SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

Products, quotients and roots of complex numbers in polar form. De Moivre’s theorem. Roots of unity

 


Polar representation of a complex number. The polar representation of the complex number


ole.gif

                        z = x + iy


is                                                                                  

                                                                                    

            z = x + iy = r (cos θ + i sin θ)


where


                      x = r cos θ

                      y = r sin θ

                       ole1.gif

                     θ = arctan (y/x)



See Figure 1. The radius vector r is called the modulus or absolute value of the complex number and the polar angle θ is called the amplitude or argument of the number. The argument θ of a complex number z is often denoted by arg z. The abbreviation cis θ is sometimes used for cos θ + i sin θ. In polar coordinates a point P is often specified by the number pair (r, θ).



 


Product of two complex numbers in polar form. The product z1z2 of the two complex numbers


                        z1 = r1 (cos θ1 + i sin θ1)

and

                        z2 = r2 (cos θ2 + i sin θ2)

is

1)                    z1z2 = r1r2[cos (θ1 + θ2) + i sin (θ1 + θ2)] ,


a result which can be easily obtained by utilizing the trigonometric identities


            sin(A ole2.gif B) = sin A cos B ole3.gif cos A sin B

            cos(A ole4.gif B) = cos A cos B ole5.gif sin A sin B .



 

Quotient of two complex numbers in polar form. The quotient z1/z2 of the two complex numbers


                        z1 = r1 (cos θ1 + i sin θ1)

and

                        z2 = r2 (cos θ2 + i sin θ2)

is


2)         z1/z2 = ( r1/r2) [cos (θ1 - θ2) + i sin (θ1 - θ2)]



 


De Moivre’s Theorem. For any complex number z = r( cos θ + i sin θ )

 

3)        zn = [ r(cos θ + i sin θ)]n = rn (cos nθ + i sin n)


This formula holds for every real value of the exponent n. For example, if the exponent is a fraction 1/n, we get


ole6.gif




Rules of arguments. If


            z1 = r1 (cos θ1 + i sin θ1)

            z2 = r2 (cos θ2 + i sin θ2)

            w = za


where a is a real number, then by 1), 2) and 3 ) above


1]        arg (z1z2) = arg z1 + arg z2


2]        arg (z1/z2) = arg z1 - arg z2


3]        arg w = a arg z

ole7.gif


Def. N-th root of a number. Let n be a positive integer. If an = b, then a is said to be the n-th root of b.




Roots of complex numbers in polar form. The n distinct n-th roots of the complex number


           z = r( cos θ + i sin θ)


can be found by substituting successively k = 0, 1, 2, ... , (n-1) in the formula



ole8.gif



Derivation


ole9.gif

The n roots are equally spaced around the circumference of a circle in the complex plane. See Figure 2. If the complex number for which we are computing the n n-th roots is z = ρ( cos θ + i sin θ) the radius of the circle will be


ole10.gif


and the first root w0 corresponding to k = 0 will be at an amplitude of α = θ/n. This root will be followed by the n-1 remaining roots at equal distances apart. The angular amplitude between each root is Δα = 360o/n.





ole11.gif

Example. Suppose we wish to compute the ten 10-th roots of z = 8(cos 150o + i sin 150o) shown in Fig. 3. The ten roots w0, w1, .... , w9 would be spaced evenly around a circle as shown in Fig. 4. The first root, w0, would be at an amplitude of α = 150/10 = 15o. The rest of the roots would be spaced at Δα = 360/10 = 36o intervals.

                                                

Roots of unity. The n n-th roots of 1 are obtained from 5) above by letting r = 1 and θ = 0. They are


ole12.gif

ole13.gif                                                             


for k = 0, 1, 2, ... , (n-1)


Let us denote the root corresponding to k = 1 by w. This root w is then given by

                                                            

ole14.gif                                                               


The n n-th roots of 1 then correspond to powers of w:


        w, w2, w3, ... ,wn


 where wn = 1


The roots are equally spaced around the circumference of a unit circle in the complex plane. See Figure 5.



Primitive roots of unity. Of the n n-th roots of 1 some of the roots may be m-th roots of 1 where m is some integer less than n. For example, the 6 sixth roots of 1 are


            r1 = w

            r2 = w2

            r3 = w3

            r4 = w4

            r5 = w5

            r6 = w6 = 1


Of these, r3 = w3 and r6 = w6 are square roots of 1 and r2 = w2, r4 = w4 and r6 = w6 are cube roots of 1. The primitive roots of 1 are those roots which are not m-th roots of 1 for some 0 < m < n. Thus in the example just given the roots r1 = w and r5 = w5 are primitive roots of 1. In other words, of the n n-th roots of 1, a particular n-th root r is a primitive root if and only if rm ole15.gif 1 for any integer m less than n.



Theorem Let w, w2, w3, ... ,wn be the n n-th roots of 1. Let m be any integer 0 < m < n and let d be the greatest common divisor (m,n) of m and n. If d > 1 then , wm is an n/d-th root of 1.


 Example. Let m = 3 and n = 6. Then d = (m,n) = (3,6) = 3 and n/d = 2. Thus w3 is a square root of 1.



Corollary. The primitive n-th roots of 1 are those and only those n-th roots w, w2, w3, ... ,wn of 1 whose exponents are relatively prime to n.






Roots of a complex number in terms of the roots of unity. Let


                 z = a + bi = r( cos θ + i sin θ )


and


                ole16.gif  



Then the n n-th roots of z are



            z0, wz0, w2z0, ... ,w k-1z0


where  


             ole17.gif  







References.

  James & James. Mathematics Dictionary.

  Brink. A First Year of College Mathematics.

  Spiegel. College Algebra.

  Hauser. Complex Variables with Physical Applications.



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]