Is it valid to express a complex number as a vector?

In summary,complex multiplication is not the same as vector multiplication. Complex numbers can be visualized as a real vector space, but the advantage of this is lost once you want to do analysis. Complex numbers are valuable for their geometry, but the average student should be aware of the field character of the complex numbers and use them cautiously.
  • #1
Mayhem
353
251
...and is it ever useful?

An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may be expressed as ##z = a+bi = \vec{r} \cdot \vec{c} = \binom{a}{b} \cdot \binom{1}{i}## where ##\vec{r}## denotes the real part and ##\vec{c}## the imaginary part.

Then in the complex plane, their composite vector ##\vec{p}## may be expressed using their magnitudes, giving ##\vec{p} = \binom{||r||}{||c||} = \binom{\sqrt{a^2+b^2}}{\sqrt{2}i}##

Math seems valid unless I made a stupid mistake. Is this ever useful?
 
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  • #3
Mayhem said:
...and is it ever useful?

An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may be expressed as ##z = a+bi = \vec{r} \cdot \vec{c} = \binom{a}{b} \cdot \binom{1}{i}## where ##\vec{r}## denotes the real part and ##\vec{c}## the imaginary part.

Then in the complex plane, their composite vector ##\vec{p}## may be expressed using their magnitudes, giving ##\vec{p} = \binom{||r||}{||c||} = \binom{\sqrt{a^2+b^2}}{\sqrt{2}i}##

Math seems valid unless I made a stupid mistake. Is this ever useful?
It depends on what you want to do.

The complex numbers are a two-dimensional real vector space ##V##: ##a+ib \mapsto (a,b).##

If we want to keep the multiplication, we have now first to define a two-dimensional, real algebra by
$$
(a\, , \,b) \circ (c\, , \,d) := (ac-bd\, , \,ad+bc)
$$
which is a multiplication that seems weird from the point of view of a vector space. Furthermore, all multiplication rules have to be proven again. Being a real vector space, we have an additional inner product defined by ##(a,b)\cdot (c,d)=ab+cd \in \mathbb{R}## which is not directly related to complex numbers and can be a source of confusion.

So, yes, you can consider the complex numbers as a real vector space ##V##, but even the natural process of complexification
$$
V_\mathbb{C}=V\otimes_\mathbb{R} \mathbb{C}
$$
will end up in a total mess if you aren't very cautious; let alone complex calculus!

The complex numbers are usually only seen as a real vector space if we want to draw them. This is in my opinion already misleading and creates misconceptions. There is little advantage in such a concept except for being an example of a two-dimensional, real vector space. This advantage will be lost the moment you want to use them as a field and do analysis.
 
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  • #4
fresh_42 said:
The complex numbers are usually only seen as a real vector space if we want to draw them. This is in my opinion already misleading and creates misconceptions. There is little advantage in such a concept except for being an example of a two-dimensional, real vector space. This advantage will be lost the moment you want to use them as a field and do analysis.
I tend to disagree with this. The geometry of complex multiplication, analysis, conformal mapping, etc., is beautifully seen in two dimensions.
 
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  • #5
FactChecker said:
I tend to disagree with this. The geometry of complex multiplication, analysis, conformal mapping, etc., is beautifully seen in two dimensions.
This likely depends on what should be visualized. I think of branching, Cauchy, Stokes, and that even ##f(z)=z^2## cannot be drawn anymore.
 
  • #6
fresh_42 said:
This likely depends on what should be visualized. I think of branching, Cauchy, Stokes, and that even ##f(z)=z^2## cannot be drawn anymore.
You're not a fan of the argand diagram?

Using the geometry of the complex plane is invaluable to the average student.
 
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  • #7
PeroK said:
You're not a fan of the argand diagram?
No.
Using the geometry of the complex plane is invaluable to the average student.
I think the field character gets lost. Even ##\mathbb{R}[x]/(x^2+1)## looks more like a vector space than a field. But the field property is essential for analysis.

However, I admit that this is a personal opinion. I haven't tried to figure out whether my lack of intuition is due to the vector space image or only a matter of the fact that complex numbers cannot be totally ordered.
 
  • #8
fresh_42 said:
This likely depends on what should be visualized. I think of branching, Cauchy, Stokes, and that even ##f(z)=z^2## cannot be drawn anymore.
I admit that I have to visualize a third dimension for the spirals of a Riemann surface, but ---
OUCH! I think I sprained a brain cell visualizing that. ;-)
 
  • #9
FactChecker said:
I admit that I have to visualize a third dimension for the spirals of a Riemann surface, but ---
OUCH! I think I sprained a brain cell visualizing that. ;-)
My professor liked to use the following image for branching:

1656424559086.jpeg


It helped a bit.

Edit: But the vector space is lost.
 
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FAQ: Is it valid to express a complex number as a vector?

What is a complex number?

A complex number is a number that contains both a real and an imaginary part. It is usually expressed in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit equal to the square root of -1.

How is a complex number represented as a vector?

A complex number can be represented as a vector in the complex plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate. This vector can also be written in the form (a, b), where a is the horizontal component and b is the vertical component.

Why is it valid to express a complex number as a vector?

It is valid to express a complex number as a vector because they both have similar properties and can be represented in the same way. Both complex numbers and vectors have magnitude and direction, and can be added, subtracted, and multiplied using similar rules.

What are the advantages of expressing a complex number as a vector?

Expressing a complex number as a vector can make it easier to visualize and understand its properties. It also allows for easier manipulation and calculation of complex numbers, as vector operations are well-defined and can be easily performed using mathematical tools such as matrices.

Are there any limitations to expressing a complex number as a vector?

While expressing a complex number as a vector can be useful, it is important to note that not all properties of complex numbers can be represented using vectors. For example, the argument or angle of a complex number cannot be represented as a vector. Additionally, vectors only work in two dimensions, whereas complex numbers can be represented in higher dimensions.

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