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1.5: Roots of Complex Numbers Recall that if

is a nonzero complex number, then it can be written in polar form as

where

and

is the angle, in radians, from the positive x-axis to the ray connecting the origin to the point .

Now, de Moivre’s formula establishes that if

Let

for

and

is a positive integer, then

be a complex number. Using de Moivre’s formula will help us to solve the equation

when

is given.

Suppose that

and

Then de Moivre’s formula gives

It follows that

by uniqueness of the polar representation and , where

is some integer. Thus .

Each value of corresponding to

gives a different value of . Any other value of merely repeats one of the values of . Thus there are exactly th roots of a nonzero complex number.

Using Euler’s formula: , the complex number \(z=r(cos\theta +isin\theta) \\) can also be written in exponential form as

Thus, the th roots of a nonzero complex number

where

can also be expressed as

.

The applet below shows a geometrical representation of the th roots of a complex number, up to around to change the value of or drag the sliders.

Juan Carlos Ponce Campuzano 1.5.1 9/8/2021

. Drag the red point

https://math.libretexts.org/@go/page/76204

Code Enter the following script in GeoGebra to explore it yourself and make your own version. The symbol # indicates comments. #Complex number Z = 1 + ί #Modulus of Z r = abs(Z) #Angle of Z theta = atan2(y(Z), x(Z)) #Number of roots n = Slider(2, 10, 1, 1, 150, false, true, false, false) #Plot n-roots nRoots = Sequence(r^(1 / n) * exp(

ί * ( theta / n + 2 * pi * k / n ) ), k,

Exercise From the exponential form (1) of the roots, show that all the th roots lie on the circle equally spaced every radians, starting with argument .

Juan Carlos Ponce Campuzano 1.5.2 9/8/2021

about the origin and are

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