What does it mean to find the nth root of 1?

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Today we're going to talk about finding the roots of 1. What does it mean to find the nth root of 1? The nth roots of 1 are the solutions to the equation ${\mathit{x}}^{\mathit{n}}-1=0$.

For square roots, this is easy. The values are $\mathit{x}=\pm \sqrt{1}=\pm 1$. For higher-order roots, it gets a bit more difficult. To find the cube roots of 1 we need to solve the equation ${\mathit{x}}^3-1=0$. We can factor this equation to get

So the first cube root is 1. Now we can use the quadratic formula to get the second and third cube roots.

Calculate the Cube Roots

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In our case a, b, and c are all equal to 1. The other two roots are calculated from these formulas:

a = 1 ; b = 1 ; c = 1;
roots = [];
roots(1) = 1;
roots(2) = (-b + sqrt(b^2 - 4*a*c))/(2*a);    % Use the quadratic formula
roots(3) = (-b - sqrt(b^2 - 4*a*c))/(2*a);

So the full set of cube roots of 1 are:

disp(roots')

   1.0000 + 0.0000i
  -0.5000 - 0.8660i
  -0.5000 + 0.8660i

Displaying Roots in the Complex Plane

Include plots in the Live Editor so students can visualize important concepts.

We can visualize the roots in the complex plane to see their location.

range = 0:0.01:2*pi;                              
plot(cos(range),sin(range),'k')                % Plot the unit circle                 
axis square; box off
ax = gca;
ax.XAxisLocation = 'origin';
ax.YAxisLocation = 'origin';
hold on
plot(real(roots), imag(roots), 'ro')           % Plot the roots

Finding Higher Order Roots

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Once you get past $$n=3$$, things get even trickier. For 4th roots we could use the quartic formula discovered by Lodovico Ferrari in 1540. But this formula is long and unwieldy, and doesn't help us find roots higher than 4. Luckily, there is a better way, thanks to a 17th century French mathematician named Abraham de Moivre.

Abraham de Moivre was born in Vitry in Champagne on May 26, 1667. He was a contemporary and friend of Isaac Newton, Edmund Halley, and James Stirling. https://en.wikipedia.org/wiki/Abraham_de_Moivre

He is best known for de Moivre's theorem that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci numbers linking the nth power of the golden ratio φ to the nth Fibonacci number. He was also the first to postulate the Central Limit Theorem, a cornerstone of probability theory.

De Moivre's theorem states that for any real x and any integer n,

How does that help us solve our problem? We also know that for any integer k,

So by de Moivre's theorem we get

Calculating the nth Roots of 1

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We can use this last equation to find the nth roots of 1. For example, for any value of n, we can use the formula above with values of $\mathit{k}=0\dots \mathit{n}-1$. We can use this MATLAB code to experiment with different values of n:

n = 6;
roots = zeros(1, n);
for k = 0:n-1
    roots(k+1) = cos(2*k*pi/n) + 1i*sin(2*k*pi/n);    % Calculate the roots
end
disp(roots')

   1.0000 + 0.0000i
   0.5000 - 0.8660i
  -0.5000 - 0.8660i
  -1.0000 - 0.0000i
  -0.5000 + 0.8660i
   0.5000 + 0.8660i

Plotting the roots in the complex plane shows that the roots are equally spaced around the unit circle at intervals of $2\pi /\mathit{n}$.

cla
plot(cos(range),sin(range),'k')                   % Plot the unit circle
hold on
plot(real(roots),imag(roots),'ro')              % Plot the roots

Finding the nth Roots of -1, i, and -i

Use additional examples to reinforce important concepts. Modify code during the lecture to answer questions or explore ideas in more depth.

We can find the roots of -1, i, and -i just by using extensions of the approach described above. If we look at the unit circle we see that the values of 1, i, -1, -i appear at angles $0$, $\pi /2$, $\pi$, and $3\pi /2$ respectively.

r = ones(1,4);
theta = [0 pi/2 pi 3*pi/2];
[x,y] = pol2cart(theta,r);
cla
plot(cos(range),sin(range),'k')           % Plot the unit circle
hold on
plot(x, y, 'ro')                          % Plot the values of 1, i, -1, and -i
text(x(1)+0.05,y(1),'1')                  % Add text labels
text(x(2),y(2)+0.1,'i')
text(x(3)-0.1,y(3),'-1')
text(x(4)-0.02,y(4)-0.1,'-i')

Knowing this, we can write the following expression for i:

Taking the nth root of both sides gives

and by de Moivre's theorem we get

Homework

Use live scripts as the basis for assignments. Give students the live script used in the lecture and have them complete exercises that test their understanding of the material.

Use the techniques described above to complete the following exercises:

Exercise 1: Write MATLAB code to calculate the 3 cube roots of i.

% Put your code here

Exercise 2: Write MATLAB code to calculate the 5 fifth roots of -1.

% Put your code here

Exercise 3: Describe the mathematical approach you would use to calculate the nth roots of an arbitrary complex number. Include the equations you used in your approach.

(Describe your approach here.)