# 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.

**►** 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

**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.

**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

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