De Moivre’s Theorem Calculator

Effortlessly raise complex numbers to any power using this powerful tool.

Calculator

Enter the components of your complex number in polar form: r(cos θ + i sin θ) and the power n.

What is De Moivre’s Theorem?

De Moivre’s Theorem, also known as De Moivre’s Formula, provides a straightforward method for computing powers of complex numbers. The theorem is a crucial link between complex numbers and trigonometry, stating that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the n-th power can be found by raising the modulus to the n-th power and multiplying the angle by n. It’s an essential tool for anyone in engineering, physics, and higher mathematics. A common mistake is trying to apply it directly to a complex number in rectangular form (a + bi); you must first convert it to polar form. This calculator helps you calculate using De Moivre’s Theorem with ease.

De Moivre’s Theorem Formula and Explanation

The formula is elegant in its simplicity. It simplifies what would otherwise be a tedious process of repeated multiplication of a complex number.

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))

This shows that to raise a complex number to a power ‘n’, you raise its magnitude ‘r’ to the power ‘n’ and multiply its angle ‘θ’ by ‘n’.

Variables Table

Variable	Meaning	Unit	Typical Range
r	The Modulus	Unitless (magnitude)	r ≥ 0
θ (theta)	The Argument or Angle	Degrees or Radians	Any real number
n	The Power	Unitless	Any integer
i	The Imaginary Unit	N/A	i² = -1

Practical Examples

Example 1: A simple integer power

Let’s calculate (1 + i)⁴.

1. Convert to Polar Form:

• Modulus (r) = √(1² + 1²) = √2 ≈ 1.414
• Angle (θ) = tan⁻¹(1/1) = 45°
• So, 1 + i = √2(cos 45° + i sin 45°)

2. Apply De Moivre’s Theorem (n=4):

• New Modulus (r⁴) = (√2)⁴ = 4
• New Angle (4 * θ) = 4 * 45° = 180°
• Result: 4(cos 180° + i sin 180°)

3. Convert back to Rectangular Form:

• Result = 4(-1 + i * 0) = -4

Example 2: A negative power

Let’s calculate [2(cos 60° + i sin 60°)]^-3.

1. Inputs:

• Modulus (r) = 2
• Angle (θ) = 60°
• Power (n) = -3

2. Apply De Moivre’s Theorem:

• New Modulus (r⁻³) = 2⁻³ = 1/8 = 0.125
• New Angle (-3 * θ) = -3 * 60° = -180°
• Result: 0.125(cos(-180°) + i sin(-180°))

3. Convert back to Rectangular Form:

• Result = 0.125(-1 + i * 0) = -0.125

For more examples, check out our guide on Euler’s formula explained, which is closely related to this theorem.

How to Use This De Moivre’s Theorem Calculator

1. Enter the Modulus (r): Input the magnitude or length of your complex number. This value must be non-negative.
2. Enter the Angle (θ): Input the angle of your complex number. Use the dropdown to specify whether you are providing the angle in degrees or radians.
3. Enter the Power (n): Provide the integer power you wish to raise the complex number to. This can be positive, negative, or zero.
4. Review the Results: The calculator automatically updates, showing the final result in standard rectangular (a + bi) form. It also provides intermediate values like the new modulus and angle, which are key to understanding the transformation.
5. Visualize: The chart shows your original complex number as a vector (in blue) and the resulting vector (in green), illustrating the scaling and rotation that occurred.

Key Factors That Affect the Calculation

• The Modulus (r): Directly impacts the magnitude of the result. If r > 1, the result’s magnitude will grow. If 0 ≤ r < 1, it will shrink. If r = 1, the point stays on the unit circle.
• The Power (n): Acts as a multiplier for both rotation and scaling. A larger `n` leads to a larger rotation and a more dramatic change in magnitude.
• The Sign of the Power (n): A positive power results in counter-clockwise rotation, while a negative power causes clockwise rotation.
• The Angle (θ): Defines the starting position of the complex number vector on the plane before the transformation is applied.
• Angle Units: It is crucial to select the correct unit (degrees or radians). Mixing them up is a common source of error when people manually calculate using De Moivre’s Theorem. Our polar to rectangular converter can help with these conversions.
• Integer Power: The standard theorem is defined for integer values of ‘n’. Fractional powers are used to find roots of complex numbers, a related but different process. See our tool on roots of unity for that topic.

Frequently Asked Questions (FAQ)

What is a complex number’s polar form?

It’s a way of representing a complex number using its distance from the origin (modulus, r) and its angle relative to the positive real axis (argument, θ).

Why must I convert to polar form first?

De Moivre’s Theorem is defined specifically for the polar representation of complex numbers. It leverages the geometric properties of multiplication (scaling magnitudes and adding angles), which is why the formula works so cleanly. A standard complex number calculator might hide this step, but it’s fundamental to the process.

What happens if the power (n) is 0?

Any non-zero complex number raised to the power of 0 is 1. The formula holds: r⁰(cos(0*θ) + i sin(0*θ)) = 1(cos(0) + i sin(0)) = 1(1 + 0) = 1.

What if my complex number is just a real number (e.g., 5)?

A positive real number ‘x’ has a polar form of x(cos 0° + i sin 0°). A negative real number ‘-x’ has a polar form of x(cos 180° + i sin 180°). The theorem works perfectly.

Can I use this theorem to find roots (e.g., the cube root)?

Yes, finding an n-th root is the same as raising to the power of (1/n). This is a generalization of the theorem used to find the n-th roots of a complex number, which typically yields ‘n’ distinct roots.

What are radians?

Radians are an alternative unit for measuring angles, based on the radius of a circle. 2π radians is equal to 360 degrees. They are the standard unit for angles in most higher-level mathematics.

How does the chart work?

It plots points on a 2D plane where the x-axis is the “real” axis and the y-axis is the “imaginary” axis. It draws a line from the origin (0,0) to the original complex number and another to the calculated result to show how the vector changed. Explore this concept with our article on phasor diagrams.

Is De Moivre’s Theorem related to Euler’s Formula?

Yes, they are very closely related. Euler’s formula states e^(iθ) = cos θ + i sin θ. De Moivre’s theorem can be seen as a direct consequence of Euler’s formula, since (e^(iθ))^n = e^(inθ), which then translates back to r^n(cos(nθ) + i sin(nθ)).

Related Tools and Internal Resources

Explore other concepts related to complex numbers and trigonometry with our other calculators and guides.