# 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.

In cases like   or   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 R & i =

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.

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i =   is called the imaginary unit. Also  i² =-l,  = -i,  = 1  etc.

►   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    then a = 0 = b but in complex numbers,

does not imply

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Equality In Complex Number: Two complex  numbers

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).

length OM is called modulus of the complex number denoted by   &   is called the argument or amplitude.

=  and

is always non-negative. Unlike real numbers

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

The unique value of   such that   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.

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#### Few Basic Questions on complex numbers–

Example-1 Compute real and imaginary part of

Solution

=

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