Complex Numbers

Complex Numbers-

Complex numbers come into existence when the square of a number is negative because we know it very well that the square of a number will always be positive doesn’t matter whether the number is positive or negative.<img src="complex number world.jpg" alt="complex number world">

In cases like  {x^2} + 1 = 0 or  2{x^3} + 5x = 0 here, if we solve, we find the square of x=-1. We say that x is not real here. Generally, these types of cases are considered as Complex numbers. Complex numbers were first observed by mathematician Girolamo Cardano (1501-1575). In his book Ars Magna, he discussed the mechanics of complex numbers in details and thus he started Complex Algebra.

Standard form of Complex Numbers-

Complex numbers are defined as expressions of the form a + ib where a,b \in  R & i = \sqrt { - 1}

It is denoted by Z  i.e. z= a + ib.

‘a’  is called as real part of z= (Re z)

and ‘b’ is called as imaginary part of z =(Im z).

i or IOTA- iota is a unique symbol. it’s the ninth letter of Latin alphabet. It’s used to denote imaginary numbers whose square root is -1.
Click here to download the book “An Imaginary Tale The Story of i” a very interesting book on iota by Paul J. Nahin.

► Zero is both purely real as well as purely imaginary but not imaginary.

Also READ this-  Permutations and Combinations-algebra tutors

i = \sqrt { - 1}   is called the imaginary unit. Also  i² =-l, {i^3} = -i, {i^4} = 1  etc.

► \sqrt a \sqrt b = \sqrt {ab}   only if at least one of either a or b is non-negative.

► Conjugate  Complex-  If  z=a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z ¯ or z*.  i.e.  z* = a – ib.

z + z*  =  2 Re(z) v

► z –  z* =  2i Im(z)

► zz* = a² + b² which is real If z lies in the 1st quadrant then lies z* in the 4th quadrant and -z* lies in the 2nd  quadrant.
 Algebraic Operations: The algebraic operations on complex numbers are similar to those on real numbers treating i as a  polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.

e.g.   z > 0,  4 + 2i < 2 + 4 i   are  meaningless .

However in real numbers, if {a^2} + {b^2} = 0   then a = 0 = b but in complex numbers,

{z_1}^2 + {\rm{ }}{z_2}^2 = 0{\rm{ }}   does not imply {z_{1{\rm{ }}}} = {z_{2{\rm{ }}}} = 0

Also READ this-  How to solve basic problems in trigonometry?(concept-1)

Equality In Complex Number: Two complex  numbers   {z_1} = {a_1} + i{b_{1}}\& {\rm{ }}{z_2} = {a_{2{\rm{ }}}} + i{b_{2{\rm{ }}}}

are equal if and only if their real  & imaginary parts coincide.

Representation  Of  A  Complex  Number  In  Various  Forms:

(a) Cartesian Form (Geometric Representation): Every complex number z = x + i y  can be represented by a point on the cartesian plane known as a complex plane (Argand diagram) by the ordered pair (x, y).

<img src="complex number.jpg" alt="complex number">

length OM is called modulus of the complex number denoted by  \left| z \right| & \theta   is called the argument or amplitude.

\left| z \right| = \sqrt {{a^2} + {b^2}}  and   \theta = {\tan ^{ - 1}}\frac{y}{x}

\left| z \right|  is always non-negative. Unlike real numbers   \left| z \right| = \left[ \begin{array}{l} z....if..z \ge 0\\ - z..if...z < 0 \end{array} \

Argument of a complex number is a many-valued function. If \theta   is the argument of a complex number then 2n\pi +\theta  where n \in  I will also be the argument of that complex number. Any two arguments of a complex number differ by 2n\pi

The unique value of  \theta  such that   - \pi < \theta \le \pi  is called the principal value of the argument. Unless otherwise stated, amp z  implies the principal value of the argument.

By specifying the modulus & argument a complex number is defined completely. For the complex number  0 + 0.i  the argument is not defined and this is the only complex number which is given by its modulus.
 There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.

Also READ this-  How to solve basic problems in trigonometry?(concept-2)

Few Basic Questions on complex numbers

Example-1 Compute real and imaginary part of  z{\rm{ }} = {\rm{ }}\frac{{i{\rm{ }} - {\rm{ }}4}}{{2i{\rm{ }} - {\rm{ }}3{\rm{ }}}}{\rm{ }}


                           z{\rm{ }} = \frac{{{\rm{ }}i{\rm{ }} - {\rm{ }}4{\rm{ }}}}{{2i{\rm{ }} - {\rm{ }}3}}

                              = \frac{{{\rm{ }}i{\rm{ }} - {\rm{ }}4{\rm{ }}}}{{2i{\rm{ }} - {\rm{ }}3}}.\frac{{{\rm{ 2}}i{\rm{ + 3 }}}}{{2i{\rm{ + }}3}}


=\frac{{{\rm{ 2}}{{\rm{i}}^2}{\rm{ + 3i - 8i - 12 }}}}{{{{(2i)}^2}{\rm{ }} - {\rm{ }}{3^2}}}


                             = \frac{{{\rm{ - 2 - 5i - 12 }}}}{{{\rm{ - 4 - 9}}}}

                             = \frac{{{\rm{ - 14 - 5i }}}}{{ - 13}}

                            = \frac{{{\rm{ - 14 }}}}{{ - 13}} - \frac{{{\rm{5i }}}}{{ - 13}}


so clearly Re(z)=14/13 and Im(z)=5/13

In the next article on complex numbers, we will learn about Polar fom of complex numbers and some other properties of complex numbers

CLICK HERE to download the Pdf of Complex numbers

<img src="demo.png" alt="demo">