Complex Numbers: Defining an Ordered System

In summary, the conversation discusses the concept of complex numbers, their definition as an ordered pair, and their use in complex analysis. The standard construction of complex numbers is explained as well as the purpose of mapping functions in the complex plane. A reference is also suggested for further explanation.
  • #1
lostcauses10x
87
0
Complex numbers?
Since the system is not an ordered pair, how then is it defined using the complex system as an ordered system to plot the z axis (Plane) to use a function?
At the point we input each point of the Real and imaginary plane into a function to get out an answer in the Z plane, is it now an ordered pair per point on input ?
 
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  • #2
A complex number is an ordered pair.
 
  • #3
In complex analysis, the relation w = f(z), where f is some function and w and z are complex number, is thought of as a mapping, whereby for each number z, the function f(z) points to, or 'maps', a different complex number w in a different complex plane.
 
  • #4
DrClaude said:
A complex number is an ordered pair.

To be precise, we construct the complex numbers from a ordered pairs of real numbers by defining addition and multiplication.
 
  • #5
pwsnafu
Where as I believe it is correct, in the link you gave: it is a statement given without any reference or source.
 
  • #6
lostcauses10x said:
pwsnafu
Where as I believe it is correct, in the link you gave: it is a statement given without any reference or source.

It's the standard construction of the complex numbers. If you want a reference, see any text on complex analysis of algebra ever published.
 
  • #7
Hey folks thanks. Simply put based of the limited ability to define i to the real set we created an set with the real and imaginary that is an analogy. of course from there end up with complex analysis etc.
Not sure what the "Mon" replies are, seem to add noting to the discussion.
 

What are complex numbers?

Complex numbers are numbers that have both a real and an imaginary component. They are written in the form a + bi, where a is the real part and bi is the imaginary part, represented by the imaginary unit i.

Why do we need complex numbers?

Complex numbers are necessary for solving certain mathematical problems that involve finding the roots of a polynomial equation. They also have many applications in physics, engineering, and other scientific fields.

How do we define an ordered system for complex numbers?

An ordered system for complex numbers is defined by assigning a value to each number based on its distance from the origin on a complex plane. This value is known as the modulus or absolute value of the complex number.

What is the relationship between complex numbers and the Cartesian coordinate system?

Complex numbers can be represented graphically on a two-dimensional plane known as the complex plane, which is similar to the Cartesian coordinate system. The real part of the complex number is plotted along the x-axis, and the imaginary part is plotted along the y-axis.

Are there any rules for performing operations with complex numbers?

Yes, there are rules for performing operations with complex numbers. For addition and subtraction, the real and imaginary parts are combined separately. For multiplication, the FOIL method is used. And for division, the complex conjugate is used to eliminate the imaginary part in the denominator.

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