A review Complex number

Complex number review 

   Complex number are usually discussed in the first part of advanced mathematics and here is a quick review about it.

Consider the equation

it has has NO SOLUTION in real number system.


But in eighteenth century mathematician invented a   new number  "i"  which is defined by the property. this in turn , led to the development of complex numbers, which are numbers of the form a+bi .
   "a "and "b" are real numbers. But it can be also observed that every real number a is also a complex number because it can be written as a=a+0i. Thus ,the real numbers  are a subset of the complex numbers.

With these properties complex number can be now defined as.
---the combination of real and imaginary number which can be expressed in the form a+bi or a+jb where i or j=-1


Powers of i or j

Note for  j^n
If n is divided by 4 and the result is 1 it follows j^4. if the result has a decimal value of (.75) if follows j^3.If (.50) it follows  j^2. If (.25) it follows j






Argand's diagram

real axis






Forms of complex numbers -complex number can be expressed in different notations.
1.) rectangularr form  -complex number is denoted by its respective horizontal and vertical components.
         a+jb                              where:   a-real value
                                                          jb-imaginary axis 

2.) polar form - complex number can be denoted by the length  and the angle of its vector
         r∠Ɵ                                          where: r- magnitude
                                                                      Ɵ - argument,degrees
3.) trigonometric form
           rcosƟ  +jsinƟ                     where: r-magnitude
                                                                    Ɵ-argument,degrees
4.) Exponential form

                                            where: r-magnitude

                                                      Ɵ-argument,degrees