De Moivre’s theorem, typically taught towards the end of Pre-Calculus courses, asserts that for any complex number \( z=a+bi \) (where \( a,b \in \mathbb{R} \) and \( i = \sqrt{-1} \)),
\( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) \) for \( n \in \mathbb{N} \), \( r = \sqrt{a^2+b^2} \), and \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) in degree measure.
We can also extend the theorem to provide the \(n^{th}\) root of \( z \) (or \( z^{\frac{1}{n}} \)) in the following manner,
\( z^{\frac{1}{n}} = r^{\frac{1}{n}}\left(\cos\left(\frac{360k}{n} + \frac{\theta}{n}\right) + i\sin\left(\frac{360k}{n} + \frac{\theta}{n}\right)\right) \) for \( n \in \mathbb{N} \), and for index \( k=\{0, 1, 2, ... n-1\} \)
Since \( k \) iterates from 0 to \( n-1 \), \( z \) has \( n \) roots.
De Moivre’s theorem was discovered in 1707 by French mathematician Abraham de Moivre, an early pioneer of the use of complex numbers in mathematics (Maths History, 2004). Complex numbers had not been considered significant until mathematicians such as de Moivre depicted their relation to existing algebraic and geometric fields of math. On its own, de Moivre’s theorem stands as a small - yet absolutely fundamental - support beam of the mathematical world of complex numbers, which has blossomed into titans of theory that shape the world as we know it. For instance, Fourier analysis, which relies on De Moivre’s theorem to solve the complex-valued integral that appears in the Fourier transform, has been instrumental in signal processing and editing (Osgood, 2007). Anything that can be expressed as a waveform - sound, light, ocean waves, etc. - can be deconstructed, understood, and even edited using Fourier analysis.
Given the axiomatic nature of de Moivre’s theorem in the mathematical realm of complex numbers, I sought to use it to generate a theorem of my own (as to my knowledge, no one else has conjured such a theorem) regarding another region of mathematics that involves complex numbers: factoring. Specifically, I sought to factor the sum and difference of \(n^{th}\) powers in an alternate form.
The factored forms of the sum or difference of \(n^{th}\) odd powers, respectively, are typically given by the following formulae:
$$ a^n + b^n = (a + b)\left(a^{n-1} - a^{n-2}b + a^{n-3}b^2 + \ldots + a^2b^{n-3} - ab^{n-2} + b^{n-1}\right) $$
$$ a^n - b^n = (a - b)\left(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \ldots + a^2b^{n-3} + ab^{n-2} + b^{n-1}\right) $$
But, using De Moivre’s theorem, it's possible to derive an alternate form for these expressions.
We begin with the polynomial \(y = x^n - a^n\), for odd \(n \in \mathbb{N}\), and \(a \in \mathbb{R}\). In order to factor such a polynomial, we will first find its zeros:
\(0 = x^n - a^n\) -> \(x^n = a^n\) -> \(x = (a^n)^{\frac{1}{n}}\)
Using exponent rules to simplify \((a^n)^{\frac{1}{n}}\) would eliminate any complex solutions to the equation. Therefore, we will instead use De Moivre’s theorem to allow values of \(x\) (including complex ones!) such that \( y = 0 = x^n - a^n \)
Since \(a \in \mathbb{R}\), we have,
\(x = (a^n)^{\frac{1}{n}} = |a^n|^{\frac{1}{n}}(\cos(\frac{360k^\circ}{n}) + i\sin(\frac{360k^\circ}{n})) = a(\cos(\frac{360k^\circ}{n}) + i\sin(\frac{360k^\circ}{n}))\), for \(k = \{0, 1, 2, \ldots, n-1\}\), and arguments in degree measure.
Now, to obtain the factors of \(y\), we have,
\( y = \prod_{k=0}^{n-1} \left( x - a(\cos(\frac{360k^\circ}{n}) + i\sin(\frac{360k^\circ}{n})) \right) \)
But this can be simplified further. The argument in each factor, \(\frac{360k^\circ}{n}\), will yield angles \(0^\circ, \frac{360(1)^\circ}{n}, \frac{360(2)^\circ}{n}, \ldots, \frac{360(n-2)^\circ}{n}, \frac{360(n-1)^\circ}{n}\). Note that, since \(n\) is odd, one can form \(\frac{n-1}{2}\) distinct pairs of angles that sum to \(360^\circ\) utilizing values of \(k\) that sum to \(n\).
For instance, for \(k=1\), \(k=n-1\), \(\frac{360(1)^\circ}{n} + \frac{360(n-1)^\circ}{n} = 360^\circ\). Since pairs of angles sum to \(360^\circ\), for a given pair \((k_1, k_2)\), where \(k_1 + k_2 = n\),
\(\frac{360k_1^\circ}{n} + \frac{360k_2^\circ}{n} = 360^\circ\) -> \(\frac{360k_2^\circ}{n} = 360^\circ - \frac{360k_1^\circ}{n}\)
Therefore, \(-\frac{360k_1^\circ}{n}\) and \(\frac{360k_2^\circ}{n}\) are coterminal angles.
By extent, for the same pair \((k_1, k_2)\), the factor \(x-a(\cos(\frac{360k_1^\circ}{n}) + i\sin(\frac{360k_1^\circ}{n}))\) must accompany another factor,
\(x-a(\cos(\frac{360k_2^\circ}{n}) + i\sin(\frac{360k_2^\circ}{n})) = x-a(\cos(-\frac{360k_1^\circ}{n}) + i\sin(-\frac{360k_1^\circ}{n})) = x-a(\cos(\frac{360k_1^\circ}{n}) - i\sin(\frac{360k_1^\circ}{n}))\)
Now, we can group the factored expressions for \((k_1, k_2)\) together using a difference of squares:
\((x-a(\cos(\frac{360k_1^\circ}{n}) + i\sin(\frac{360k_1^\circ}{n})))(x-a(\cos(\frac{360k_1^\circ}{n}) - i\sin(\frac{360k_1^\circ}{n}))) = (x-a\cos(\frac{360k_1^\circ}{n}))^2 - (a\sin(\frac{360k_1^\circ}{n}))^2\)
Expanding, and using the identity \(\cos^2(u) + \sin^2(u) = 1\), we have,
\(x^2 - 2ax\cos(\frac{360k_1^\circ}{n}) + a^2 = x^2 - 2ax\cos(\frac{360k_1^\circ}{n}) + a^2\)
For every given pair \((k_1, k_2)\) in the factored form of \(y\), of which we know there are \(\frac{n-1}{2}\), we can express the product of the \(k_1\) and \(k_2\) terms as,
\(x^2 - 2ax\cos(\frac{360k_1^\circ}{n}) + a^2\)
This leads to the final factored form of \(y\):
\(y = x^n - a^n = (x-a) \prod_{k=1}^{\frac{n-1}{2}} \left(x^2 - 2ax\cos(\frac{360k^\circ}{n}) + a^2\right)\)
This is the alternate factored difference of \(n^{th}\) odd powers we set out to find. Similarly, to find the sum of \(n^{th}\) odd powers, we substitute \(-a\) for \(a\), leading to:
\(y = x^n + a^n = (x + a) \prod_{k=1}^{\frac{n-1}{2}} \left(x^2 + 2ax\cos(\frac{360k^\circ}{n}) + a^2\right)\)
The most fascinating aspect of this assertion is the potential for evaluating many trigonometric series based on this polynomial equality. For instance, it might be possible to prove:
\(\sum_{n=1}^{k} \cos\left(\frac{360n^\circ}{2k+1}\right) = -\frac{1}{2}\) for any \(k \in \mathbb{N}\)
This proof and further exploration of the theorem’s implications could unveil new insights into the relationships between trigonometry, polynomials, and complex numbers.
De Moivre’s theorem, crafted over 300 years ago, has indeed enabled significant mathematical advancements and discoveries. Its application spans numerous fields, underscoring the deep interconnectivity of mathematical concepts.