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India
Class XI

Complex Roots of Unity

Lesson

We can use de Moivre's theorem to find solutions to problems like $z^n=1$zn=1. This category of problem is called a complex root of unity problem

Let's have a look at how we can use de Moivre's theorem in this way. 

de Moivre's theorem states that if $z=\left(r,\theta\right)$z=(r,θ) then $z^n=\left(r^n,n\theta\right)$zn=(rn,nθ)

So if we want to solve $z^3=1$z3=1 then we start by stating de Moivre's.

Let, $z=\left(r,\theta\right)$z=(r,θ)

then $z^3=\left(r,\theta\right)^3=\left(r^3,3\theta\right)$z3=(r,θ)3=(r3,3θ)

We can write the number $1$1 in modulus argument form like this: $\left(1,2n\pi\right)$(1,2nπ)

So we can write $\left(r^3,3\theta\right)=\left(1,2n\pi\right)$(r3,3θ)=(1,2nπ)

Therefore by equating like parts we can see that:

$r^3=1$r3=1 and $3\theta=2n\pi$3θ=2nπ

$r=1$r=1 and $\theta=\frac{2n\pi}{3}$θ=2nπ3

 

Now we can find the solutions by letting $n=0,1,2,...$n=0,1,2,...

If $n=0$n=0 $z_1=\left(1,0\right)=1$z1=(1,0)=1
If $n=1$n=1 $z_2=\left(1,\frac{2\pi}{3}\right)=\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$z2=(1,2π3)=cos2π3+isin2π3=12+32i
If $n=2$n=2 $z_3=\left(1,\frac{4\pi}{3}\right)=\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$z3=(1,4π3)=cos4π3+isin4π3=1232i

So the $3$3 solutions to $z^3=1$z3=1 are$1,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}-\frac{\sqrt{3}}{2}i$1,12+32i,1232i

 

Activity

Try this for yourself before you look at the solution. 

Solve the complex roots of unit for $z^n=1$zn=1 for $n=3,4,5$n=3,4,5 ($3$3 already completed as an example above).

Then plot the roots for $z^n=1$zn=1 (for $n=3,4,5$n=3,4,5) on separate Argand diagrams.

What do you notice?

See here for the solution.

As you will have discovered through the activity above, roots of unity occur in conjugate pairs and are evenly spaced around the plane.  For $z^n=1$zn=1, they are evenly spaced by an angle of $\frac{2\pi}{n}$2πn.

Roots of unity are cyclic around the region, starting position equal to $\frac{2\pi}{n}$2πn if $z^n=1$zn=1 and $\frac{\pi}{n}$πn if $z^n=-1$zn=1.

 

Worked Examples

Question 1

Find the cube roots of $1$1 in trigonometric form (with the arguments in degrees).

Write all roots on the same line, separating each one with a comma.

Question 2

Find the cube roots of $1+\sqrt{3}i$1+3i in trigonometric form (with the arguments in degrees).

Write all roots on the same line, separating each one with a comma.

Question 3

Find the cube roots of $8\left(\cos90^\circ+i\sin90^\circ\right)$8(cos90°+isin90°) in rectangular form.

Write all roots on the same line, separating each one with a comma.

Outcomes

11.A.CNQE.1

Need for complex numbers, especially √-1, to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.

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