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Complex Number

Complex Number

Complex numbers are numbers containing a real part and an imaginary part. The real part is equal to an ordinary real number in value, while the imaginary part is equal to an imaginary value with an imaginary unit √-1 in unit and an ordinary real number in magnitude. A complex number z is therefore usually expressed as x+𝑖y algebraically.

Complex Plane

Rectangular Coordinates

The expression of a complex number of the form z=x+𝑖y can be identified as the two elements of a double tuple

• x=Re z and x is called the real part of complex number z
• y=Im z and y is called the imaginary part of complex number z

Both x and y are real numbers and 𝑖 is the imaginary unit. The set of complex numbers can therefore be represented in the complex plane ℂ, with both vertical and horizontal axes are real number value. While real numbers can be considered as complex numbers whose imaginary part is equal to zero. Real numbers is therefore the subset of the complex numbers. and the complex plane can be identified with ℝ².

Polar Coordinates

Consider z=x+𝑖y∈ℂ, z≠0. The coordinates of z can also be described by the distance r from the origin, r=|z| and the angle 𝜃 between the positive x-axis and the line segment from orign 0 to complex number z. In other words, (r, 𝜃) are the polar coordinates of z.

Through geometric conversion, the Cartesian coordinates can also be expressed as polar representation of z in terms of polar coordinates.

x=rcos𝜃
y=rsin𝜃
∵z=x+𝑖y⇒z=rcos𝜃+𝑖rsin𝜃⇒z=r(cos𝜃+𝑖sin𝜃)

Exponential Notation

Exponential notation e𝑖𝜃 is a more convenient notation or compact notation for complex number, cos 𝜃+𝑖sin 𝜃.

ex=1+x+x2/2!+x3/3!+x4/4!+x5/5!+x6/6!+x7/7!+⋯
⇒e𝑖x=1+𝑖x+(𝑖x)2/2!+(𝑖x)3/3!+(𝑖x)4/4!+(𝑖x)5/5!+(𝑖x)6/6!+(𝑖x)7/7!+⋯
⇒e𝑖x=1+𝑖x+𝑖2x2/2!+𝑖3x3/3!+𝑖4x4/4!+𝑖5x5/5!+𝑖6x6/6!+𝑖7x7/7!+⋯
∵𝑖2=-1, 𝑖3=-𝑖, 𝑖4=1, 𝑖5=i, ⋯
⇒e𝑖x=1+𝑖x-x2/2!-𝑖x3/3!+x4/4!+𝑖x5/5!-x6/6!-𝑖x7/7!+⋯
⇒e𝑖x=(1-x2/2!+x4/4!-x6/6!+⋯)+(𝑖x-𝑖x3/3!+𝑖x5/5!-𝑖x7/7!+⋯)
⇒e𝑖x=(1-x2/2!+x4/4!-x6/6!+⋯)+𝑖(x-x3/3!+x5/5!-x7/7!+⋯)
∵cos x=1-x2/2!+x4/4!-x6/6!+⋯ and
 sin x=x-x3/3!+x5/5!-x7/7!+⋯
⇒e𝑖x=cos x+𝑖sin x

Therefore exponential notation can be used as the polar form of complex numbers

z=r(cos𝜃+𝑖sin𝜃)=re𝑖x

Similarly,

e𝑖𝜃=e𝑖(𝜃+2𝜋)=e𝑖(𝜃+4𝜋)=⋯=e𝑖(𝜃+2k𝜋), k∈ℤ

For examples,

e𝑖𝜋/2=cos(𝜋/2)+𝑖sin(𝜋/2)=i
e𝑖𝜋=cos(𝜋)+𝑖sin(𝜋)=-1
e2𝜋𝑖=cos(2𝜋)+𝑖sin(2𝜋)=1
e-𝑖𝜋/2=cos(-𝜋/2)+𝑖sin(-𝜋/2)=-𝑖
e𝑖𝜋/4=cos(𝜋/4)+𝑖sin(𝜋/4)=(1+𝑖)/√2

Algebraic and Geometric of Complex Number

Addition of Complex Numbers

Since the real unit of real part is 1 and the imaginary unit of imagibary part is 𝑖, the real and imaginary parts of a complex number should be manipulated accordingly.

Algebraically, the addition of two complex numbers z=x+𝑖y and w=u+𝑖v is

z+w=(x+𝑖y)+(u+𝑖v)=(x+u)+𝑖(y+v)

In other words, Re(z+w)=Re x+Re w and Im(z+w)=Im z+Im w

Geometrically, the addition of two complex numbers corresponds to the vector addition of the two corresponding complex number vectors.

Modulus of Complex Number

By definition, the modulus of a complex number z=x+𝑖y is the length or magnitude of the vector z:

|z|=√(x²+y²)
⇒|z|²=x²+y²

Multiplication of Complex Numbers

The multiplication of two complex numbers z=x+𝑖y and w=u+𝑖v can be manipulated as an ordinary multiplication:

zw=(x+𝑖y)(u+𝑖v)=xu+𝑖xv+𝑖yu+𝑖²yv
∵𝑖=√-1; ∴𝑖²=-1
⇒zw=(x+𝑖y)(u+𝑖v)=xu+𝑖xv+𝑖yu+𝑖²yv=xu-yv+𝑖(xv+yu)

Algebraically, the multiplication of two complex numbers z=x+𝑖y and w=u+𝑖v is

zw=(x+𝑖y)(u+𝑖v)=xu-yv+𝑖(xv+yu)∈ℂ

The usual properties hold:

• associative: (z₁z₂)z₃=z₁(z₂z₃)

  (z₁z₂)z₃=((x+𝑖y)(u+𝑖v))(p+𝑖q)=((xu-yv)+𝑖(xv+yu))(p+𝑖q)=((xu-yv)p-(xv+yu)q)+𝑖((xu-yv)q+(xv+yu)p)
  ⇒(z₁z₂)z₃=(xup-yvp-xvq-yuq)+𝑖(xuq-yvq+xvp+yup)=(x(up-vq)-y(vp+uq))+𝑖(x(uq+vp)+y(up-vq))
  ⇒(z₁z₂)z₃=x(up-vq)+𝑖²y(vp+uq)+𝑖x(uq+vp)+𝑖y(up-vq)=(x+𝑖y)((up-vq)+𝑖(uq+vp))
  ⇒(z₁z₂)z₃=(x+𝑖y)(up+𝑖²vq+𝑖uq+𝑖vp)=(x+𝑖y)((u+𝑖v)(p+𝑖q))=z₁(z₂z₃)

• commutative: z₁z₂=z₂z₁
  

  z₁z₂=(x+𝑖y)(u+𝑖v)=(xu-yv)+𝑖(xv+yu)=xu-yv+𝑖xv+𝑖yu=xu+𝑖yu+𝑖²yv+𝑖xv
  ⇒z₁z₂=u(x+𝑖y)+𝑖v(x+𝑖y)=(u+𝑖v)(x+𝑖y)=z₂z₁

• distributive: z₁(z₂+z₃)=z₁z₂+z₁z₃ or  (z₂+z₃)z₁=z₂z₁+z₃z₁ commutatively
  

  z₁(z₂+z₃)=(x+𝑖y)((u+𝑖v)+(p+𝑖q))=(x+𝑖y)((u+p)+𝑖(v+q))=(x(u+p)-y(v+q))+𝑖(x(v+q)+y(u+p))=xu+xp-yv-yq+𝑖xv+𝑖xq+𝑖yu+𝑖yp
  ⇒z₁(z₂+z₃)=xu+𝑖xv+𝑖²yv+𝑖yu+xp+𝑖xq+𝑖²yq+𝑖yp=x(u+𝑖v)+𝑖y(𝑖v+u)+x(p+𝑖q)+𝑖y(𝑖q+p)
  ⇒z₁(z₂+z₃)=(x+𝑖y)(u+𝑖v)+(x+𝑖y)(p+𝑖q)=z₁z₂+z₁z₃
  

Multiplication of Imaginary Unit 𝑖

By definition, an imaginary unit 𝑖 is equal to √-1. Therefore 𝑖²=-1. The multiplication of imaginary unit is

i=0+1i⇒i²=(0+1i)(0+1i)=(0*0+𝑖²*1*1+𝑖(0*1+1*0)=(0*0-1*1+𝑖(0*1+1*0)=-1

Therefore

• 𝑖=√-1
• 𝑖²=𝑖*𝑖=-1
• 𝑖³=𝑖²*𝑖=-1*𝑖=-𝑖
• 𝑖⁴=𝑖²*𝑖²=-1*-1=1
• 𝑖⁵=𝑖⁴*𝑖=1*𝑖=𝑖
• 𝑖⁶=𝑖⁵*𝑖=𝑖*𝑖=-1

Complex Conjugate of Complex Numbers

By definition, if complex number z=x+𝑖y then z̅=x-𝑖y is the complex conjugate of z

The properties of complex conjugate is:

• z̿=z
  z=x+𝑖y⇒z̅=x-𝑖y⇒z̿=x-𝑖y=x+𝑖y=z
• z+w=z̅+w̅
• z/w=z̅/w̅; w≠0
• |z|=|z̅|
• zz̅=(x+𝑖y)(x-𝑖y)=x²+y²=|z|²=|z̅|²
• 1/z=z̅/zz̅=z̅/|z|²;  z≠0
• if z∈ℝ then z=z̅
• Re z=(z+z̅)/2; Im z=(z-z̅)/2i
  z+z̅=(x+𝑖y)+(x-𝑖y)=2x⇒x=(z+zÌ…)/2=Re z
  z-z̅=(x+𝑖y)-(x-𝑖y)=𝑖2y⇒y=(z+zÌ…)/2𝑖=Im z

Division of Complex Numbers

The division of complex numbers z/w can be performed by making use of the complex conjugate of complex number w, since 1/z=z̅/|z|². Suppose that z=x+𝑖y and w=u+𝑖v.

z/w=(x+𝑖y)/(u+𝑖v)=(x+𝑖y)(u-𝑖v)/(u+𝑖v)(u-𝑖v)=((xu+yv)+𝑖(-xv+yu))/(u²+v²+𝑖(-uv+vu))
⇒z/w=(x+𝑖y)/(u+𝑖v)=((xu+yv)/(u²+v²))+𝑖((yu-xv)/(u²+v²))

More Properties of Complex Numbers

• |z*w|=|z|*|w|
• |z|=0 if and only if z=0
• -|z|≤Re z≤|z|
• -|z|≤Im z≤|z|
• |z+w|≤|z|+|w|; Triangle Inequality
• |z-w|≥|z|-|w|; Reverse Triangle Inequality

Argument of Complex Numbers

The argument of a complex number z is the counterclockwise angle 𝜃 measured from the real positive axis to the line segment from orign 0 to complex number z. The argument of a complex number is not unique and argument is a multi-valued function.

By definition, the principal argument of z, Arg z, is the value of 𝜃 for which -𝜋<𝜃≤𝜋 and the argument of z is

arg z={Arg z+2𝜋k:k=0,±1,±2,⋯},z≠0.

Since z=x+𝑖y=r(cos𝜃+𝑖sin𝜃), if r=1 then

Arg 𝑖=𝜋/2
Arg 1=0
Arg(-1)=𝜋
Arg(-𝑖)=-𝜋/2
Arg(1-𝑖)=-𝜋/4

Properties of Exponential Notation

• |e𝑖𝜃|=1
  |e𝑖𝜃|=|cos𝜃+𝑖sin𝜃|=√(cos²𝜃+sin²𝜃)=√1=1
• e𝑖𝜃=e-𝑖𝜃
  e𝑖𝜃=cos𝜃+𝑖sin𝜃=cos𝜃-𝑖sin𝜃=cos𝜃+𝑖sin(-𝜃)=cos(-𝜃)+𝑖sin(-𝜃)=e-𝑖𝜃
• 1/(e𝑖𝜃)=e-𝑖𝜃
  1/(e𝑖𝜃)=(e𝑖𝜃)/((e𝑖𝜃)(e𝑖𝜃))=(e-𝑖𝜃)/((e𝑖𝜃)(e-𝑖𝜃))=(e-𝑖𝜃)/1=e-𝑖𝜃
• e𝑖(𝜃+𝜑)=e𝑖𝜃e𝑖𝜑
  e𝑖(𝜃+𝜑)=cos(𝜃+𝜑)+𝑖sin(𝜃+𝜑)=cos𝜃cos𝜑-sin𝜃sin𝜑+𝑖(sin𝜃cos𝜑+cos𝜃sin𝜑)
  ⇒e𝑖(𝜃+𝜑)=cos𝜃cos𝜑+𝑖²sin𝜃sin𝜑+𝑖sin𝜃cos𝜑+𝑖cos𝜃sin𝜑=(cos𝜃+𝑖sin𝜃)(cos𝜑+𝑖sin𝜑)
  ⇒e𝑖(𝜃+𝜑)=e𝑖𝜃e𝑖𝜑

Properties of Argument Function

• arg(z̅)=-arg(z)
  arg(z̅)=arg(re𝑖𝜃)=arg(re-𝑖𝜃)=-𝜃=-arg(re𝑖𝜃)=-arg(z)
• arg(1/z)=-arg(z)
  arg(1/z)=arg(1/(e𝑖𝜃))=arg(e𝑖𝜃)=arg(z̅)=-arg(z)
• arg(z₁z₂)=arg(z₁)+arg(z₂)
  arg(z₁z₂)=arg(e𝑖𝜃e𝑖𝜑)=arg(e𝑖(𝜃+𝜑))=𝜃+𝜑=arg(e𝑖𝜃)+arg(e𝑖𝜑)

Multiplication in Polar Form 

Consider z₁=r₁e𝑖𝜃₁ and z₂=r₂e𝑖𝜃₂, the multiplication in polar form is

z₁z₂=r₁e𝑖𝜃₁r₂e𝑖𝜃₂=r₁r₂e𝑖(𝜃₁+𝜃₂)

De Moivre's Formula 

De Moivre's Formula states that for any complex number (and, in particular, for any real number) x and integer n it holds that

(cos(x)+𝑖sin(x))ⁿ=cos(nx)+𝑖sin(nx)

By polar form

e𝑖𝜃e𝑖𝜃=e𝑖(𝜃+𝜃)=e𝑖(2𝜃)
(e𝑖𝜃)³=e𝑖(2𝜃)e𝑖𝜃=e𝑖(3𝜃)
(e𝑖𝜃)ⁿ=e𝑖n𝜃
also true for negative n, (e𝑖𝜃)⁻ⁿ=(1/(e𝑖𝜃))ⁿ=(e-𝑖𝜃)ⁿ
⇒(cos(x)+𝑖sin(x))ⁿ=(e𝑖x)ⁿ=e𝑖nx=cos(nx)+𝑖sin(nx)

Consequences of De Moivre's formula

De Moivre's formula can be used to derive equations for sine and cosine

For examples, n=3

(cos(x)+𝑖sin(x))³=cos³(x)+3cos²(x)(𝑖sin(x))+3cos(x)(𝑖sin(x))²+(𝑖sin(x))³
⇒(cos(x)+𝑖sin(x))³=cos³(x)-3cos(x)sin²(x)+𝑖(3cos²(x)sin(x)-sin³(x))=cos(3x)+𝑖sin(3x)
⇒cos(3x)=cos³(x)-3cos(x)sin²(x) and sin(3x)=3cos²(x)sin(x)-sin³(x)

Nth Root of Complex Number

By definition, let w be a complex number. An nth root of w is a complex number z such that zⁿ=w.

By polar form, let w=𝜌e𝑖𝜑 , and z=re𝑖𝜃 , then

 zⁿ=w
⇒(re𝑖𝜃)ⁿ=𝜌e𝑖𝜑 
⇒rⁿe𝑖n𝜃=𝜌e𝑖𝜑 
⇒rⁿ=𝜌, and e𝑖n𝜃=e𝑖𝜑 
⇒r=ⁿ√𝜌, and n𝜃=𝜑+2k𝜋, k∈ℤ
⇒𝜃=𝜑/n+2k𝜋/n, k=0,1,2,⋯,n-1
⇒w1/n=ⁿ√𝜌 e𝑖(𝜑/n+2k𝜋/n), k=0,1,2,⋯,n-1

Nth Root of Unity

By definition, the nth roots of 1 are called the nth roots of unity.

    

By polar form, let 1=1e𝑖0 , then

11/n=ⁿ√1 e𝑖(0/n+2k𝜋/n), k=0,1,2,⋯,n-1
⇒11/n=e𝑖(2k𝜋/n), k=0,1,2,⋯,n-1