Complex Number

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Introduction

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A complex number is an extended version of real number in the form
x + iy; 𝑥∈ℝ
Euler (1707 – 1783) introduced the imaginary unit ‘i’ (read as iota) for √-1 with property
\( i^2+ 1 = 0\)
Therefore, imaginary unit i is the solution of an equation
\( x^2+ 1 = 0\)

Acomplex is written in the STANDARD form as z = x+iy where x and y are real numbers

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Note

1. The cartesian form of complex number can be written z=x+iy [standard form] or z=(x,y) [order pair form]
2. 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
3. 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.
4. 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.

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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
\( i^2=-1\).
A complex number is visually represented in Argand diagram. Here i is operator giving anticlockwise quarter turn such that \( i^2=-1\).

NOTE

1. We should NOT mean i as non-existence number nor a number that exist only in imagination
2. i is a number that denotes imaginary unit (0,1)
3. i is an anticlockwise quarter turn operator for (x,y), thus
   \( i^2=(0,1) (0,1) = i (0,1) =(-1,0) =-1 (1,0)=-1 \)

Positive powers of i

In general, for any integer k,
\( i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i\).
For example:
\( i^{23} = i^{4.5+3} = (i^4)5. i ^3= 1^5. (-i) = -i.\)

The verification on the positive powers of i are as follows.
\( i^2 = -1\)
\(i^3 = i2. I = (-1).i = - i\)
\(i^4 = (i^2)^2 = (-1)^2 = 1\)
\(i^5 = i^{4 + 1}= i^4. i = 1. i = i\)
\(i^6 = i^{4 + 2} = i^4. i^2= 1. 1 = 1\) and so on

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Absolute Value

In a complex number
\(z= x + iy \)
The absolute value is Modulus, a non-negative real number denoted by
| z | and defined by
\( |𝑧|=\sqrt{x^2+y^2} \)
Geometrically, 
|𝑧| is distance of z from origin

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Note
Due to order property, z₁ <z₂ is meaningless unless z₁ and z₂ both are real.
However,| z₁ | <| z₂ | means z₁ is closer than z₂.
Distance between \( z_1 =x_1 +iy_1 \) and \( z_2 =x_2 +iy_2 \) is denoted by
\( | z_1 -z_2 |\)
and defined by
\( | z_1 -z_2 |=\sqrt{( x_1 -x_2 )^2 +( y_1 -y_2 )^2 }\)

NOTE:
\( ( x+iy )^2 =( x+iy )( x-iy )\)

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Conjugate

The conjugate of a complex number z=x+iy is denoted \( \bar{z} \) and defined by
\( \bar{z}=x-iy\)
Geometrically, \( \bar{z} \) is reflection of \( z\) on real axis

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Properties of Conjugate numbers

1. If \( z_1 =x_1 +iy_1 \) and \( z_2 =x_2 +iy_2 \) then \( \overline{z_1 +z_2 }=\bar{z_1} +\bar{z_2} \)
   Proof
   \( \overline{z_1 +z_2 }=\overline{( x_1 +iy_1 )+( x_2 +iy_2 )}\)
   or \( \overline{z_1 +z_2 }=\overline{( x_1 +x_2 )+i( y_1 +y_2 )}\)
   or \( \overline{z_1 +z_2 }=( x_1 +x_2 )-i( y_1 +y_2 )\)
   or \( \overline{z_1 +z_2 }=( x_1 -iy_1 )+( x_2 -iy_2 )\)
   or \( \overline{z_1 +z_2 }=\bar{z}_1 +\bar{z}_2 \)
2. The sum of a complex number z and its conjugate \( \bar{z}\) is twice of its real part
   Proof
   If \( z=x+iy\) be a complex number then \( \bar{z}=x-iy\) , where
   \( z+\bar{z}=2x\)
   or \( x=\frac{z+\bar{z}}{2}\)
   or \( Rez=\frac{z+\bar{z}}{2}\)
3. The difference of z and its conjugate \( \bar{z}\) is twice of its imaginary part
   Proof
   If z=x+iy be a complex number then \( \bar{z}=x-iy\) , where
   \( z-\bar{z}=2iy\)
   or \( y=\frac{z-\bar{z}}{2i}\)
   or \( Imz=\frac{z-\bar{z}}{2i}\)
4. 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 \( z.\bar{z}=|z|^2 \)

Proof
Given z=x+iy be a complex number, then \( \bar{z}=xiy\) .
Now,
\( z.\bar{z}=( x+iy )( x-iy )\)
or \( z.\bar{z}=( x )^2 -( iy )^2\)
or \( z.\bar{z}=x^2 +y^2 \) (i)
Also
\( | z |=\sqrt{x^2 +y^2} \)
or \( |z|^2 =x^2 +y^2\)  (ii)
From (i) and (ii), we get
\( z.\bar{z}=|z|^2\)

Question

Theorem: Triangle inequality: \( | z_1 +z_2 |\le | z_1 |+| z_2 |\)

Proof>br> We know that
\( | z_1 +z_2 |^2 =( z_1 +z_2 )( \overline{z_1 +z_2 } )\)
or \( | z_1 +z_2 |^2 =( z_1 +z_2 )( \bar{z}_1 +\bar{z}_2 )\)
or \( | z_1 +z_2 |^2 =( z_1 \bar{z}_1 +z_1 \bar{z}_2 +z_2 \bar{z}_1 +z_2 \bar{z}_2 )\)
or \( | z_1 +z_2 |^2 =( z_1 \bar{z}_1 +z_1 \bar{z}_2 +\bar{\bar{z}}_2 \bar{z}_1 +z_2 \bar{z}_2 )\)
or \( | z_1 +z_2 |^2 =( z_1 \bar{z}_1 +z_1 \bar{z}_2 +\overline{z_1 \bar{z}_2 }+z_2 \bar{z}_2 )\)
or \( | z_1 +z_2 |^2 =z_1 \bar{z}_1 +2Re( z_1 \bar{z}_2 )+z_2 \bar{z}_2 \)
or \( | z_1 +z_2 |^2 \le z_1 \bar{z}_1 +2| z_1 \bar{z}_2 |+z_2 \bar{z}_2 \)
or \( | z_1 +z_2 |^2 =|z_1|^2 +2| z_1 || \bar{z}_2 |+|z_2|^2 \)
or \( | z_1 +z_2 |^2 =( | z_1 |+| z_2 | )^2 \)
or \( | z_1 +z_2 |\le | z_1 |+| z_2 |\)
This completes the proof.

Corollary

1. \( | z_1 +z_2 |\ge | z_1 |-| z_2 |\)

   
   Proof
   or\( | z_1 |=| z_1 +z_2 +( -z_2 ) |\)
   or \( \le | z_1 +z_2 |+| -z_2 |\)
   or \( =| z_1 +z_2 |+| z_2 |\)
   Thus, \( | z_1 +z_2 |\ge | z_1 |-| z_2 |\)

2. \( | z_1 -z_2 |\le | z_1 |+| z_2 |\)

   
   Proof
   \( | z_1 +z_2 | \le | z_1 |+| z_2 |\)
   Now, replacing \( z_2 \) by \( -z_2 \) we get
   \( | z_1 -z_2 |\le | z_1 |+| -z_2 |\)
   Thus, \( | z_1 -z_2 |\le | z_1 |+| z_2 |\)

3. \( | z_1 -z_2 |\ge | z_1 |-| z_2 |\)

   
   Proof
   \( | z_1 +z_2 | \ge | z_1 |-| z_2 |\)
   Now, replacing \( z_2 \) by \( -z_2 \) we get
   \( | z_1 -z_2 |\ge | z_1 |-| -z_2 |\)
   Thus, \( | z_1 -z_2 |\ge | z_1 |-| z_2 |\)

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Algebra of complex number

Complex plane looks like an ordinary two-dimensional 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.

1. Equality
   Two complex numbers \( z_1 =x_1 +iy_1 \) and \( z_2 =x_2 +iy_2 \) are equal if
   \( x_1 =x_2 \text{ and } y_1 =y_2 \)
2. Addition
   Sum of two complex numbers\( z_1 =x_1 +iy_1 \) and \( z_2 =x_2 +iy_2 \) is defined as
   \( z_1 +z_2 =( x_1 +iy_1 )+( x_2 +iy_2 )=( x_1 +x_2 )+i( y_1 +y_2 )\)
   According to definition,\( z_1 +z_2 \) corresponds to resultant vector addition.
3. Multiplication
   Product/multiplication of two complex numbers \( z_1 =x_1 +iy_1 \) and \( z_2 =x_2 +iy_2 \) is defined as
   \( z_1 .z_2 =( x_1 +iy_1 ).( x_1 +iy_2 )\)
   or \( z_1 .z_2 =x_1 x_2 +ix_1 y_2 +ix_2 y_1 +{{i}^{2}}y_1 y_2 \)
   or \( z_1 .z_2 =( x_1 x_2 -y_1 y_2 )+i( x_1 y_2 +x_2 y_1 )\)
   The product is neither scalar nor the vector product of ordinary vector analysis. This departure is due to \( i^2=-1\)

Also, complex number C is a field. Thus, the complex number satisfy all Field axioms as below.

1. Closer: \( \forall a,b \in G \implies a \ast b \in G \)
2. Additive associative
   \( \forall a,b,c \in G \implies (a+ b) + c=a + (b + c) \)
3. Additive identity
   \( \forall a \in G \implies a \ast e =e \ast a =a \)
   Additive identity (0,0)
4. Additive Inverse
   \( \forall a \in G \implies a \ast (a^{-1}) =(a^{-1}) \ast a =e\)
   Additive inverse of z=x+iy is -z=-x-iy
5. Additive commutative
   \( \forall a,b \in G \implies a \ast b =b \ast a \)
6. Distributive
   \( \forall a,b,c \in G \implies (a + b)\bullet c =a c+bc\)
7. Multiplicative associative
   \( \forall a,b,c \in G \implies (a. b) . c=a . (b . c) \)
8. Multiplicative identity
   \( \forall a \in G \implies a \ast e =e \ast a =a\)
   Multiplicative identity (1,0)
9. Multiplicative inverse
   \( \forall a \in G \implies a \ast (a^{-1}) =(a^{-1}) \ast a =e\)
   Multiplicative inverse of non-zero complex number z=x+iy is
   \( z^{-1} =\frac{x}{x^2 +y^2} +i\frac{-y}{x^2 +y^2} \)
   Proof
   Let z=x+iy be non-zero complex number and \( z^{-1} =u+iv\) be its multiplicative inverse, then
   \( zz^{-1} =( 1,0 )\)
   or \( ( x+iy )( u+iv )=( 1,0 )\)
   or \( ( xu-yv )+i( xv+uy )=( 1,0 )\)
   Comparing real and imaginary parts separately, we get
   ux-vy=1 and uy+vx=0
   Solving for u and v , we get
   \( u=\frac{x}{x^2 +y^2} \) and \( v=\frac{-y}{x^2 +y^2} \)
   
   Hence,
   \( z^{-1} =\frac{x}{x^2 +y^2} +i\frac{-y}{x^2 +y^2} \)
10. Multiplicative commutative
    \( \forall a,b \in G \implies a \ast b =b \ast a \)

Algebric Structure

Axioms	Structure	Example
1-2	Semi Group	Example
1-3	Monoid	Example
1-4	Group	Example
1-5	Abelian Group	Example
1-6	Ring	Example
1-7	Assocoative Ring	Example
1-8	Assocoative Ring with Unity	Example
1-9	Division Ring (Skew Field)	Example
1-10	Field	Example

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Polar form

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Let \(z=x+iy\) be a complex number with magnitude r and amplitude \( \theta \) , then
\( r=| z |=\sqrt{x^2 +y^2} \) , and \( \theta \tan^{-1}( \frac{y}{x} )\) Using right angle triangle, we get that
\( \cos \theta=\frac{b}{h}\)
or \( \cos \theta=\frac{x}{r}\)
or \( x= r \cos \theta \)
Similarly,
\( \sin \theta=\frac{p}{h}\)
or \( \sin \theta=\frac{y}{r}\)
or \( y= r \sin \theta \)
Hence, polar form of complex number is
\( z=x+iy\)
or \( z= r\cos \theta +ir \sin \theta \)
or \( z=r( \cos \theta +i\sin \theta )\)
NOTE
A complex number z=x+iy is represented by polar coordinate \( (r,\theta)\) , as
\( z=r \cos \theta +i\sin \theta \)

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Examples: polar and rectangular form

Express the followig complex numbers in trigonometric (polar) forms.

1. -1
   

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   Solution,
   Given that,
   z = -1
   or z = -1 + 0i
   Comparing -1 + 0i with x + iy, We get
   
   x = -1 and y = 0
   Then, the magnitude of z is given by
   
   \( r = \sqrt{x^2 + y^2} \)
   or \(r = \sqrt{(-1)^2 + (0)^2}\)
   so r = 1
   Again, the amplitude of z is giveb by
   \( \theta = \tan^{-1} (\frac{y}{x} )\)
   or \( \theta = \tan^{-1}( \frac{0}{-1} )\)
   or \( \theta = -\tan^{-1} (\frac{0}{1}) \)
   or \( \theta = -\tan^{-1} (0) \)
   According to quadrant rule, (-,+) belong to second quadrant

   (-,+)→(II) ✅	❌(I)←(+,+)
   (-,-)→(III)❌	❌(IV)←(+,-)

   Therefore
   θ = 180
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = cos180 + isin180
2. z=-5+5i
   

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   Solution,
   Given that,
   z =-5+5i
   or z = 5(-1+1i)
   Comparing 5(-1+1i) with x + iy, We get
   x = -1 and y = 1
   Then, the magnitude of z is given by
   
   \( r = 5 \sqrt{(x)^2 + (y)^2} \)
   or \(r = 5 \sqrt{(-1)^2 + (1)^2}\)
   so \(r = 5 \sqrt{2}\)
   Again, the amplitude of z is giveb by
   \( \theta = \tan^{-1} (\frac{y}{x} )\)
   or \( \theta = \tan^{-1}( \frac{1}{-1} )\)
   or \( \theta = -\tan^{-1} (\frac{1}{1}) \)
   or \( \theta = -\tan^{-1} (1) \)
   According to quadrant rule, (-,+) belong to second quadrant

   (-,+)→(II) ✅	❌(I)←(+,+)
   (-,-)→(III)❌	❌(IV)←(+,-)

   Therefore
   θ = 135
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = cos135 + i sin135
   or\( z=5\sqrt{2}( \cos \frac{3\pi }{4}+\sin \frac{3\pi }{4}i )\)
3. z=3i
   

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   Solution,
   Given that,
   z = 3i
   or z = 3(0 + i)
   Comparing 3(0 + i) with x + iy, We get
   x = 0 and y = 1
   Then, the magnitude of z is given by
   
   \( r = 3 \sqrt{(x)^2 + (y)^2} \)
   or \(r = 3 \sqrt{(0)^2 + (1)^2}\)
   so \(r = 3\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   Since \( x = 0 \), the amplitude θ directly corresponds to:
   \( \theta = \frac{\pi}{2} \) or \( \theta = 90^\circ \)
   Therefore,
   θ = 90°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = 3(cos90° + i sin90°)
   
4. z = 1 + i
   

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   Solution,
   Given that,
   z = 1 + i
   or z = 1(1 + i)
   Comparing 1(1 + i) with x + iy, We get
   x = 1 and y = 1
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(1)^2 + (1)^2}\)
   so \(r = \sqrt{2}\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{1}{1} \right) \)
   or \( \theta = \tan^{-1} (1) \)
   According to quadrant rule, (+,+) belong to first quadrant
   

   (-,+)→(II) ❌	✅(I)←(+,+)
   (-,-)→(III)❌	❌(IV)←(+,-)

   Therefore,
   θ = 45°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = √2(cos45° + i sin45°)
   
5. z = 1 - i
   

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   Solution,
   Given that,
   z = 1 - i
   or z = 1(1 - i)
   Comparing 1(1 - i) with x + iy, We get
   x = 1 and y = -1
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(1)^2 + (-1)^2}\)
   so \(r = \sqrt{2}\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{-1}{1} \right) \)
   or \( \theta = \tan^{-1} (-1) \)
   According to quadrant rule, (+,-) belong to fourth quadrant
   

   (-,+)→(II) ❌	❌(I)←(+,+)
   (-,-)→(III)❌	✅(IV)←(+,-)

   Therefore,
   θ = 315°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = √2(cos(315°) + i sin(315°))
   
6. z = -1 - i
   

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   Solution,
   Given that,
   z = -1 - i
   or z = 1(-1 - i)
   Comparing 1(-1 - i) with x + iy, We get
   x = -1 and y = -1
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(-1)^2 + (-1)^2}\)
   so \(r = \sqrt{2}\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{-1}{-1} \right) \)
   or \( \theta = \tan^{-1} (1) \)
   According to quadrant rule, (-,-) belong to third quadrant
   

   (-,+)→(II) ❌	❌(I)←(+,+)
   (-,-)→(III)✅	❌(IV)←(+,-)

   Therefore,
   θ = 180° + 45° = 225°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = √2(cos225° + i sin225°)
   
7. z = -1 + i
   

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   Solution,
   Given that,
   z = -1 + i
   or z = 1(-1 + i)
   Comparing 1(-1 + i) with x + iy, We get
   x = -1 and y = 1
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(-1)^2 + (1)^2}\)
   so \(r = \sqrt{2}\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{1}{-1} \right) \)
   or \( \theta = \tan^{-1} (-1) \)
   According to quadrant rule, (-,+) belong to second quadrant
   

   (-,+)→(II) ✅	❌(I)←(+,+)
   (-,-)→(III)❌	❌(IV)←(+,-)

   Therefore,
   θ = 180° - 45° = 135°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = √2(cos135° + i sin135°)
   
8. z = -1 - √3i
   

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   Solution,
   Given that,
   z = -1 - √3i
   or z = 1(-1 - √3i)
   Comparing 1(-1 - √3i) with x + iy, We get
   x = -1 and y = -√3
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(-1)^2 + (-√3)^2}\)
   so \(r = \sqrt{1 + 3} = \sqrt{4} = 2\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{-√3}{-1} \right) \)
   or \( \theta = \tan^{-1} (√3) \)
   According to quadrant rule, (-,-) belong to third quadrant
   

   (-,+)→(II) ❌	❌(I)←(+,+)
   (-,-)→(III)✅	❌(IV)←(+,-)

   Therefore,
   θ = 180° + 60° = 240°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = 2(cos240° + i sin240°)
   
9. z = -1 + √3i
   

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   Solution,
   Given that,
   z = -1 + √3i
   or z = 1(-1 + √3i)
   Comparing 1(-1 + √3i) with x + iy, We get
   x = -1 and y = √3
   Then, the magnitude of z is given by
   
   \( r = \sqrt{(x)^2 + (y)^2} \)
   or \(r = \sqrt{(-1)^2 + (√3)^2}\)
   so \(r = \sqrt{1 + 3} = \sqrt{4} = 2\)
   Again, the amplitude of z is given by
   \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
   or \( \theta = \tan^{-1} \left( \frac{√3}{-1} \right) \)
   or \( \theta = \tan^{-1} (-√3) \)
   According to quadrant rule, (-,+) belong to second quadrant
   

   (-,+)→(II) ✅	❌(I)←(+,+)
   (-,-)→(III)❌	❌(IV)←(+,-)

   Therefore,
   θ = 180° - 60° = 120°
   Therefore, the polar form of z is
   z = r(cosθ + isinθ)
   or z = 2(cos120° + i sin120°)
   
10. z = -2√3 + 2i
    

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    Solution,
    Given that,
    z = -2√3 + 2i
    or z = 2(-√3 + i)
    Comparing 2(-√3 + i) with x + iy, We get
    x = -√3 and y = 1
    Then, the magnitude of z is given by
    
    \( r = 2 \times \sqrt{(x)^2 + (y)^2} \)
    or \(r = 2 \times \sqrt{(-√3)^2 + (1)^2}\)
    so \(r = 2 \times \sqrt{3 + 1} = 2 \times \sqrt{4} = 2 \times 2 =4\)
    Again, the amplitude of z is given by
    \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
    or \( \theta = \tan^{-1} \left( \frac{1}{-√3} \right) \)
    or \( \theta = \tan^{-1} \left( -\frac{1}{√3} \right) \)
    According to quadrant rule, (-,+) belong to second quadrant
    

    (-,+)→(II) ✅	❌(I)←(+,+)
    (-,-)→(III)❌	❌(IV)←(+,-)

    Therefore,
    θ ≈ 180° - 30° = 150°
    Therefore, the polar form of z is
    z = r(cosθ + isinθ)
    or z = 4(cos150° + i sin150°)
    
11. z = -2 + 2√3i
    

    Solution 👉 Click Here

    Solution,
    Given that,
    z = -2 + 2√3i
    or z = 2(-1 + √3i)
    Comparing 2(-1 + √3i) with x + iy, We get
    x = -1 and y = √3
    Then, the magnitude of z is given by
    
    \( r = 2 \times \sqrt{(x)^2 + (y)^2} \)
    or \(r = 2 \times \sqrt{(-1)^2 + (√3)^2}\)
    so \(r = 2 \times \sqrt{1 + 3} = 2 \times \sqrt{4} = 2 \times 2 = 4\)
    Again, the amplitude of z is given by
    \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
    or \( \theta = \tan^{-1} \left( \frac{√3}{-1} \right) \)
    or \( \theta = \tan^{-1} \left( -√3 \right) \)
    According to quadrant rule, (-,+) belong to second quadrant
    

    (-,+)→(II) ✅	❌(I)←(+,+)
    (-,-)→(III)❌	❌(IV)←(+,-)

    Therefore,
    θ ≈ 180° - 60° = 120°
    Therefore, the polar form of z is
    z = r(cosθ + isinθ)
    or z = 4(cos120° + i sin120°)
    
12. z = -3 - √3i
    

    Solution 👉 Click Here

    Solution,
    Given that,
    z = -3 - √3i
    or z = (-3) + (-√3)i
    Comparing (-3) + (-√3)i with x + iy, We get
    x = -3 and y = -√3
    Then, the magnitude of z is given by
    
    \( r = \sqrt{(x)^2 + (y)^2} \)
    or \(r = \sqrt{(-3)^2 + (-√3)^2}\)
    so \(r = \sqrt{9 + 3} = \sqrt{12} = 2\sqrt{3}\)
    Again, the amplitude of z is given by
    \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
    or \( \theta = \tan^{-1} \left( \frac{-√3}{-3} \right) \)
    or \( \theta = \tan^{-1} \left( \frac{1}{√3} \right) \)
    According to quadrant rule, (-,-) belong to third quadrant
    

    (-,+)→(II) ❌	❌(I)←(+,+)
    (-,-)→(III)✅	❌(IV)←(+,-)

    Therefore,
    \( \theta = 180° + 30° = 210° \)
    Therefore, the polar form of z is
    \( z = r(\cos \theta + i \sin \theta) \)
    or \( z = 2\sqrt{3}(\cos 210° + i \sin 210°) \)
    
13. z = √3 - 3i
    

    Solution 👉 Click Here

    Solution,
    Given that,
    z = √3 - 3i
    or z = √3 + (-3)i
    Comparing √3 + (-3)i with x + iy, We get
    x = √3 and y = -3
    Then, the magnitude of z is given by
    
    \( r = \sqrt{(x)^2 + (y)^2} \)
    or \(r = \sqrt{(\sqrt{3})^2 + (-3)^2}\)
    so \(r = \sqrt{3 + 9} = \sqrt{12} = 2\sqrt{3}\)
    Again, the amplitude of z is given by
    \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
    or \( \theta = \tan^{-1} \left( \frac{-3}{\sqrt{3}} \right) \)
    or \( \theta = \tan^{-1} \left( -\sqrt{3} \right) \)
    According to quadrant rule, (+, -) belong to fourth quadrant
    

    (-,+)→(II) ❌	❌(I)←(+,+)
    (-,-)→(III)❌	✅(IV)←(+,-)

    Therefore,
    \( \theta = 360° - 60° = 300° \)
    Therefore, the polar form of z is
    \( z = r(\cos \theta + i \sin \theta) \)
    or \( z = 2\sqrt{3}(\cos 300° + i \sin 300°) \)
    

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The essence of Polar form

The relevance of complex number in polar form is that multiplication and division are simpler with this form than the Cartesian form.

Let \( z_1 =r_1 (\cos \theta_1 +i\sin \theta_1 )\) and \( z_2 =r_2 (\cos \theta_2 +i\sin \theta_2 )\) be two complex numbers, then
\( z_1 z_2 =r_1 r_2 (\cos (\theta_1 +\theta_2 )+i\sin (\theta_1 +\theta_2 ))\)
Proof
Given that \( z_1 =r_1 (\cos \theta_1 +i\sin \theta_1 )\) and \( z_2 =r_2 (\cos \theta_2 +i\sin \theta_2 )\) . Thus
\( z_1 z_2 = r_1 (\cos \theta_1 +i\sin \theta_1 ) \times r_2 (\cos \theta_2 +i\sin \theta_2 ) \)
or \( z_1 z_2 = r_1 r_2 (\cos \theta_1 +i\sin \theta_1 )(\cos \theta_2 +i\sin \theta_2 )\)
or \( z_1 z_2 = r_1 r_2 [\cos \theta_1 (\cos \theta_2 +i\sin \theta_2 )+i\sin \theta_1 (\cos \theta_2 +i\sin \theta_2 ) ]\)
or \( z_1 z_2 = r_1 r_2 [\cos \theta_1 \cos \theta_2 +i\cos \theta_1\sin \theta_2 +i\sin \theta_1\cos \theta_2 -\cos \theta_1\sin \theta_2 ]\)
or \( z_1 z_2 = r_1 r_2 [(\cos \theta_1 \cos \theta_2-\cos \theta_1\sin \theta_2 ) +i(\cos \theta_1\sin \theta_2 +\sin \theta_1\cos \theta_2) ]\)
or \( z_1 z_2 = r_1 r_2 [\cos (\theta_1 +\theta_2) +i \sin (\theta_1+ \theta_2)]\)

Theorem

\( arg( z_1 z_2 )=arg z_1 +argz_2 \)

Proof

Let \( z_1 =r_1 ( \cos \theta_1 +i \sin \theta_1 )\) and \( z_2 =r_2 ( \cos \theta_2 +i\sin \theta_2 )\) then
\( z_1 .z_2=r_1 .r_2 [ \cos ( \theta_1 +\theta_2 )+i\sin ( \theta_1 +\theta_2 ) ]\)
Thus,
\( arg( z_1 z_2 )=argz_1 +argz_2 \)

Note:
Any complex number z has infinite arguments; all differ by multiple of \( 2\pi \) . The principal value is in the interval \( [ -\pi ,\pi ]\)

Some important property

1. Let \( z_1 \text{ and } z_2 \) be two complex number then \( arg( z_1 .z_2 )=argz_1 +argz_2 \)
   The argument of product of two complex number is sum of their arguments.
   Proof
   Let \( z_1 =r_1 e^{ i\theta_1}\) and \( z_2 =r_2 e^{ i\theta_2}\) then
   \( z_1 z_2 =( r_1 e^{ i\theta_1} )( r_2 e^{ i\theta_2} )\)
   or \( z_1 z_2 =( r_1 r_2 )e^{i( \theta_1 +\theta_2 )}\)
   Hence,
   \( arg( z_1 .z_2 )=\theta_1 +\theta_2 =argz_1 +argz_2 \)
2. Let\( z_1 \text{ and }z_2 \) be two complex number then,
   \( arg( \frac{z_1 }{z_2 } )=argz_1 -argz_2 \)
   
   The argument of quotient of two complex number is difference of their arguments.
3. Argument of complex number of the form \( z=a+i.0, a > 0\) is 0
4. Argument of complex number of the form \( z=a+i.0, a < 0\) is \( \pi \)
5. Argument of complex number of the form \( z=0+i.b, b > 0\) is \( \frac{\pi }{2}\)
6. Argument of complex number of the form \( z=0+i.b, b < 0\) is \( -\frac{\pi }{2}\)

Drag the point for intaraction : 👇

If \( z =r( \cos \theta +i\sin \theta )\) then show that \( z^{-1} =\frac{1}{r}( \cos \theta -i\sin \theta )\)

Question

Let \( z=r( \cos \theta +i\sin \theta )\) be a complex number and its inverse is \( z^{-1} =R( \cos \phi +i\sin \phi )\) then
\( zz^{-1} =1\)
or \( r( \cos \theta +i\sin \theta )R( \cos \phi +i\sin \phi )=( 1,0 )\)
or \( rR\{ \cos ( \theta +\phi )+i\sin ( \theta +\phi ) \} =( 1,0 )\)
or \( rR\cos ( \theta +\phi )=1 \text{ and } rR\sin ( \theta +\phi )=0\)
Here
\( rR\sin ( \theta +\phi )=0\)
or \(\sin ( \theta +\phi )=0\)
or \( \theta +\phi =0\)
or \( \theta = - \phi \)
So,
\( rR\cos ( \theta +\phi )=1 \)
or \( rR=1\)
or \( R=\frac{1}{r}\)
Thus,
\( z^{-1} =\frac{1}{r}( \cos \theta -i\sin \theta )\)

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De-Moivre’s theorem

Let \( z=r \cos \theta +i\sin \theta \) be a complex number then \( z^n =r^n (\cos n\theta +i\sin n\theta )\) where n is a positive integer.
Proof

1. Case 1: n=1
   Then
   \( z =r (\cos \theta +i\sin \theta )\)
   or \( z^1 =r^1 (\cos 1.\theta +i\sin 1.\theta )\)
   So,
   or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=1
   
2. Case 2: n=2
   \( z^2 =z.z\)
   or \( z^2 = r (\cos \theta +i\sin \theta ) \times r (\cos \theta +i\sin \theta ) \)
   or \( z^2 = r^2 (\cos \theta +i\sin \theta )(\cos \theta +i\sin \theta )\)
   or \( z^2 = r^2 [\cos \theta (\cos \theta +i\sin \theta )+i\sin \theta (\cos \theta +i\sin \theta )\)
   or \( z^2 = r^2 [\cos \theta \cos \theta +i\cos \theta\sin \theta +i\sin \theta\cos \theta -\cos \theta\sin \theta ]\)
   or \( z^2 = r^2 [(\cos \theta \cos \theta-\sin \theta \sin \theta ) +i(\cos \theta\sin \theta +\sin \theta\cos \theta) ]\)
   or \( z^2 = r^2 [\cos (\theta +\theta) +i \sin (\theta+ \theta)]\)
   or \( z^2 = r^2 [\cos 2\theta +i \sin 2\theta]\)
   So,
   or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=2
   
3. Case 3: We assume the same formula is true for n = k, so we have
   \( (\cos\theta + i\sin\theta)^k = r^k(\cos(k\theta) + i\sin(k\theta))\)
   So,
   or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=k
   
4. Case 4: Now, we prove for n = k + 1,
   \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^k(\cos\theta + i\sin\theta)^k r (\cos\theta + i\sin\theta)\)
   or \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^k(\cos(k\theta) + i\sin(k\theta)) r(\cos\theta + i\sin\theta)\)
   or \( [r(\cos\theta + i\sin\theta)]^{k + 1} = r^{k+1}[(\cos(k\theta) \cos\theta-\sin(k\theta)\sin\theta )+i (\cos\theta\sin(k\theta) + \sin\theta\cos(k\theta))]\)
   or \( [r(\cos\theta + i\sin\theta)]^{k + 1} =r^{k+1}[\cos(k\theta+\theta)+i \sin(k\theta+\theta )]\)
   or \( [r(\cos\theta + i\sin\theta)]^{k + 1} =r^{k+1}[\cos(k+1)\theta+i \sin(k+1)\theta]\)
   So,
   or \( z^n =r^n (\cos n\theta +i\sin n\theta )\) when n=k+1
   
5. Using case 1-case 4, for any number \( n \in Z\) , we have
   \( [r(\cos\theta + i\sin\theta)]^n =r^n[\cos(n\theta)+i \sin(n\theta)]\)

Example 1

Compute \( (1+i)^6\)
Solution
Since
\( 1+i= \sqrt{2} \left ( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right ) \)
or \( 1+i= \sqrt{2} \left ( \cos \frac{\pi }{4} +i\sin \frac{\pi }{4} \right ) \)
We get
\(r=\sqrt{2} \) and \( \theta=\frac{\pi }{4}\)
Thus,
\( ( 1+i )^6 =\sqrt{2}^6 \left [ \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right ]^6\)
or \( ( 1+i )^6 =\sqrt{2}^6[ \cos \left (6 \frac{\pi }{4} \right )+i\sin \left (6 \frac{\pi }{4} \right )]\)
or \( ( 1+i )^6 =8 (\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2})\)
or \( ( 1+i )^6 =-8i\)

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nth root of Complex number

If \( Z=r(\cos \theta +i\sin \theta )\) be a complex number then the nth root of z is
\( \sqrt[n]{Z}=\sqrt[n]{r} \left ( \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ) \)
Proof
Given that Z is a complex number. Also let, nth root of Z is W such that \( W=R(\cos \phi +i\sin \phi )\)
Now we have
\( \sqrt[n]{Z}=W\)
or \( W^n=Z\)
or \( [R(\cos \phi +i\sin \phi )]^n=r(\cos \theta +i\sin \theta )\)
or \( R^n(\cos (n\phi) +i\sin (n\phi) )=r(\cos \theta +i\sin \theta )\)
Equating real and Imaginary parts, we get
\( R^n=r\) and \( \cos (n\phi)= \cos \theta\) and \( \sin (n\phi) =\sin \theta \)
or \( R=\sqrt[n]{r}\) and \( n\phi= \theta +2k \pi \)
or \( R=\sqrt[n]{r}\) and \( \phi= \frac{(\theta +2k\pi )}{n}\)
Thus, nth root of \( Z=r(\cos \theta +i\sin \theta )\) is
\( W=R(\cos \phi +i\sin \phi )\)
or \( W=\sqrt[n]{r} \left ( \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ) \)

square roots of i

Example 1

Find the square roots of i

We know that
\( \sqrt[n]{z}=\sqrt[n]{r} \left [ \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ] \)

Solution
Since \( i=\cos \frac{\pi }{2} +i\sin \frac{\pi }{2} \) , we get
r=1 and \( \theta=\frac{\pi }{2}\)
Hence the first square roots of i is
\( w_1[i=0] =\cos \frac{\frac{\pi }{2}+0}{2} +i\sin \frac{\frac{\pi }{2}+0}{2} =\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\)
The second square root of i is

\( w_2[i=1] =\cos \frac{\frac{\pi }{2}+2\pi }{2} +i\sin \frac{\frac{\pi }{2}+2\pi }{2} =\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}=\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\)

Example 2

In electrical engineering, a circuit has an impedance represented by the complex number. Z=8+6i ohms. The engineers need to design a component with an impedance that, when cubed, matches the original impedance.

1. Calculate the magnitude ∣Z∣ and angle θ of the original impedance.
2. Determine the cube root of the original impedance in polar form.
3. Design a new component with an impedance Zn such that (Zn)^3 matches the original impedance.
4. Express the new impedance in rectangular form and calculate its magnitude and angle.

Find the cube roots of 8+6i

We know that
\( \sqrt[n]{z}=\sqrt[n]{r} \left [ \cos \frac{(\theta +2k \pi )}{n} +i\sin \frac{(\theta +2k\pi )}{n} \right ] \)

Solution
Since \( 8+6i= 10 \left ( \cos 36.87 +i\sin 36.87 \right ) \) , we get
r=1 and \( \theta=36.87 \)
Hence the first cube roots of 8+6i is
\( w_1[i=0] =\cos \frac{36.87+0}{3} +i\sin \frac{36.87+0}{3} \)
The second cube root of 8+6i is
\( w_2[i=1] =\cos \frac{36.87+2 \pi}{3} +i\sin \frac{36.87+2 \pi}{3} \)
The third cube root of 8+6i is
\( w_2[i=2] =\cos \frac{36.87+4 \pi}{3} +i\sin \frac{36.87+4 \pi}{3} \)

10th roots of 1

The figure shows 10th root of 1.
Drag the value of k=0,1,2,...,9

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Properties of cube root of unity

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Exponenrial Form

Show/Hide 👉 Click Here

Euler's formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function. It states that for any real number x:
\( e^{i \theta}=\cos \theta+i\sin \theta\)
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions.

When x = π, Euler's formula evaluates to \( e^{i\pi}+1=0\), which is known as Euler's identity.

Given Euler’s exponential form,
\( e^{i\theta}=\cos \theta +i\sin \theta \)
Thus, complex number \( z=r( \cos \theta +i\sin \theta )\) is defined as
\( z=re^{i\theta}\)
The significance of exponential form of complex number is that we can easily compute conjugate and inverse.
For example,
\( \bar{z}=re^{- i\theta}\) and \( z^{-1} =\frac{1}{r} e^{- i\theta}\)

Proof of the Formula

1. Function Method
   

   Consider the function f(θ) given by
   \( f(\theta )=\frac {\cos \theta +i\sin \theta }{e^{i\theta }}\)
   or\( f(\theta )= e^{- i\theta } ( \cos \theta +i\sin \theta) \)
   Differentiating gives by the product rule, we have
   \( f' (\theta )=0\)
   Thus, f(θ) is a constant.
   Since f(0) = 1, then f(θ) = 1 for all θ, and thus
   \( 1= e^{- i\theta } ( \cos \theta +i\sin \theta)\)
   or\( e^{ i\theta }= \cos \theta +i\sin \theta\)
   This completes the proof.

2. Series Method
   

   We know that
   \(e^x=\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} \)
   \(\cos x= 1- \frac{x^2}{2!}+ \frac{x^4}{4!}-\)
   \(\sin x= x-\frac{x^3}{3!} +\frac{x^5}{5!}-... \)
   Using power series expansion, we get
   \(e^{ i\theta }= \sum_{n=0}^\infty \frac{(i\theta)^n}{n!}\)
   or \(e^{ i\theta }= 1+ \frac{(i\theta)^1}{1!}+ \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!}+ \frac{(i\theta)^4}{4!}+...\)
   or \(e^{ i\theta }= 1+ i\theta- \frac{\theta^2}{2!} -i \frac{\theta^3}{3!}+ \frac{\theta^4}{4!}+...\)
   or \(e^{ i\theta }= \left ( 1- \frac{\theta^2}{2!}+ \frac{\theta^4}{4!}-...\right ) +i \left (\theta-\frac{\theta^3}{3!} +... \right ) \)
   or \(e^{ i\theta } = \cos \theta +i\sin \theta\)
   This completes the proof.

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