## Complex Numbers

**Complex Numbers DEFINITION:** Complex numbers are definited 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).

**Every Complex Number Can Be Regarded As **

Purely real Purely imaginary Imaginary

If b = 0 If a = 0 If b ≠ 0

**Note:**

- The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
- Zero is both purely real as well as purely imaginary but not imaginary.
- i = is called the imaginary unit. Also i² = −1 ; i
^{3}= −i ; i4 = 1 etc. - only if atleast one of either a or b is non-negative.

### Conjugate Complex | Complex Numbers

If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z. i.e. = a − ib.

**Note:**

- z + = 2 Re(z)
- z − = 2i Im(z)
- z = a² + b² which is real
- If z lies in the 1
^{st}quadrant then lies in the 4^{th}quadrant and lies in the 2^{nd}quadrant.

### Algebraic Operations | Complex Numbers

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} + z_{2}^{2} = 0 does not imply z_{1} = z_{2} = 0.

### Equality in Complex Number

Two complex numbers z_{1} = a_{1} + ib_{1} & z_{2} = a_{2} + ib_{2} are equal if and only if their real & imaginary parts coincide.

### Representation of Complex Number in Various Forms

**Cartesian Form (Geometric Representation):**

Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y). length OP is called modulus of the complex number denoted by |z| & θ is called the argument or amplitude

e.g. |z| =

θ = tan^{−1}

(angle made by OP with positive x−axis)

**Note:**1. |z| is always non-negative. Unlike real numbers is not correct

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

3. The unique value of θ such that – π < θ ≤ π is called the principal value of the argument.

4. Unless otherwise stated, amp z implies principal value of the argument.

5. 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.

6. There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.**Trignometric / Polar Representation:**

z = r (cos θ + i sin θ) where | z | = r ; arg z = θ ; = r (cos θ − i sin θ)

**Note:**cos θ + i sin θ is also written as CiS θ.

Also**Exponential Representation:**

z = re^{iθ}; | z | = r ; arg z = θ ; = re^{-iθ}

### Important Properties of Conjugate/ Moduli/ Amplitude | Complex Numbers

If z , z_{1} , z_{2} ∈ C then ;

- z + = 2 Re (z) ; z − = 2 i Im (z) ; ; ;

; ; ; z_{2}≠ 0 - |z
_{1}+ z_{2}|^{2}+ |z_{1}– z_{2}|^{2}= 2 [|z_{1}|^{2}+ |z_{2}|^{2}]

||z_{1}| − |z_{2}|| ≤ |z_{1}+ z_{2}| ≤ |z_{1}| + |z_{2}| - (i) amp (z
_{1}. z_{2}) = amp z_{1}+ amp z_{2}+ 2 kπ. k ∈ I

(ii) amp = amp z_{1}– amp z_{2}+ 2kπ; k ∈ I

(iii) amp(z^{n}) = n amp(z) + 2kπ .

where proper value of k must be chosen so that RHS lies in (− π , π ].

### Vectorial Representation Of A Complex Number

Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then, & || = |z|

Note:

- If = z = re
^{iθ}then = z_{1}= re^{i(θ + φ)}= z . e^{iφ}. If and are of unequal magnitude then φ - If A, B, C & D are four points representing the complex numbers z
_{1}, z_{2}, z_{3}& z_{4}then is purely real ;

is purely imaginary ]

- If z
_{1}, z_{2}, z_{3}are the vertices of an equilateral triangle where z0 is its circumcentre then (a) (b)

### Demoivre’S Theorem

Statement: cos nθ + i sin nθ is the value or one of the values of (cos θ + i sin θ)^{n} ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity

**Note:** Continued product of the roots of a complex quantity should be determined using theory of equations.

### Cube Root Of Unity | Complex Numbers

- The cube roots of unity are 1,
- If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general 1 + w
^{r}+ w²^{r}= 0 ; where r ∈ I but is not the multiple of 3. - In polar form the cube roots of unity are:

- The three cube roots of unity when plotted on the Argand plane constitute the vertices of an equilateral triangle.
- The following factorisation should be remembered:

(a, b, c ∈ R & ω is the cube root of unity)

a^{3} − b^{3} = (a − b) (a − ωb) (a − ω^{²}b); x^{2} + x + 1 = (x − ω) (x − ω^{2});

a^{3} + b^{3} = (a + b) (a + ωb) (a + ω^{2}b);

a^{3} + b^{3} + c^{3} − 3abc = (a + b + c)(a + ωb + ω²c)(a + ω²b + ωc)

### n^{th} Roots Of Unity | Complex Numbers

If 1 ,_{1} ,α_{2} , α_{3} ….. α_{n − 1} are the n, nth root of unity then:

- They are in G.P. with common ratio e
^{i(2π/n)}& - if p is not an integral multiple of n

= n if p is an integral multiple of n - (1 − α
_{1}) (1 − α_{2}) …… (1 − α_{n – 1}) = n &

(1 + α_{1}) (1 + α_{2}) ……. (1 + α_{n}− 1) = 0 if n is even and 1 if n is odd. - 1 . α
_{1}. α_{2}. α_{3}……… α_{n − 1}= 1 or −1 according as n is odd or even.

### The Sum Of The Following Series Should Be Remembered:

### Straight Lines & Circles In Terms Of Complex Numbers:

- If z
_{1}& z_{2}are two complex numbers then the complex number z =mn

divides the joins of z_{1}& z_{2}in the ratio m : n.

**Note:**

(i) If a , b , c are three real numbers such that az_{1}+ bz_{2}+ cz_{3}= 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z_{1}, z_{2}& z_{3}are collinear.

(ii) If the vertices A, B, C of a ∆ represent the complex nos. z_{1}, z_{2}, z_{3}respectively, then:

(a) Centroid of the ∆ ABC =

(b) Orthocentre of the ∆ ABC = OR

(c) Incentre of the ∆ ABC = (az_{1}+ bz_{2}+ cz_{3}) ÷ (a + b + c)

(d) Circumcentre of the ∆ ABC = :

(Z_{1}sin 2A + Z_{2}sin 2B + Z_{3}sin 2C) ÷ (sin 2A + sin 2B + sin 2C) - amp(z) = θ is a ray emanating from the origin inclined at an angle θ to the x− axis.
- |z − a| = |z − b| is the perpendicular bisector of the line joining a to b.
- The equation of a line joining z
_{1}& z_{2}is given by;

z = z_{1}+ t (z_{1}− z_{2}) where t is a parameter. - z = z
_{1}(1 + it) where t is a real parameter is a line through the point z_{1}& perpendicular to oz_{1}. - The equation of a line passing through z
_{1}& z_{2}can be expressed in the determinant form as

This is also the condition for three complex numbers to be collinear. - Complex equation of a straight line through two given points z
_{1}& z_{2}can be written as

which on manipulating takes the form as where r is real and α is a non zero complex constant. - The equation of circle having center z
_{0}& radius ρ is: |z − z_{0}| = ρ or

which is of the form , r is real centre − α & radius

. Circle will be real if . - The equation of the circle described on the line segment joining z
_{1}& z_{2}as diameter is:

- Condition for four given points z
_{1}, z_{2}, z_{3}& z_{4}to be concyclic is, the number . is real. Hence the equation of a circle through 3 non collinear points z_{1}, z_{2}& z_{3}can be taken as is real ⇒

**Reflection points for a straight line:** Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z_{1} & z_{2} will be the reflection points for the straight line if and only if; where r is real and α is non zero complex constant.

Inverse points w.r.t. a circle:

Two points P & Q are said to be inverse w.r.t. a circle with center ‘O’ and radius ρ, if :

- the point O, P, Q are collinear and on the same side of O.
- OP. OQ = ρ
^{2}.

Note that the two points z_{1} & z_{2} will be the inverse points w.r.t. the circle

if and only if .

### Ptolemy’s Theorem:

It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides.

i.e. |z_{1} − z_{3}| |z_{2} − z_{4}| = |z_{1} − z_{2}| |z_{3} − z_{4}| + |z_{1} − z_{4}| |z_{2} − z_{3}|.

### Logarithm Of A Complex Quantity

- Log
_{e}(α + iβ) = ½Log_{e}(α² + β²) + i( 2nπ + tan^{−1}where n ∈ I. - i
^{i}represents a set of positive real numbers given by