where and how do we use complex numbers and calculus??

Question

complex numbers are all abstracts,,not even used to count things,,lol,,
calculus i don't know what do u do differentiating and integrating variables,,

what are their profitable uses??

Answer 1

Used WIDELY in the following to name but a few:-
Physics
Electrical Engineering
Electronics
Civil Engineering
Chemical Engineering
Mechanical Engineering.

What about the theory behind mobile phones?
Radio waves?
Guess what ----complex numbers!

Answer 2

I am currently in AP Calculus as a High School Senior and I have been wondering the same thing.
My teacher was talking about this with the class and he started naming off jobs and most of them were engineering areas. However, the medical fields do have some calculus in them. Like for instance, determining the maximum and minimum effective dosages for a cancer patient undergoing chemotherapy.
Also, someone who can interpret graphs really well, as needed in Calculus because Calculus is the study of motion or change. This could be usesful for someone working for a large company trying to figure the rate at which a product sold or did not sell. Also, the rate at which stock fell or rose.


The last thing I can think of now is my personal favorite: Related Rates. Where you are given some information, like "Sand is pouring out of a cone at a rate of 3ft/s, how fast is the sand accumulating after 30 seconds?"

Related Rates offers real life questions that require you to have an understanding of Calculus.

Answer 3

http://mathworld.wolfram.com/ComplexNumber.html
The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of -1, . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.

The set of complex numbers is implemented in Mathematica as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.

Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).


Through the Euler formula, a complex number

(1)

may be written in "phasor" form

(2)

Here, is known as the complex modulus (or sometimes the complex norm) and is known as the complex argument or phase. The plot above shows what is known as an Argand diagram of the point , where the dashed circle represents the complex modulus of and the angle represents its complex argument. Historically, the geometric representation of a complex number as simply a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, "imaginary" numbers became accepted partly through their visualization.

Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in the complex plane, since points in a plane also lack a natural ordering.

The absolute square of is defined by , with the complex conjugate, and the argument may be computed from

(3)

The real and imaginary parts are given by

(4)
(5)

de Moivre's identity relates powers of complex numbers for real by

(6)

A power of complex number to a positive integer exponent can be written in closed form as

(7)

The first few are explicitly

(8)
(9)
(10)
(11)

(Abramowitz and Stegun 1972).

Complex addition

(12)

complex subtraction

(13)

complex multiplication

(14)

and complex division

(15)

can also be defined for complex numbers. Complex numbers may also be taken to complex powers. For example, complex exponentiation obeys

(16)

where is the complex argument.
http://mathworld.wolfram.com/ComplexArgument.html