Introduction
A complex number is an extended version of real number in the form
x + iy; π₯ββ
Euler (1707 β 1783) introduced the imaginary unit βiβ (read as iota) for β1 with property
Therefore, imaginary unit i is the solution of an equation
Acomplex is written in the STANDARD form as z = x+iy where x and y are real numbers
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Note
 The cartesian form of complex number can be written z=x+iy [standard form] or z=(x,y) [order pair form]
 In z = (x,y,), x is real part denoted as Re (z), and y is imaginary part denoted as Im (z)
In z = (x,y), Re (z) = x and Im (z) =y  A complex number z = x+iy, is purely real if y = 0 i.e. Re (z) = 0, and purely imaginary if x = 0 i.e. Im (z) = 0.
 A complex number z = x+iy, is zero if x = y = 0 i.e. Re (z) = Im (z) = 0
The introduction of complex numbers gives rise to the fundamental theorem of algebra. In the 16th century, Italian mathematician Gerolamo Cardano used complex numbers to find solutions to cubic equations.
Then Italian mathematician Rafael Bombelli developed the rules for addition, subtraction, multiplication, and division of complex numbers. A more abstract formalism for complex numbers was developed by the Irish mathematician William Rowan Hamilton.
Meaning of i
In the complex number system, i is called imaginary unit. Tthe value of i is (0, 1). Thus, we can write i=(0,1).
If we expand a complex number z=(x,y) as z=(1,0)x+(0,1)y then (1,0) is unit of real part denoted by 1 and (0,1) is unit of imaginary part and denoted by i. Here i is imaginary unit with
.
A complex number is visually represented in Argand diagram. Here i is operator giving anticlockwise quarter turn such that .
NOTE
 We should NOT mean i as nonexistence number nor a number that exist only in imagination
 i is a number that denotes imaginary unit (0,1)
 i is an anticlockwise quarter turn operator for (x,y), thus
Positive powers of i
In general, for any integer k,
.
For example:
The verification on the positive powers of i are as follows.
and so on
Absolute Value
In a complex number
The absolute value is Modulus, a nonnegative real number denoted by
 z  and defined by
Geometrically,
π§ is distance of z from origin
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Note
Due to order property, z_{1} <z_{2} is meaningless unless z_{1} and z_{2} both are real.
However, z_{1}  < z_{2}  means z_{1 }is closer than z_{2}.
Distance between and is denoted by
and defined by
NOTE:
Conjugate
The conjugate of a complex number z=x+iy is denoted and defined by
Geometrically, is reflection of on real axis
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Properties of Conjugate numbers
 If and then
Proof
or
or
or
or  The sum of a complex number z and its conjugate is twice of its real part
Proof
If be a complex number then , where
or
or  The difference of z and its conjugate is twice of its imaginary part
Proof
If z=x+iy be a complex number then , where
or
or  The real and imaginary parts of a complex number can be extracted using its conjugate
Theorem: An important property
Let z=x+iy be a complex number then
Proof
Given z=x+iy be a complex number, then .
Now,
or
or
(i)
Also
or
(ii)
From (i) and (ii), we get
Question
Theorem: Triangle inequality:
Proof>br> We know that
or
or
or
or
or
or
or
or
or
This completes the proof.
Corollary

Proof
or
or
or
Thus, 
Proof
Now, replacing by we get
Thus, 
Proof
Now, replacing by we get
Thus,
Algebra of complex number
Complex plane looks like an ordinary twodimensional plane of z=( x,y ), but z=( x,y ) is a single number losing order axioms. Fundamental operations on complex number are defined as below.
 Equality
Two complex numbers and are equal if
 Addition
Sum of two complex numbers and is defined as
According to definition, corresponds to resultant vector addition.  Multiplication
Product/multiplication of two complex numbers and is defined as
or
or
The product is neither scalar nor the vector product of ordinary vector analysis. This departure is due to
Also, complex number C is a field. Thus, the complex number satisfy all Field axioms as below.
 Closer:
 Additive associative
 Additive identity
Additive identity (0,0)  Additive Inverse
Additive inverse of z=x+iy is z=xiy  Additive commutative
 Distributive
 Multiplicative associative
 Multiplicative identity
Multiplicative identity (1,0)  Multiplicative inverse
Multiplicative inverse of nonzero complex number z=x+iy is
Proof
Let z=x+iy be nonzero complex number and be its multiplicative inverse, then
or
or
Comparing real and imaginary parts separately, we get
uxvy=1 and uy+vx=0
Solving for u and v , we get
and
Hence,
 Multiplicative commutative
Algebric Structure
Axioms  Structure  Example 
12  Semi Group  Example 
13  Monoid  Example 
14  Group  Example 
15  Abelian Group  Example 
16  Ring  Example 
17  Assocoative Ring  Example 
18  Assocoative Ring with Unity  Example 
19  Division Ring (Skew Field)  Example 
110  Field  Example 
Polar form
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Let be a complex number with magnitude r and amplitude , then
and
Hence,
or
Thus, a complex number z=x+iy is represented by polar coordinate , as
Here,
r is the length of z , and is argument of z with
, and
Example
Find polar form of complex number z=5+5i
Solution
Given that
z=5+5i
or
z=5( 1+i)
or
or
The relevance of complex number in polar form is that multiplication and division are simpler with this form than the Cartesian form.
Let and be two complex numbers, then
Proof
Given that and . Thus
or
or
or
or
or
Theorem
Proof
Let and thenThus,
Note:
Any complex number z has infinite arguments; all differ by multiple of . The principal value is in the interval
Some important property
 Let be two complex number then
The argument of product of two complex number is sum of their arguments.
Proof
Let and then
or
Hence,
 Let be two complex number then,
The argument of quotient of two complex number is difference of their arguments.  Argument of complex number of the form is 0
 Argument of complex number of the form is
 Argument of complex number of the form is
 Argument of complex number of the form is
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If then show that
Question
Let be a complex number and its inverse is then
or
or
or
or
Thus,
Exponenrial Form
Euler’s formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function. It states that for any real number x:
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions.
When x = Ο, Euler’s formula evaluates to , which is known as Euler’s identity.
Given Eulerβs exponential form,
Thus, complex number is defined as
The significance of exponential form of complex number is that we can easily compute conjugate and inverse.
For example,
and
Proof of the Formula
 Function Method
Consider the function f(ΞΈ) given by
or
Differentiating gives by the product rule, we have
Thus, f(ΞΈ) is a constant.
Since f(0) = 1, then f(ΞΈ) = 1 for all ΞΈ, and thus
or
This completes the proof.  Series Method
We know that
Using power series expansion, we get
or
or
or
or
This completes the proof.
DeMoivreβs theorem
Let be a complex number then where n is a positive integer.
Proof
 Case 1: n=1
Then
or
So,
or when n=1
 Case 2: n=2
or
or
or
or
or
or
or
So,
or when n=2
 Case 3: We assume the same formula is true for n = k, so we have
So,
or when n=k
 Case 4: Now, we prove for n = k + 1,
or
or
or
or
So,
or when n=k+1
 Using case 1case 4, for any number , we have
Example 1
Compute
Solution
Since
or
We get
and
Thus,
or
or
or
nth root of Complex number
If be a complex number then the nth root of z is
Proof
Given that Z is a complex number. Also let, nth root of Z is W such that
Now we have
or
or
or
Equating real and Imaginary parts, we get
and and
or
and
or
and
Thus, nth root of is
or
square roots of i
Find the square roots of i
We know that
Solution
Since , we get
r=1 and
Hence the first square roots of i is
The second square root of i is