De Moivre’s Theorem Calculator: Calculate (1+i)^23

De Moivre’s Theorem Calculator

Calculate powers of complex numbers like (1 + i)²³ and beyond.

Enter Complex Number (a + bi)ⁿ

Real Part (a)

The ‘a’ value in a + bi. For (1 + i)²³, this is 1.

Imaginary Part (b)

The ‘b’ value in a + bi. For (1 + i)²³, this is 1.

Power (n)

The exponent ‘n’. For (1 + i)²³, this is 23.

Angle Unit for Intermediate Steps

Choose the unit for displaying the angle (argument) θ.

Primary Result (Rectangular Form)

-4096 – 4096i

Intermediate Values

Formula Explanation

Complex Plane Visualization

Visualization of the original complex number (blue) and the result (red). Vectors are scaled to fit.

What is De Moivre’s Theorem?

De Moivre’s Theorem, also known as De Moivre’s Formula, is a fundamental theorem in mathematics that provides a straightforward way to calculate powers of complex numbers. The theorem states that for any complex number in polar form, z = r(cos(θ) + i sin(θ)), and any integer n, the n^th power of z is given by zⁿ = rⁿ(cos(nθ) + i sin(nθ)). This is incredibly useful because it simplifies the otherwise tedious process of repeatedly multiplying a complex number by itself.

This theorem is used extensively in engineering, physics, and higher mathematics, particularly for tasks involving trigonometry and wave analysis. It elegantly connects complex numbers with trigonometric functions, allowing for the derivation of many trigonometric identities. Anyone needing to find powers or roots of complex numbers will find this theorem, and our calculate 1 i 23 using de moivre’s theorem calculator, indispensable.

The Formula and Explanation

To use De Moivre’s Theorem, you must first convert the complex number from its rectangular form (a + bi) to its polar form (r(cos(θ) + i sin(θ))).

Calculate the Modulus (r): This is the magnitude of the complex number, or its distance from the origin on the complex plane. The formula is: r = √(a² + b²).
Calculate the Argument (θ): This is the angle the number makes with the positive real axis. The formula is: θ = atan2(b, a).
Apply the Theorem: Once you have r and θ, you apply the theorem for the power n: zⁿ = rⁿ(cos(nθ) + i sin(nθ)).

Variables in De Moivre’s Theorem
Variable	Meaning	Unit (Auto-inferred)	Typical Range
z	The complex number	Unitless (a + bi)	Any complex number
r	The Modulus (magnitude)	Unitless	r ≥ 0
θ	The Argument (angle)	Radians or Degrees	-π to π or -180° to 180°
n	The power (exponent)	Integer	Any integer

Practical Examples

Example 1: Calculate (1 + i)²³

This is the default calculation for our tool, a classic problem to calculate 1 i 23 using de moivre’s theorem.

Inputs: a = 1, b = 1, n = 23
Polar Conversion:
- r = √(1² + 1²) = √2 ≈ 1.414
- θ = atan2(1, 1) = π/4 radians or 45°
Apply De Moivre’s Theorem:
- rⁿ = (√2)²³ = 2^11.5 ≈ 5792.6
- nθ = 23 * (π/4) = 5.75π radians or 23 * 45° = 1035°
Convert Back to Rectangular:
- a_final = 5792.6 * cos(1035°) = -4096
- b_final = 5792.6 * sin(1035°) = -4096
Result: -4096 – 4096i

Example 2: Calculate (2 – 3i)⁵

Inputs: a = 2, b = -3, n = 5
Polar Conversion:
- r = √(2² + (-3)²) = √13 ≈ 3.606
- θ = atan2(-3, 2) ≈ -0.983 radians or -56.31°
Apply De Moivre’s Theorem:
- rⁿ = (√13)⁵ ≈ 609.33
- nθ = 5 * (-56.31°) = -281.55°
Convert Back to Rectangular:
- a_final = 609.33 * cos(-281.55°) = 122
- b_final = 609.33 * sin(-281.55°) = 597
Result: 122 + 597i (approximately)

How to Use This Calculator

Using our De Moivre’s Theorem Calculator is simple and intuitive.

Enter the Real Part (a): Input the real component of your complex number.
Enter the Imaginary Part (b): Input the coefficient of ‘i’.
Enter the Power (n): Input the integer exponent you want to raise the complex number to.
Select Angle Unit: Choose whether you want the intermediate angle (θ) to be displayed in radians or degrees. This does not affect the final result.
Interpret the Results: The calculator automatically updates, showing the final answer in rectangular form (a + bi), along with the crucial intermediate steps like the modulus (r) and argument (θ). You can find more details using a Roots of a Complex Number Calculator.

Key Factors That Affect the Result

The final value of a complex number exponentiation depends on several key factors:

The Real and Imaginary Parts (a, b): These values determine the starting point of the vector on the complex plane. Changing them alters both the initial magnitude (r) and angle (θ).
The Modulus (r): The initial length of the vector. The final magnitude will be rⁿ, so it grows or shrinks exponentially with n.
The Argument (θ): The initial angle. The final vector will be rotated to a new angle of n*θ.
The Power (n): This is the most dynamic factor. A large ‘n’ will dramatically increase the magnitude (if r > 1) and cause the vector to rotate many times around the origin. A negative ‘n’ will calculate the inverse power.
Angle Quadrant: The quadrant of the initial angle determines the signs of the resulting real and imaginary parts after rotation.
Unit System (Radians/Degrees): While our calculator lets you view the angle in either unit, all internal calculations for trigonometric functions in JavaScript use radians. Understanding the conversion (180° = π radians) is key to a Complex Number to Polar Form Calculator.

Frequently Asked Questions (FAQ)

What is a complex number?: A complex number is a number that comprises both a real part and an imaginary part, written in the form a + bi, where ‘i’ is the imaginary unit (√-1).
Why is De Moivre’s theorem useful?: It provides a quick method for finding powers and roots of complex numbers, which is much faster than manual repeated multiplication. It’s essential in many areas of science and engineering.
Can this theorem be used to find roots?: Yes, finding an n-th root is the same as raising to the power of 1/n. The theorem can be generalized to handle fractional exponents, though it yields multiple answers (n roots for the n-th root).
How do I handle the angle units?: Our calculator lets you choose your preferred display unit. Internally, all math is done in radians as required by standard trigonometric functions in programming. You can learn more about Complex Numbers In Polar Form online.
What are the limitations?: The main limitation is floating-point precision. For extremely large powers or numbers, small rounding errors can accumulate, though for most practical purposes, the results are highly accurate. De Moivre’s formula as stated is for integer powers.
What is the polar form of a complex number?: Polar form represents a complex number by its distance from the origin (modulus, r) and the angle it makes with the positive real axis (argument, θ). It’s written as r(cosθ + i sinθ).
How exactly is (1 + i)^23 calculated?: First, 1 + i is converted to its polar form, which is √2(cos(45°) + i sin(45°)). Then, De Moivre’s theorem is applied: (√2)^23 * (cos(23*45°) + i sin(23*45°)). This simplifies to 5792.6 * (cos(1035°) + i sin(1035°)), which equals -4096 – 4096i.
What is Euler’s Formula?: Euler’s formula is a related concept, stating that e^iθ = cos(θ) + i sin(θ). This provides the foundation for the exponential form of a complex number (z = re^iθ) and is intimately connected to De Moivre’s theorem.

Enter Complex Number (a + bi)n