De Moivre’s formula (or) De Moivre’s theorem is related to complex numbers. We can expand the power of a complex number just like how we expand the power of any binomial. But De Moivre’s formula simplifies the process of finding the power of a complex number much simple. To apply De Moivre’s formula, the complex number first needs to be converted into polar form. [1]
This theorem holds utmost importance in the universe of complex numbers as it helps connect the field of trigonometry to the intricacies of complex numerals. It also helps obtain relationships between various trigonometric functions of different angles. It is so-called because this theorem was propounded by one of the most notable mathematicians in history, De Moivre, who contributed a lot to the fields of probability, algebra, etc. The theorem is also referred to as De Moivre’s Formula or De Moivre’s Identity. [2]
DeMoivre’s Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for [7]
Euler’s formula is ubiquitous in mathematics, physics, chemistry, and engineering. [2]
De Moivre’s formula is a precursor to Euler’s formula
which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
One can derive de Moivre’s formula using Euler’s formula and the exponential law for integer powers,
since Euler’s formula implies that the left side is equal to (cos x + sin x) n while the right side is equal to [8]
Thanks to Abraham de Moivre we have this useful formula:
[ r(cos θ + i sin θ) ] n = r n (cos nθ + i sin nθ)
But what does it do?
The formula allows us multiply a complex number by itself (as many times as we want) in one go! [3]
De Moivre’s Theorem is an essential theorem when working with complex numbers. This theorem can help us easily find the powers and roots of complex numbers in polar form. [4]
Let’s use a simple example to see how to use De Moivre’s Theorem.
Many of the References and Additional Reading websites and Videos will assist you with understanding and applying De Moivre’s Theorem.
cis is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. The notation is less commonly used in mathematics than Euler’s formula, e ix , which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries. [5]
Euler’s formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler’s formula states that for any real number x:
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x (“cosine plus i sine”). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler’s formula. [1]
Euler’s formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation “our jewel” and “the most remarkable formula in mathematics”. [2]
When x = π, Euler’s formula may be rewritten as e iπ + 1 = 0 or e iπ = -1, which is known as Euler’s identity. [6]
