Complex Number Calculator

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complex number powers and roots

Powers and roots of complex numbers (De Moivre)

On the Powers and roots tab, choose the power zⁿ, the n roots or the power of i. De Moivre’s formula is applied step by step, and the roots are drawn on the circle of the Argand plane.

De Moivre, roots on the circle and the cycle of i

By De Moivre’s formula, if z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos nθ + i sin nθ) for any integer n, positive or negative. In (1 + i)⁸ we have r = √2 and θ = 45°, so r⁸ = 16 and 8θ = 360°, which gives 16. For rectangular inputs, the tool also computes the power by exact multiplication: (1 + i)² = 2i, (3 + 4i)² = −7 + 24i and (1 − i)⁴ = −4 come out as integers, with no floating-point residue.
Every nonzero complex number has exactly n distinct nth roots: ⁿ√r ∠ (θ + 360°·k)/n, for k from 0 to n − 1. They lie on the circle of radius ⁿ√r, spaced 360°/n apart, and form a regular polygon. The cube roots of 8 are 2, −1 + √3 i and −1 − √3 i; the fourth roots of 1 are 1, i, −1 and −i. The sum of the n roots is zero for n ≥ 2.
The powers of i are periodic: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, i⁴ = 1. For any exponent n, the remainder of n divided by 4 is enough, which holds even for n with hundreds of digits and for negative n (i⁻¹ = −i). For example, 2027 = 4·506 + 3, so i²⁰²⁷ = i³ = −i.

Worked powers and roots

Eighth power

Input: (1+i)^8
Expected output: 16

r⁸ = 16 and 8·45° = 360°.

Square of a 3-4-5 number

Input: (3+4i)^2
Expected output: −7 + 24i

9 − 16 = −7 and 2·3·4 = 24.

Fourth power

Input: (1-i)^4
Expected output: −4

(1 − i)² = −2i, and (−2i)² = −4.

Third power

Input: (2+2i)^3
Expected output: −16 + 16i

(2 + 2i)² = 8i, and 8i·(2 + 2i) = 16i − 16.

Fourth roots of unity

Input: 1, n = 4
Expected output: 1, i, −1, −i

Angles 0°, 90°, 180° and 270° on the unit circle.

Very large power of i

Input: i^2027
Expected output: −i

2027 = 4·506 + 3, so it equals i³.

Full tool FAQ

How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number. In (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i. The numerator is (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i, and the denominator is 1² + 2² = 5. The result is −1 + 2i.

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Does De Moivre’s formula work for fractional exponents?

For the exponent 1/n it generates the n roots, but you have to list all of them, adding 360°·k to the angle; with a single value you get only the principal root. That is why the tool keeps the integer power apart from the list of roots.

How many roots does a complex number have?

Exactly n distinct nth roots, if the number is nonzero: 2 square roots, 3 cube roots and so on. Zero has a single root, itself. The tool accepts indices from 1 to 360.

Why do the roots sum to zero?

Because they form a regular polygon centered at the origin and the vectors cancel. Algebraically, the roots of zⁿ = w are those of zⁿ − w, whose zⁿ⁻¹ coefficient is zero for n ≥ 2, and by Vieta’s formulas the sum is zero.

Next steps

De Moivre, roots on the circle and the cycle of i

By De Moivre’s formula, if z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos nθ + i sin nθ) for any integer n, positive or negative. In (1 + i)⁸ we have r = √2 and θ = 45°, so r⁸ = 16 and 8θ = 360°, which gives 16. For rectangular inputs, the tool also computes the power by exact multiplication: (1 + i)² = 2i, (3 + 4i)² = −7 + 24i and (1 − i)⁴ = −4 come out as integers, with no floating-point residue.

Every nonzero complex number has exactly n distinct nth roots: ⁿ√r ∠ (θ + 360°·k)/n, for k from 0 to n − 1. They lie on the circle of radius ⁿ√r, spaced 360°/n apart, and form a regular polygon. The cube roots of 8 are 2, −1 + √3 i and −1 − √3 i; the fourth roots of 1 are 1, i, −1 and −i. The sum of the n roots is zero for n ≥ 2.

The powers of i are periodic: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, i⁴ = 1. For any exponent n, the remainder of n divided by 4 is enough, which holds even for n with hundreds of digits and for negative n (i⁻¹ = −i). For example, 2027 = 4·506 + 3, so i²⁰²⁷ = i³ = −i.

Worked powers and roots

Eighth power

Input: (1+i)^8
Expected output: 16

r⁸ = 16 and 8·45° = 360°.

Square of a 3-4-5 number

Input: (3+4i)^2
Expected output: −7 + 24i

9 − 16 = −7 and 2·3·4 = 24.

Fourth power

Input: (1-i)^4
Expected output: −4

(1 − i)² = −2i, and (−2i)² = −4.

Third power

Input: (2+2i)^3
Expected output: −16 + 16i

(2 + 2i)² = 8i, and 8i·(2 + 2i) = 16i − 16.

Fourth roots of unity

Input: 1, n = 4
Expected output: 1, i, −1, −i

Angles 0°, 90°, 180° and 270° on the unit circle.

Very large power of i

Input: i^2027
Expected output: −i

2027 = 4·506 + 3, so it equals i³.

Full tool FAQ

How do you divide complex numbers?

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Does De Moivre’s formula work for fractional exponents?

How many roots does a complex number have?

Exactly n distinct nth roots, if the number is nonzero: 2 square roots, 3 cube roots and so on. Zero has a single root, itself. The tool accepts indices from 1 to 360.