Exploring the Relationship between Complex Numbers and Vectors

In summary: This means that the two structures have the same underlying mathematical structure, but the complex numbers have an extra operation of multiplication by a complex number that makes them different from just a 2-dimensional vector space. So, in summary, the relationship between complex numbers and vectors in a plane is that they have the same mathematical structure in terms of addition and multiplication by a real number, but the complex numbers have an additional operation of multiplication by a complex number that sets them apart from just a 2-dimensional vector space.
  • #1
Ratzinger
291
0
What is the relationship between complex numbers and vectors in a plane?

I read they have the same mathematical structure. What does that mean and how far does that sameness go?

If the complex numbers are all ordered pair that obey (a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac-bd,ad-bc), can we then equate these ordered pairs with 2-dim vectors? I believe not. What does then (a,b)(c,d)=(ac-bd,ad-bc) mean?

Could someone help?

thanks
 
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  • #2
As a set, the complex numbers are in bijection with R^2. They also carry the same additive structure as R^2, and they behave properly with respect to multiplication by elements of R, so the complex numbers are a 2 dimensional real vector space. They also carry an extra structure, multiplication of complex numbers, that the plane does not naturally carry. What this means is that the complex numbers are an algebra over the reals, 2-dimensional. More than that they are a division algebra, meaning that you can divide elements, ie form inverses under multiplication, and even more than that they are a commutative division algebra, i.e. a field too.

So, think of the complex numbers as a real vector space with extra structure, that of an algebra.

Really, this is the best way to think about complex numbers, if you ask me. No one can deny they exist...
 
  • #3
No, they don't have the same "mathematical structure". If we associate the vector [itex]x\vec{i}+ y\vec{j}[/itex] with the complex number x+ iy, then addition and scalar multiplication are the same (so they are "isomorphic" as vector spaces) but the complex numbers have a multiplication that is not defined for vectors in R2.
 
  • #4
HallsofIvy said:
No, they don't have the same "mathematical structure". If we associate the vector [itex]x\vec{i}+ y\vec{j}[/itex] with the complex number x+ iy, then addition and scalar multiplication are the same (so they are "isomorphic" as vector spaces) but the complex numbers have a multiplication that is not defined for vectors in R2.
So is the property of multiplication the *only* thing that separates complex numbers from vectors in R^2?
 
  • #5
Swapnil said:
So is the property of multiplication the *only* thing that separates complex numbers from vectors in R^2?
Anyone...?
 
  • #6
Swapnil said:
So is the property of multiplication the *only* thing that separates complex numbers from vectors in R^2?
Essentially. (The exact answer depends on the fine print of what you mean by R² and C)
 
  • #7
In technical terms, there is an "isomorphism" between the vector space, R2, and the set of complex numbers with the operations of addition and multiplication by a real number (NOT multiplication by a complex number).
 

1. What is a complex number?

A complex number is a number that contains two parts - a real part and an imaginary part. The real part is a normal number that we are familiar with, while the imaginary part is a multiple of the imaginary unit i, which is defined as the square root of -1. Complex numbers are written in the form a + bi, where a is the real part and bi is the imaginary part.

2. What is a vector?

A vector is a mathematical object that has both magnitude and direction. It is usually represented by an arrow pointing in a specific direction, with the length of the arrow representing the magnitude of the vector. Vectors can be used to represent physical quantities such as displacement, velocity, and force.

3. How are complex numbers and vectors related?

Complex numbers and vectors are related because they both have both magnitude and direction. In fact, complex numbers can be thought of as a special type of vector, with the real part representing the horizontal component and the imaginary part representing the vertical component. This connection between complex numbers and vectors is what allows us to use vector operations (such as addition, subtraction, and multiplication) to manipulate complex numbers.

4. How do we represent complex numbers and vectors geometrically?

Complex numbers can be represented geometrically on the complex plane, which is similar to the traditional coordinate plane but with the real numbers on the horizontal axis and the imaginary numbers on the vertical axis. Vectors can also be represented on the coordinate plane, with the starting point at the origin and the magnitude and direction determined by the coordinates of the endpoint of the vector.

5. What are some real-world applications of the relationship between complex numbers and vectors?

The relationship between complex numbers and vectors has many practical applications in fields such as engineering, physics, and computer science. For example, in electrical engineering, complex numbers are used to represent electrical currents, while in quantum mechanics, complex numbers are used to describe the state of particles. In computer graphics, vectors are used to represent the position and direction of objects in a 3D space, allowing for the creation of realistic images. Additionally, the use of complex numbers and vectors in signal processing allows for the analysis and manipulation of signals in various forms, such as audio and video signals.

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