Views On Complex Numbers

[fresh_42]

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.

Separating complex numbers into real and imaginary parts is exactly what this article wants to put into the second row of consideration, and viewing complex numbers as elements of one field in the first place rather than reducing them to a simple, real vector space.

This narrowed view as $a+ib$ is in my opinion what hides the beauty of complex analysis, or the algebraic background of complex numbers. It is a widespread disease and not really helpful. This article was all about
$$
\mathbb{C} \neq \mathbb{R}^2.
$$

 

[FactChecker]

This narrowed view as $a+ib$ is in my opinion what hides the beauty of complex analysis, or the algebraic background of complex numbers. It is a widespread disease and not really helpful. This article was all about
$$
\mathbb{C} \neq \mathbb{R}^2.
$$

The geometry of the complex plane is very important in many applications of complex analysis. IMO, a lot is lost when it is only considered algebraically.

 

[fresh_42]

The geometry of the complex plane is very important in many applications of complex analysis. IMO, a lot is lost when it is only considered algebraically.

It is a tool, and shouldn't be the central view that it often unfortunately is. The complex plane supports the perspective of a two-dimensional real vector space. You can throw complex analysis in the trash with this limited view (sounded better in German).
$$
\begin{pmatrix}0\\1\end{pmatrix}\cdot \begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\end{pmatrix}
$$
is crucial!

 

[martinbn]

Separating complex numbers into real and imaginary parts is exactly what this article wants to put into the second row of consideration, and viewing complex numbers as elements of one field in the first place rather than reducing them to a simple, real vector space.

It is not reducing, it is using it.

This narrowed view as $a+ib$ is in my opinion what hides the beauty of complex analysis, or the algebraic background of complex numbers. It is a widespread disease and not really helpful.

Why is it not helpful? It definitely is for some things.

This article was all about
$$
\mathbb{C} \neq \mathbb{R}^2.
$$

This is not true without any context. The opposite is true.

Ps. Do you have a similar objection for the Gaussian integers?

 

[fresh_42]

I am simply of the opinion that the complex numbers are primarily a field and not a real vector space. Putting the perspective as a vector field in front is in my opinion stupid. You can be of different opinion. I already know that old habits die hard, regardless of how questionable they are.

The Gaussian integers are primarily a ring and not a $\mathbb{Z}$-module. I guess, I have the same objections.

 

[fresh_42]

This [$\mathbb{C}\neq \mathbb{R}^2$, ed.] is not true without any context. The opposite is true.

What??? This is a clear misinformation!
$$
\underbrace{\mathbb{C}}_{\nearrow \text{ field }} \quad \neq \quad \underbrace{\mathbb{R}^2}_{\text{ no field }\nwarrow}
$$

 

[dextercioby]

Yes, but he probably meant vector spaces over $\mathbb R$.

 

[fresh_42]

Yes, but he probably meant vector spaces over $\mathbb R$.

This is at the same level as saying they are both additive groups. $\mathbb{C}$ and $\mathbb{R}^2$ have a common understanding as, resp. field, two-dimensional real vector space. Of course, one can always right them as $\mathbb{C}=\left(\mathbb{C},s_1,\ldots,s_n\right)$ or $\mathbb{R}^2=\left( \mathbb{R}^2, t_1, \ldots , t_m \right)$ where $s_i , t_j$ represent the countless structures they can carry but nobody does this.

 

[FactChecker]

Complex analysis is usually a required course for Engineers. The geometry of analytic functions is critical to their use. IMO, approaching complex analysis from an abstract algebra perspective would be a mistake. I would follow Ahlfors.

 

[Agent Smith]

Then what is the difference from the real numbers?

Square roots of negative numbers are imaginary numbers e.g. $\sqrt {-3}$

 

[Agent Smith]

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


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


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.


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.

That interesting property being $e^{i \pi} + 1 = 0$?

We've already defined imaginary numbers as ... square roots of negative numbers, right?

As far as I know, an imaginary circle should have imaginary dimensions.

P.S. Danke for the advice.

 

[FactChecker]

Many people consider Euler's formula, $e^{i\theta} = \cos(\theta)+i\sin(\theta)$, to be the most important equation in mathematics. I like its geometric meaning.

 

[martinbn]

Square roots of negative numbers are imaginary numbers e.g. $\sqrt {-3}$

If you ignore multiplication and consider only addition, the imaginari numbers are indistiguishable from the reals.