What Is De Moivre’s Theorem?
De Moivre’s Theorem is one of the most practical shortcuts in complex-number math. It tells you how to raise a complex number to an integer power (and how to take roots) by moving from rectangular form to polar form. Instead of repeatedly multiplying (a + bi) by itself, you scale the magnitude and multiply the angle. That’s why the theorem shows up in trigonometry, signal processing, electrical engineering, and any topic that treats rotation as multiplication.
If a complex number is written in polar/trigonometric form as z = r(cos θ + i sin θ), then for any integer n: zn = rn(cos(nθ) + i sin(nθ)). In one line, a difficult-looking power becomes two easy operations: raise r to the n and multiply θ by n.
Why Polar Form Makes Complex Arithmetic Easier
Rectangular form (a + bi) is great for addition and subtraction because you combine real parts and imaginary parts directly. But multiplication and powers are not as friendly in rectangular form. Polar form is designed for multiplication because it separates a complex number into a size and a direction. When you multiply two complex numbers in polar form, magnitudes multiply and angles add. Powers are repeated multiplication, so they turn into repeated angle addition—exactly what De Moivre captures.
A useful mental model is rotation. Multiplying by a complex number with magnitude 1 rotates the complex plane by θ. If the magnitude is not 1, it also stretches or shrinks. Raising z to the n repeats that rotate-and-scale action n times.
Rectangular and Polar: How the Conversion Works
A complex number z = a + bi can be converted to polar form by computing: r = √(a² + b²) and θ = atan2(b, a). The magnitude r is always non-negative. The angle θ points from the origin to the point (a, b) and is commonly taken as the principal argument.
To convert back from polar to rectangular form, use: a = r cos θ and b = r sin θ. If your angle differs by a full turn (360° or 2π), you will still land on the same point because sine and cosine repeat.
What Does “Principal Argument” Mean?
The argument (angle) of a complex number is not unique, because θ, θ + 2π, θ − 2π, and so on represent the same direction. Many textbooks define a principal argument Arg(z) in a fixed interval such as (−π, π] or [0, 2π). Your calculator might output one convention while your notes use another. That is normal. As long as angles differ by a multiple of 2π, they represent the same complex number.
How De Moivre’s Theorem Computes Powers
When you compute zn, De Moivre gives you the trigonometric form immediately:
- Convert z to polar form: z = r(cos θ + i sin θ)
- Compute rn and nθ
- Return zn = rn(cos(nθ) + i sin(nθ))
The final step is optional: if you need a + bi, expand with cosine and sine. This calculator outputs both forms (unless you choose otherwise), so you can match the format your homework, lab report, or code expects.
Example: (3 + 4i)2
The complex number 3 + 4i has magnitude r = 5 and angle θ ≈ 53.130102° (or ≈ 0.927295 rad). Squaring it gives r2 = 25 and angle 2θ. Converting back to rectangular form yields the familiar result (3 + 4i)² = −7 + 24i. The power “feels” like a rotation combined with scaling.
Negative Powers: What If n Is Less Than Zero?
Negative powers are just reciprocals. z−n = 1 / zn. In polar form, that means r−n and angle −nθ, because reciprocals invert magnitude and reverse rotation. This calculator supports negative integer exponents in Power mode and will report the result in both forms.
One caution: z must not be 0 for negative powers. Dividing by zero is undefined, and the calculator will flag it as an error.
How De Moivre’s Theorem Produces n-th Roots
Roots are where De Moivre becomes especially powerful. The n-th roots of z are the complex numbers w such that wn = z. In polar form, you take the n-th root of the magnitude and divide the angle by n, but you also must include all possible rotations by full turns:
wk = r1/n(cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)), for k = 0, 1, 2, …, n−1.
This is the core reason complex roots come in sets: adding 2π to an angle doesn’t change z, but dividing by n creates distinct angles separated by 2π/n. The roots sit evenly on a circle of radius r1/n, forming a regular n-gon when plotted.
Why Does a Complex Number Have Exactly n Roots?
For nonzero z, the equation wn = z has n solutions because the complex plane wraps angles in a periodic way. The roots are separated by a constant angular step 2π/n, and after n steps you return to the starting angle. This calculator lists all roots in a table so you can see the structure at a glance.
Degrees vs Radians: Which Is Better for De Moivre?
Both are valid; the “best” choice depends on context. If your problem statement gives angles like 45° or 120°, degrees will be natural. If your formulas come from calculus or physics, radians are standard. The important part is consistency: enter θ in the unit you select. When in doubt, remember that π radians equals 180°.
Common Checks That Catch Mistakes Fast
De Moivre problems often go wrong for simple reasons: an angle unit mismatch, a sign error in b, or forgetting that roots have multiple solutions. These checks help you confirm you’re on track:
- Convert back: Rectangular → polar → rectangular should reproduce z (up to rounding).
- Magnitude logic: |zn| should be |z|n.
- Angle logic: Arg(zn) should be n·Arg(z), modulo 2π.
- Root logic: Each root wk raised to n should return z (small rounding differences are normal).
How to Interpret “Equivalent Angles” in Results
If your output angle is −120° but your book reports 240°, both are the same direction because they differ by 360°. Similarly, an angle like 7π/6 can be written as −5π/6. The calculator may choose a principal angle, but the rectangular form confirms equivalence.
What If Your Input Is Already on the Unit Circle?
When r = 1, the complex number lies on the unit circle and represents a pure rotation. Then z = cos θ + i sin θ. De Moivre becomes especially clean: zn = cos(nθ) + i sin(nθ). This is one reason the theorem is used to derive trigonometric identities and simplify cyclic patterns.
Using the Calculator for Homework, Engineering, and Signals
In many engineering settings (especially AC circuit analysis and signals), complex numbers encode amplitude and phase. De Moivre fits naturally: powering a complex exponential multiplies phase and scales amplitude. Roots split phase into evenly spaced options, which maps to multi-solution phase ambiguity. If you’ve ever seen “phasors” or “polar impedance,” you’ve already seen the same idea in applied form.
What If You Only Need Rectangular Form?
Some contexts need a + bi exclusively—for example, if you’re adding complex results from multiple components. This tool lets you output rectangular only, polar only, or both. A practical workflow is: compute in polar for the power/root step, then convert to rectangular to combine results cleanly.
Limitations and Safe Use Notes
This calculator is designed for educational and planning use. It uses standard floating-point arithmetic, which is accurate for typical classroom and engineering calculations, but extremely large exponents or very large magnitudes can overflow. Rounding is also expected when converting between polar and rectangular forms. If you’re using results in high-stakes work, keep track of measurement precision and validate with independent checks.
FAQ
De Moivre’s Theorem Calculator – Frequently Asked Questions
Learn how complex powers and roots work, why roots come in sets, and how to interpret angles in degrees or radians.
De Moivre’s Theorem says that if z = r(cos θ + i sin θ), then z^n = r^n(cos(nθ) + i sin(nθ)) for integer n. It turns complex powers into simple operations on magnitude and angle.
Use it when you need powers like (a+bi)^n or n-th roots of complex numbers. It’s especially efficient for large exponents, root tables, and problems written in polar form.
No. You can enter z as a+bi. The calculator converts it to polar form internally using r = √(a²+b²) and θ = atan2(b, a).
Degrees are common in geometry-style work; radians are standard in calculus and many engineering formulas. The calculator converts correctly either way—just pick the unit that matches your problem statement.
A complex number has n distinct n-th roots (unless it is 0). Their angles are evenly spaced: θk = (θ + 2kπ)/n for k = 0…n−1. That’s why roots appear as a table.
The principal root is usually the k=0 root using the principal argument θ (often chosen in (−π, π] or [0, 2π)). Other roots differ by rotations of 2π/n.
Yes. Negative powers are computed as 1/z^|n|. In polar form, that becomes r^(−|n|)(cos(−|n|θ) + i sin(−|n|θ)), which the calculator handles automatically.
Angles are periodic. θ, θ+2π, and θ−2π represent the same direction, so different but equivalent angles can produce identical rectangular results after cosine and sine.
For powers, 0^n is 0 for positive n. For roots, the only n-th root of 0 is 0 (repeated). The calculator simplifies this case safely.
No. Calculations and history are stored only in your browser session. Nothing is sent to a server or saved in a database.