De Moivre's Formula

Powers of Complex Numbers

De Moivre's formula generalizes Euler's formula to powers, showing how raising a complex number to a power multiplies its angle. Watch how transforms rotations on the unit circle.

De Moivre's Formula

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

Using Euler's formula, this can also be written as: $(e^{i\theta})^n = e^{in\theta}$

$\theta$

The angle (in radians) of the original complex number on the unit circle.

$n$

The power to which we raise the complex number.

$\cos\theta + i\sin\theta$

The original complex number in trigonometric form, equivalent to $e^{i\theta}$ .

$\cos(n\theta) + i\sin(n\theta)$

The result after raising to power n, equivalent to $e^{in\theta}$ .

Prerequisites & Learning Path

Before diving deep into De Moivre's Formula, it's helpful to have a solid understanding of foundational concepts. Here are the key prerequisites that will help you master this identity.

Complex Numbers

Understanding of complex number arithmetic and representation.

Required

Calculus

Strong foundation in real analysis and calculus.

Required

Euler's Formula

Understanding of the connection between exponentials and trigonometry.

Recommended

Historical Timeline

16th century

Cardano and others begin working with square roots of negative numbers.

18th century

Euler develops fundamental relationships between complex numbers and trigonometry.

19th century

Cauchy and Riemann establish the rigorous foundations of complex analysis.

20th century

Complex analysis becomes central to many areas of mathematics and physics.

The complete history of De Moivre's formula

Named after Abraham de Moivre (1667–1754), a French mathematician who worked extensively with probability and complex numbers. De Moivre discovered the relationship while studying probability and trigonometric identities, though the formula may have been known to others.

The modern proof using Euler's formula makes the connection to complex exponentials explicit. Euler's exponential form $e^{i\theta} = \cos\theta + i\sin\theta$ provides a more elegant foundation, showing that De Moivre's formula is simply a consequence of the properties of exponentials.

De Moivre's work on probability led him to study binomial expansions and trigonometric identities, where he discovered patterns that would later be formalized as his eponymous formula.

What each symbol means: a deep dive

$\theta$

The angle (in radians) of the original complex number on the unit circle. It represents rotation: $\theta = 0$ is the positive real axis, $\theta = \pi/2$ is the positive imaginary axis, and $\theta = \pi$ is the negative real axis. In quantum mechanics, θ often represents a phase angle that determines interference patterns.

$n$

The power to which we raise the complex number. When n is a positive integer, it multiplies the angle: $(e^{i\theta})^n = e^{in\theta}$ . For fractional powers like $n = 1/2$ , we get square roots. Negative powers reverse the rotation direction. In physics, n can represent the number of rotations or the order of symmetry.

$\cos\theta + i\sin\theta$

The original complex number in trigonometric form, equivalent to $e^{i\theta}$ . This represents a point on the unit circle. The real part ( $\cos\theta$ ) is the horizontal coordinate, and the imaginary part ( $\sin\theta$ ) is the vertical coordinate. This form connects geometry (circles) to algebra (complex numbers).

$\cos(n\theta) + i\sin(n\theta)$

The result after raising to power n, equivalent to $e^{in\theta}$ . The angle has been multiplied by n, so if we started at $30° = \pi/6$ , raising to the power 3 gives us $90° = \pi/2$ . This is the key insight: exponentiation multiplies angles, just as multiplication of real numbers multiplies magnitudes.

Complete proof: step by step

Why this proof method works (A beginner's guide)

You might wonder: why do we use Euler's formula to prove De Moivre's formula? The answer is that Euler's formula gives us a way to work with complex exponentials that makes exponentiation behave naturally. Just like how $(2^3)^4 = 2^{12}$ in regular arithmetic, we want $(e^{i\theta})^n$ to equal $e^{in\theta}$ .

The key insight is that complex exponentials follow the same rules as real exponentials: when you raise a power to another power, you multiply the exponents. This is why $(e^{i\theta})^n = e^{in\theta}$ works—it's the same rule you learned in algebra, just extended to complex numbers!

Step 1: Start with Euler's formula

We begin with Euler's formula, which connects exponential and trigonometric functions:

e^{i\theta} = \cos\theta + i\sin\theta

Step 2: Raise both sides to the power n

Now we raise both sides of Euler's formula to the power $n$ :

(e^{i\theta})^n = (\cos\theta + i\sin\theta)^n

Step 3: Apply the exponential rule

Using the property of exponentials that $(a^b)^c = a^{bc}$ , we simplify the left side:

(e^{i\theta})^n = e^{in\theta}

This works because complex exponentials follow the same rules as real exponentials—when you raise a power to another power, you multiply the exponents.

Step 4: Apply Euler's formula to the right side

Now we apply Euler's formula again, but this time with $n\theta$ instead of $\theta$ :

e^{in\theta} = \cos(n\theta) + i\sin(n\theta)

Step 5: Conclude De Moivre's formula

Since both sides equal $e^{in\theta}$ , we can equate them:

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

This is De Moivre's formula! It tells us that raising a complex number on the unit circle to a power $n$ is equivalent to multiplying its angle by $n$ . ✓

The result

This is a fundamental identity in complex analysis! De Moivre's formula connects complex number multiplication to angle addition, making it essential for understanding rotations, oscillations, and periodic phenomena. It's the foundation for Fourier transforms, signal processing, and quantum mechanics.

Examples for beginners

If you're new to complex numbers and De Moivre's formula, here are some concrete examples that show how it works:

Example 1: Square (n = 2) - Double angle

When n = 2, De Moivre's formula gives: $(\cos\theta + i\sin\theta)^2 = \cos(2\theta) + i\sin(2\theta)$ . This doubles the angle. For example, if $\theta = 30° = \pi/6$ , then $(\cos 30° + i\sin 30°)^2 = \cos 60° + i\sin 60°$ . This is useful for deriving double-angle trigonometric identities like $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ .

Example 2: Cube (n = 3) - Triple angle

When n = 3: $(\cos\theta + i\sin\theta)^3 = \cos(3\theta) + i\sin(3\theta)$ . This triples the angle. If $\theta = 20°$ , then raising to the power 3 gives us $60°$ . This leads to triple-angle formulas that are used in trigonometry and calculus.

Example 3: Negative powers - Reverse rotation

For negative powers, we get: $(\cos\theta + i\sin\theta)^{-n} = \cos(-n\theta) + i\sin(-n\theta) = \cos(n\theta) - i\sin(n\theta)$ . This rotates in the opposite direction. If $n = 2$ and $\theta = 30°$ , then raising to the power -2 gives us $-60°$ , which is the same as rotating $60°$ clockwise instead of counterclockwise.

Example 4: Fractional powers - Finding roots

When $n = 1/2$ , we get the square root: $(\cos\theta + i\sin\theta)^{1/2} = \cos(\theta/2) + i\sin(\theta/2)$ . This halves the angle. This is how we find the square roots of complex numbers. For example, the square root of i (which is at $90°$ ) is at $45°$ , which is $\frac{1 + i}{\sqrt{2}}$ .

Common Mistakes

When working with De Moivre's Formula, students often encounter several common pitfalls. Understanding these mistakes can help you avoid them and apply the identity correctly.

Forgetting to check domain restrictions

Incorrect:

Applying De Moivre's Formula without verifying that all variables satisfy the required conditions (e.g., denominators not zero, square roots of non-negative numbers).

Correct approach:

Always check the domain of each variable before applying the identity. Verify that all conditions are met.

Why this matters: Many identities have implicit domain restrictions that must be satisfied for the identity to hold.

Order of operations errors

Incorrect:

Incorrectly applying operations when using De Moivre's Formula, especially with fractions or exponents.

Correct approach:

Follow the correct order of operations: parentheses, exponents, multiplication/division, addition/subtraction.

Why this matters: Order of operations is critical for correctly applying mathematical identities.

Treating complex numbers like real numbers

Incorrect:

Applying real number properties incorrectly to De Moivre's Formula when working with complex numbers.

Correct approach:

Remember that complex numbers have different properties than real numbers, especially regarding ordering and square roots.

Why this matters: Complex analysis requires careful attention to the unique properties of complex numbers.

Quantum Implications

De Moivre's formula is fundamental to quantum mechanics, where complex exponentials describe how quantum states evolve and rotate in Hilbert space. Quantum rotation operators use powers of complex exponentials:

\hat{R}_z(\phi) = e^{-i\hat{J}_z\phi/\hbar} = \cos(\phi/2) - i\sin(\phi/2)\hat{\sigma}_z

When we apply this rotation n times, De Moivre's formula tells us that the total rotation is n times the original angle. This is crucial for understanding how quantum gates work in quantum computing.

In quantum computing, qubit rotations are described by powers of complex exponentials. For example, a rotation gate $R(\theta)$ applied n times gives $R(n\theta)$ , which follows directly from De Moivre's formula. This allows quantum algorithms to efficiently perform multiple rotations.

The formula also appears in the quantum Fourier transform, where powers of roots of unity (which come from De Moivre's formula) are used to decompose quantum states into frequency components. This is the basis of Shor's algorithm for factoring large numbers.

Philosophical Implications

De Moivre's formula reveals a deep pattern: exponentiation in the complex plane multiplies angles. This suggests that mathematical operations have geometric meanings that transcend their algebraic definitions.

The formula connects three seemingly different concepts: trigonometry (angles and rotations), algebra (powers and exponents), and geometry (the unit circle). This unity suggests that mathematics has an underlying structure that we discover rather than invent.

Some philosophers argue that the elegance of De Moivre's formula (and its connection to Euler's formula) indicates a pre-existing mathematical reality. Others see it as a consequence of our chosen definitions and notation. The truth likely lies somewhere in between: the relationships exist, but our language shapes how we express them.

The relationship between mathematical patterns and physical reality is particularly striking in the case of De Moivre's formula. The fact that exponentiation in the complex plane has a simple geometric meaning (angle multiplication) suggests that mathematical operations might reflect deeper structures in reality. When we observe that quantum rotations follow the same pattern as complex exponentials, we must ask: Is this because we've discovered a fundamental truth about how rotations work in nature, or because we've invented a mathematical language that happens to describe physical phenomena? The remarkable fit between mathematical structure and physical behavior suggests that mathematics might be more than just a human invention—it could be a discovery of patterns that exist independently of human thought.

Patents and practical applications

Finding roots

De Moivre's formula is essential for finding the n-th roots of complex numbers, which connects to the roots of unity. To find the n-th root, we divide the angle by n.

Trigonometric identities

Can be used to derive multiple-angle formulas for sine and cosine by expanding the binomial and equating real and imaginary parts.

Quantum mechanics

Powers of complex exponentials appear in quantum phase evolution and rotation operators. The formula describes how quantum states rotate in Hilbert space.

Signal processing

Used in Fourier analysis and digital signal processing for frequency domain operations, where powers represent frequency multiplication.

Is it fundamental?

Is De Moivre's formula fundamental, or does it reveal something about our mathematical framework? It shows that complex exponentiation has a simple geometric meaning: multiply the angle. This principle appears throughout mathematics and physics, from Fourier analysis to quantum mechanics. But does this indicate fundamental truth, or mathematical elegance?

The formula appears in signal processing (frequency domain operations), quantum computing (rotation gates), number theory (roots of unity), and differential equations (solutions involving complex exponentials). We might question: does this ubiquity suggest that rotations and oscillations are fundamental to nature, or does it reflect that complex numbers provide an especially convenient language for describing these phenomena? Are we discovering structure, or creating useful representations?

Open questions and research frontiers

Quantum error correction

How can De Moivre's formula help design better quantum error correction codes? The relationship between powers and rotations might reveal new symmetries in quantum codes.

Higher-dimensional extensions

Can De Moivre's formula be extended to quaternions or octonions? These higher-dimensional number systems might reveal new patterns in rotation and symmetry.

Computational complexity

The Fast Fourier Transform uses roots of unity (derived from De Moivre's formula). Are there faster algorithms waiting to be discovered using similar principles?

Topological phases

Topological insulators have wavefunctions with complex phases. Does De Moivre's formula reveal new topological invariants or phases of matter?

People and milestones

Abraham de Moivre (1667–1754) discovered the relationship while working on probability theory and trigonometric identities. His work laid the foundation for complex analysis, though the modern exponential form came later with Euler.

Leonhard Euler (1707–1783) provided the exponential form that makes De Moivre's formula elegant and easy to prove. His work unified trigonometry and complex analysis.

Modern contributors include: Richard Feynman, who used complex exponentials extensively in quantum mechanics; Peter Shor, whose factoring algorithm relies on roots of unity (derived from De Moivre's formula); and countless engineers who use the formula in signal processing and communications.

Related Identities

External References

Discovered Patterns

Research Notes

Loading notes...

Power (n)

Original:

e^{i\theta}

Result:

e^{i\cdot1.0\theta}

Visualization Guide:

━ Blue circle — The unit circle, representing all complex numbers with magnitude 1. In De Moivre's formula, we raise a complex number on the unit circle to a power, which multiplies its angle by that power while keeping the magnitude at 1.

━ Green dot and line — Represents the original complex number $e^{i\theta}$ at angle θ. The green line connects the origin to the green dot, showing the vector representation of the original complex number. As θ rotates, the green dot traces out the unit circle.

━ Orange dot and line — Represents the result after raising to power n: $e^{in\theta} = (e^{i\theta})^n$ . The orange dot is at angle nθ, which is n times the original angle. The orange line shows the result vector. Notice how the orange dot rotates n times faster than the green dot—this demonstrates De Moivre's formula in action.

━ White axes — The coordinate system with the real axis (horizontal) and imaginary axis (vertical). The real axis represents $\cos(n\theta)$ and the imaginary axis represents $\sin(n\theta)$ for the result.

How it works: De Moivre's formula states that raising a complex number on the unit circle to a power n multiplies its angle by n. If the original is at angle θ, the result is at angle nθ. This is because $(e^{i\theta})^n = e^{in\theta}$ , and the angle of $e^{in\theta}$ is nθ.

Controls: Adjust the power n using the slider. When n = 1, the orange and green dots coincide (no change). When n = 2, the orange dot rotates twice as fast as the green dot. When n = 3, three times as fast, and so on. Try n = 2 to see the square, n = 3 for the cube, and n = 4 for the fourth power.

Powers of Complex Numbers

Prerequisites & Learning Path

Complex Numbers

Calculus

Euler's Formula

Historical Timeline

The complete history of De Moivre's formula

What each symbol means: a deep dive

Complete proof: step by step

Examples for beginners

Try solving a related problem:

Common Mistakes

Forgetting to check domain restrictions

Order of operations errors

Treating complex numbers like real numbers

Quantum Implications

Philosophical Implications

Patents and practical applications

Is it fundamental?

Open questions and research frontiers

People and milestones

Related Identities

External References

Discovered Patterns

Research Notes