1.5: Roots of Complex Numbers
- Page ID
- 76204
Recall that if \(z=x+iy\) is a nonzero complex number, then it can be written in polar form as
\(z=r(cosθ+isinθ)\)
where \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta \) is the angle, in radians, from the positive x-axis to the ray connecting the origin to the point \(z\).
Now, de Moivre’s formula establishes that if \(z=r(cosθ+isinθ)\) and \(n\) is a positive integer, then
\(z^{n}=r^{n}(cosn\theta +isinn\theta )\)
Let \(w\) be a complex number. Using de Moivre’s formula will help us to solve the equation
\(z^{n}=w\)
for \(z\) when \(w\) is given.
Suppose that \(w=r(cosθ+isinθ)\) and \(z=\rho(cos\psi +isin\psi )\) Then de Moivre’s formula gives
\(z^{n}=\rho ^{n}(cosn\psi +isinn\psi )\)
It follows that
\(\rho ^{n}=r=\left | w \right |\)
by uniqueness of the polar representation and
\(n\Psi =\theta +k(2\pi )\),
where \(k\) is some integer. Thus
\(z=\sqrt[n]{r}[cos(\frac{\theta }{n}+\frac{2k\pi }{n})+isin(\frac{\theta }{n}+\frac{2k\pi }{n})]\).
Each value of \(k=0,1,2,…,n−1\) gives a different value of \(z\). Any other value of \(k\) merely repeats one of the values of \(z\) corresponding to \(k=0,1,2,…,n−1\). Thus there are exactly \(n\)th roots of a nonzero complex number.
Using Euler’s formula:
\(e^{i\theta }=cos\theta +isin\theta \),
the complex number \(z=r(cos\theta +isin\theta) \\) can also be written in exponential form as
\(z=re^{i\theta }\)
Thus, the \(n\)th roots of a nonzero complex number \(z≠0\) can also be expressed as
\(\begin{eqnarray}\label{expform}
z=\sqrt[n]{r}\;\mbox{exp}\left[i\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)\right]
\end{eqnarray}\)
where \(k=0,1,2,...,n-1\).
The applet below shows a geometrical representation of the \(n\)th roots of a complex number, up to \(n=10\). Drag the red point around to change the value of \(z\) or drag the sliders.
- Code
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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, 0, n-1)
Exercise \(\PageIndex{1}\)
From the exponential form (1) of the roots, show that all the \(n\)th roots lie on the circle \(\left | z \right |=\sqrt[n]{r}\) about the origin and are equally spaced every \(\frac{2\pi }{n}\) radians, starting with argument \(\frac{\theta }{n}\).