Comparison on calculus of real and complex numbers

Comparison on calculus of real and complex numbers

Project Description:

You will do a review on similarities and differences between Calculus of Real and Complex numbers. The main goal of this project is to observe the main differences of calculus of complex numbers in C over the real numbers and what advantages this differences brings to other fields of mathematics and to science.

Obj. #1: The aim of this objective is to compare the set of complex numbers and the set of real numbers from algebraic structural and from geometric perspective (such as closedness, nth-roots, ’rational’ powers, infinity as extending the number system, roots of polynomials,…).

?!!! Write a report on Obj. #1, about 5 – 6 pages with proper citations, and on a separate page(s) the references (only the ones used in the report), in an MS Word file.

The Objective #1 is to compare the set of complex numbers and the set of real numbers from algebraic structural and from geometric perspective, such as

a.Closedness (Complex number system is closed under the operation of taking roots. That is, for any z complex number z^(1/n) is also a complex number. Does the real number system has a similar closure property? Explain, why?)

b.n-th roots (Geometrically, the nth rooth of a complex number z can be interpreted as the vertices of a polygon with n sides. You can provide examples of this observation. On the real numbers there is no similar geometric representation.)

c.’rational’ powers (the n-th root finding formula enables us to define z^(m/n), where m and n are positive integers with no common factors. The set of values of (z^(1/n))^m has the same set of values (z^m)^ (1/n) defined as the set of values of z^(m/n). Is this true for real numbers?)

d.Infinity as extending the number system (in real number system we have positive and negative infinity but in complex number system we have only infinity, the ideal point. How do we define the ideal point? (by stereographic projection!) )

e.Roots of polynomials (If a polynomial’s all coefficients are all real then if it has a complex root z_1 then z_1’s conjugate is also a root of that polynomial. However, if a polynomial has at least one of the coefficients as a complex numbers then if the polynomial has a complex root z_1 then z_1’s conjugate does not have to be a root of that polynomial. Why? Provide also examples. Moreover, is the synthetic division works for complex coefficient polynomials? )

f.Concept of order (Real numbers are ordered, but complex numbers are not. Statements such as z_1<z_2 have no meaning in the complex numbers. Why complex numbers has no such order as real number system?)

g.Moreover, for example solutions for e^x=-5 or sin(x)=10 when x is real are impossible, do not even make sense but when x is complex both of the equations have solutions.

The report is already written, just needed to be fixed the paper by the following instruction.

“I am attaching the reviewed file to this email. I also have some other feedback which I wrote below. Please, look at the file and do the pointed by me changes on it. Moreover, you missed to add the following concepts which I asked you to consider in my previous feedback. Please, add them and expend them by doing some literature research.

1. Infinity as extending the number system (in real number system we have positive and negative infinity but in complex number system we have only infinity, the ideal point. How do we define the ideal point? (by stereographic projection!) )

2. You considered this concept but not completely. Roots of polynomials (If a polynomial’s all coefficients are all real then if it has a complex root z_1 then z_1’s conjugate is also a root of that polynomial. However, if a polynomial has at least one of the coefficients as a complex numbers then if the polynomial has a complex root z_1 then z_1’s conjugate does not have to be a root of that polynomial.

a.On this concept you did not write about the question Why if a polynomial has at least one of the coefficients as a complex numbers then if the polynomial has a complex root z_1 then z_1’s conjugate does not have to be a root of that polynomial?

b.You also did not check whether the synthetic division (for polynomials with real coefficients) works for complex coefficient polynomials? Provide example(s).)

3. You considered this concept but not completely. Concept of order (Real numbers are ordered, but complex numbers are not. Statements such as z_1<z_2 have no meaning in the complex numbers.

a.On this concept you did not answer what I asked you to. Why complex numbers has no such order as real number system?)”

Thank you.

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