Interpretation of Complex numbers - confused

In summary, Complex numbers are typically represented in a two-dimensional plane with one real and one imaginary dimension. The quaternions, on the other hand, consist of one real number and three imaginary numbers. However, in our world with three mutually orthogonal real dimensions, the imaginary dimension would have to be a fourth dimension. This leads to the concept of a four-dimensional vector with three real components and one imaginary component. This vector is represented by V=(x,y,z,it), where "t" is the fourth dimensional real number. The "regular" inner product of this vector would give the Minkowski metric V^2=(x^2+y^2+z^2-t^2). The idea of three different planes with a real and
  • #1
closet mathemetician
44
0
I know that generally complex numbers are represented in a two-dimensional plane with one real and one imaginary dimension. I also know that we have the quaternions, consisting of one real number and three imaginary numbers.

The imaginary axis is always perpendicular to the real axis. The problem is that in our world we have three mutually orthogonal real dimensions. So if the imaginary dimension has to be perpendicular to all three real dimensions, that would make the imaginary dimension a fourth dimension.

Now we have the problem that there are three different planes with a real dimension and an imaginary dimension so we have three complex planes. It seems to me that you would have a four-dimensional vector, with three real components and one imaginary component:

V=(x,y,z,it)

using "t" as the fourth dimensional real number by which to multiply the imaginary unit.


The "regular" inner product of V with itself would give the Minkowski metric

V^2=(x^2+y^2+z^2-t^2)


I have never seen the imaginary dimension presented this way. Why not? Can anyone explain?
 
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  • #2
Quoting myself,

closet mathemetician said:
Now we have the problem that there are three different planes with a real dimension and an imaginary dimension so we have three complex planes.

Could this be the basis for quaternions, i.e., three complex planes? However, there is really only one imaginary dimension involved with all three planes.
 

Related to Interpretation of Complex numbers - confused

1. What are complex numbers?

Complex numbers are numbers that contain both a real and imaginary component. They are usually written in the form a + bi, where a is the real part and bi is the imaginary part.

2. How do you interpret complex numbers?

Complex numbers can be interpreted geometrically as points in the complex plane, with the real axis representing the real part and the imaginary axis representing the imaginary part. They can also be interpreted algebraically as expressions that can be operated on using rules of arithmetic.

3. What is the difference between a real number and a complex number?

A real number is a number that can be represented on the number line and does not have an imaginary component. A complex number, on the other hand, has both a real and imaginary component and cannot be represented on the number line.

4. How do you add, subtract, multiply, and divide complex numbers?

To add or subtract complex numbers, you simply combine the real and imaginary components separately. To multiply complex numbers, you use the FOIL method (first, outer, inner, last). To divide complex numbers, you multiply the numerator and denominator by the complex conjugate of the denominator.

5. Why do people get confused about interpreting complex numbers?

Complex numbers can be confusing because they involve a combination of real and imaginary components, which may be difficult for some people to visualize. Additionally, the rules for operating on complex numbers may seem counterintuitive at first, leading to confusion.

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