparison of additive groups, but that is guesswork from my side. Could you elaborate on what you meant @martinbn in post #7?

 

[bhobba]

Apart from Euler's relation, which IMHO is one of the greatest formulas in all of math, the thing I find interesting about complex numbers is the remarkable theorems of complex analysis.

My favourite is Analytic Continuation and its use is summing divergent or otherwise meaningless sums. It may seem a pure math pastime, but Borel Summation (which is justified by analytic continuation - reversing the sum and integral can only be done on a region, but by analytic continuation can be extended to the whole complex plane) is very useful in differential equations:

https://math.osu.edu/~costin/adiab.pdf

For this reason, Borel Summation is included in many Engineering Mathematics textbooks such as Advanced Engineering Mathematics by Lopez and Benders Lectures on Mathematical Physics (a treasure trove of all sorts of interesting stuff). But, naughty, naughty, they do not explain why you can reverse the sum and integral.



Thanks
Bill

 

[docnet]

Definitely reading this!

 

[martinbn]

Could you elaborate on what you meant @martinbn in post #7?

A topological manifold is a topological space, which is locally homwomorphic to ##\mathbb R^n##. This ##n## is the dimenssion of the topological manifold. As such, ##\mathbb C## is two dimenssional.

 

[fresh_42]

A topological manifold is a topological space, which is locally homwomorphic to ##\mathbb R^n##. This ##n## is the dimenssion of the topological manifold. As such, ##\mathbb C## is two dimenssional.

This argument is circular. You start with a real Riemann plane and conclude that it is two-dimensional.

 

[martinbn]

This argument is circular. You start with a real Riemann plane and conclude that it is two-dimensional.

This is not an argument. I am not trying to prove anything. This is just another way to look at things.

 

[fresh_42]

This is not an argument. I am not trying to prove anything. This is just another way to look at things.

Just a remark on the Krull dimension. I haven't done the math but Wiki says that ##\dim_K(\mathbb{C},\text{Hausdorff})=0## and ##\dim_K(\mathbb{C},\text{Zariski})=1.##

That projected article about dimensions gets longer the closer I look at it.

 

[martinbn]

Just a remark on the Krull dimension. I haven't done the math but Wiki says that ##\dim_K(\mathbb{C},\text{Hausdorff})=0## and ##\dim_K(\mathbb{C},\text{Zariski})=1.##

That projected article about dimensions gets longer the closer I look at it.

Which wiki page is that?

 

[fresh_42]

Which wiki page is that?

https://de.wikipedia.org/wiki/Krulldimension#Beispiele

 

[Klystron]

Feedback on this complex numbers Insights article:

I liked the emphasis on Euler's Equation and veiwing complex numbers as fields. Since I first encountered Euler and complex numbers learning EM fields as a youngster, the descriptions seem natural and explanatory without topological confusion.

I agree an Insights article, or a series of articles, on mathematical dimensions to be useful and topical based on reading many threads and even some linked papers that do exhibit confusion.

 

[martinbn]

https://de.wikipedia.org/wiki/Krulldimension#Beispiele

That is not what meant. I meant the Krull dimension of a ring, then any field has dimension 0.

 

[fresh_42]

That is not what meant. I meant the Krull dimension of a ring, then any field has dimension 0.

Added to the list. It's hard to find a dimension without ideals.

I have meanwhile 21 keywords. And I haven't even looked up the history which should be exciting, too, especially topology and fractals, let alone ...

... some linked papers that do exhibit confusion.

And Greg said that he does not like articles split into parts.

 

[mathwonk]

that is a rather interesting confusion about whether the algebraic dimension of C is zero or one, using krull dimension. of course as a ring (field) it has only one prime ideal so krull dim zero, but as an algebraic set it is isomorphic to the maximal spectrum of the polynomial ring C[Z], which has krull dim one. fun!

one can have such lengthy arguments about words when one does not give their definitions.

 

[Hans de Vries]

Thank you for the article. Just passing by for this:

The complex algebra is identical to the algebra of 2x2 real matrices (homomorphic both ways)

\begin{equation}\nonumber
\begin{array}{c}
\mathbf{w} = (a + ib)\,\mathbf{u} ~+~ (c + id)\,\mathbf{u}^*
\\ \\
\equiv
\\ \\
\left[
\begin{array}{c}
\mathbf{w}_r \\
\mathbf{w}_i
\end{array}
\right]
=
\left(\,
\left[\,
\begin{array}{cc}
a & 0 \\
0 & a
\end{array}
\,\right]
+
\left[\,
\begin{array}{cc}
0 &\!\! -b \\
b & 0
\end{array}
\,\right]
+
\left[\,
\begin{array}{cc}
c & 0 \\
0 &\!\! -c
\end{array}
\,\right]
+
\left[\,
\begin{array}{cc}
0 & d \\
d & 0
\end{array}
\,\right]
\,\right)
\left[
\begin{array}{c}
\mathbf{u}_r \\
\mathbf{u}_i
\end{array}
\right]
\end{array}
\end{equation}

Here we used all four (2x2) orthonormal generators of the 2x2 matrices which are either ##\sqrt{I}## or ##\sqrt{-I}##.

Any real 2x2 matrix can be decomposed into these four generators using the element-wise dot product of the arbitrary real 2x2 matrix and the generators.

The i-generator is the only one which is a ##\sqrt{-I}## generator. The other three are all ##\sqrt{I}##.

( Note: Interestingly the last two generators are anti-commutative ##i\mathbf{u}^* =-( i\mathbf{u})^* ## within an otherwise "commutative algebra" )

 

[FactChecker]

You can ignore the geometric aspects of complex analysis in two dimensions, but IMO that misses a lot of the beauty of the subject. I might be biased because my field within complex analysis was geometric function theory.

 

[Agent Smith]

Why not study pure complex numbers. Drop the real part and simply investigate the imaginary part. It is a number line too, just like the real number line, oui? Is there an imaginary π, a πi? Don't we have LaTex on this site? An imaginary circle, looks like will have to have a real circumference or diameter.

 

[martinbn]

Why not study pure complex numbers. Drop the real part and simply investigate the imaginary part. It is a number line too, just like the real number line, oui? Is there an imaginary π, a πi? Don't we have LaTex on this site? An imaginary circle, looks like will have to have a real circumference or diameter.

When you multiply two of those you get a real number ##i\cdot i =-1##.

 

[FactChecker]

When you multiply two of those you get a real number ##i\cdot i =-1##.

So you're one of those people who believe in multiplication? It's a Deep State trick! ;-)

 

[martinbn]

So you're one of those people who believe in multiplication? It's a Deep State trick! ;-)

Multiplication is the key to survival.

 

[Agent Smith]

When you multiply two of those you get a real number ##i\cdot i =-1##.

Jene sais pas, but we could exclude these anomalies from our study, oui? Even division seems to be a problem since ##\frac{ai}{bi} = \frac{a}{b}##. We would be restricted to addition and subtraction i.e. the set of purely imaginary numbers would be closed under addition and subtraction only. Is ##0## both real AND imaginary? The complex and real lines do intersect at ##(0, 0)##.

 

[martinbn]

Jene sais pas, but we could exclude these anomalies from our study, oui? Even division seems to be a problem since ##\frac{ai}{bi} = \frac{a}{b}##. We would be restricted to addition and subtraction i.e. the set of purely imaginary numbers would be closed under addition and subtraction only. Is ##0## both real AND imaginary? The complex and real lines do intersect at ##(0, 0)##.

Then what is the difference from the real numbers?

 

[pbuk]

Why not study pure complex numbers. Drop the real part and simply investigate the imaginary part.

You can if you wish. The first thing you should find out is that these are simply called imaginary numbers.

Is there an imaginary π, a πi?

I am not sure what you mean by an 'imaginary π' but the number ## i \pi ## has a very interesting property.

Don't we have LaTex on this site?

You can see ## \LaTeX ## Math expressions in some of the posts in this thread. There is a tutorial at https://www.physicsforums.com/help/latexhelp.

An imaginary circle...

When we want to think about something in Mathematics the first thing we need to do is define exactly what it is we want to think about. I am not sure what you think you mean by an 'imaginary circle' but I do think that you will find it difficult to find a definition that works.

 

[Vanadium 50]

So you're one of those people who believe in multiplication?

It's in the Bible and everything - be fruitful and multiply!

The concept being discussed is called closure. If I have c = a ⊗ b where ⊗ is addition, subtraction, multiplication or division, and a and b are both real, so is c. If a and b are both complex, so is c. If a and b are purely imaginary, c might or might not be - it is not closed.

This makes purely imaginary numbers less useful.