Find Cube Root Of Complex Number Calculator

Enter the real and imaginary parts of the complex number (z = a + bi) to find its three cube roots.

The three cube roots of the complex number.

Root (k)	Rectangular Form (x + yi)	Polar Form (r^1/3, angle in degrees)
0
1
2

What is the Cube Root of a Complex Number?

The cube root of a complex number is a set of three complex numbers that, when multiplied by themselves three times (cubed), result in the original complex number. Unlike real numbers, which have only one real cube root, every non-zero complex number has exactly three distinct cube roots in the complex number system. Our cube root of complex number calculator helps you find these three roots quickly.

If you have a complex number `z = a + bi`, its cube roots are generally not just `(a)^(1/3) + (b)^(1/3)i`. Finding them involves converting the complex number to its polar form and then applying De Moivre’s theorem for roots.

Who should use it?

This cube root of complex number calculator is useful for students studying complex numbers in algebra, trigonometry, or engineering, as well as engineers and scientists who encounter complex number calculations in fields like electrical engineering, quantum mechanics, and fluid dynamics.

Common Misconceptions

A common misconception is that a complex number has only one cube root, similar to how a positive real number has one positive real cube root. However, within the complex number system, there are always three distinct cube roots for any non-zero complex number. These three roots are equally spaced on a circle in the complex plane.

Cube Root of a Complex Number Formula and Mathematical Explanation

To find the cube roots of a complex number `z = a + bi`, we first convert it to its polar form `z = r(cos(θ) + i sin(θ))`, where:

• `r = |z| = sqrt(a^2 + b^2)` is the modulus (magnitude) of z.
• `θ = atan2(b, a)` is the argument (angle) of z, usually in radians (-π < θ ≤ π).

According to De Moivre’s theorem for roots, the n-th roots of `z` are given by:

z_k = r^1/n [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]

For cube roots, n = 3, so the three roots (k=0, 1, 2) are:

z₀ = r^1/3 [cos(θ/3) + i sin(θ/3)]

z₁ = r^1/3 [cos((θ + 2π)/3) + i sin((θ + 2π)/3)]

z₂ = r^1/3 [cos((θ + 4π)/3) + i sin((θ + 4π)/3)]

Here, r^1/3 is the principal (real) cube root of the positive real number r.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Real part of the complex number z	None	-∞ to +∞
b	Imaginary part of the complex number z	None	-∞ to +∞
r	Modulus (magnitude) of z	None	0 to +∞
θ	Argument (angle) of z	Radians or Degrees	-π to π rad, -180° to 180°
k	Index for the roots	None	0, 1, 2 (for cube roots)
zk	The k-th cube root	Complex number	–

Practical Examples (Real-World Use Cases)

Example 1: Finding the cube roots of 8i

Let z = 0 + 8i. So, a=0, b=8.

r = sqrt(0² + 8²) = 8

θ = atan2(8, 0) = π/2 radians (90 degrees)

r^1/3 = 8^1/3 = 2

The angles for the roots are θ/3 = π/6 (30°), (θ+2π)/3 = 5π/6 (150°), (θ+4π)/3 = 9π/6 = 3π/2 (270° or -90°).

z₀ = 2(cos(π/6) + i sin(π/6)) = 2(√3/2 + i * 1/2) = √3 + i

z₁ = 2(cos(5π/6) + i sin(5π/6)) = 2(-√3/2 + i * 1/2) = -√3 + i

z₂ = 2(cos(3π/2) + i sin(3π/2)) = 2(0 + i * -1) = -2i

Using the cube root of complex number calculator with a=0 and b=8 will confirm these results.

Example 2: Finding the cube roots of 1 + i

Let z = 1 + i. So, a=1, b=1.

r = sqrt(1² + 1²) = √2 ≈ 1.414

θ = atan2(1, 1) = π/4 radians (45 degrees)

r^1/3 = (√2)^1/3 = 2^1/6 ≈ 1.122

The angles are π/12 (15°), 9π/12 (75°), 17π/12 (255° or -105°).

z₀ ≈ 1.122(cos(15°) + i sin(15°)) ≈ 1.122(0.966 + 0.259i) ≈ 1.084 + 0.291i

z₁ ≈ 1.122(cos(75°) + i sin(75°)) ≈ 1.122(0.259 + 0.966i) ≈ 0.291 + 1.084i

z₂ ≈ 1.122(cos(255°) + i sin(255°)) ≈ 1.122(-0.259 – 0.966i) ≈ -0.291 – 1.084i (or using 17π/12)

The cube root of complex number calculator provides precise values.

How to Use This Cube Root of Complex Number Calculator

1. Enter Real Part (a): Input the real part of your complex number into the first field.
2. Enter Imaginary Part (b): Input the imaginary part (the coefficient of ‘i’) into the second field.
3. View Results: The calculator automatically updates and displays the modulus (r), argument (θ in radians and degrees), and the three cube roots in both rectangular (x + yi) and polar forms in the table. The primary result area highlights the first root.
4. See Visualization: The chart below the table plots the original complex number (if within range) and its three cube roots on the complex plane. The roots lie on a circle centered at the origin.
5. Reset: Click the “Reset” button to clear the inputs and results, restoring default values.
6. Copy Results: Click “Copy Results” to copy the inputs, intermediate values, and the three roots to your clipboard.

Understanding the results: The three roots are equally spaced by 120 degrees (2π/3 radians) around a circle of radius r^1/3.

Key Factors That Affect Cube Root of Complex Number Results

The cube roots of a complex number are determined entirely by the real and imaginary parts of the original number. Here’s how:

• Real Part (a): This influences both the modulus (r) and the argument (θ), thus affecting the magnitude and angle of the roots.
• Imaginary Part (b): Similarly, this also affects r and θ, changing the roots.
• Modulus (r): The magnitude of each root is the cube root of the original modulus (r^1/3). A larger original modulus means the roots are further from the origin.
• Argument (θ): The angles of the roots are derived from the original argument (θ/3, (θ+2π)/3, (θ+4π)/3). Changing the original angle rotates the set of roots.
• Using Radians vs. Degrees: Ensure consistency. The formulas use radians, but our cube root of complex number calculator shows angles in both for convenience.
• Principal Value of the Argument: The `atan2(b, a)` function typically returns θ in (-π, π]. This principal value determines the angle of the first root (k=0).

Frequently Asked Questions (FAQ)

How many cube roots does a complex number have?

Every non-zero complex number has exactly three distinct cube roots.

What if the complex number is real (b=0)?

If z is real and positive (e.g., 8), one root is real and positive (2), and the other two are complex conjugates. If z is real and negative (e.g., -8), one root is real and negative (-2), and the other two are complex conjugates. Our cube root of complex number calculator handles these cases.

Are the cube roots always complex?

Not always. If the original number is real, at least one cube root will be real. The other two might be complex.

What do the cube roots look like on the complex plane?

The three cube roots form the vertices of an equilateral triangle inscribed in a circle of radius r^1/3 centered at the origin.

Can I find the fourth or fifth roots with this calculator?

No, this cube root of complex number calculator is specifically for cube roots (n=3). The principle for nth roots is similar but involves n roots spaced by 2π/n.

What is De Moivre’s Theorem?

De Moivre’s Theorem relates complex numbers and trigonometry. For roots, it gives the formula used by this cube root of complex number calculator to find the nth roots of a complex number in polar form.

Why use atan2(b, a) instead of atan(b/a)?

atan2(b, a) correctly determines the quadrant of the angle θ based on the signs of a and b, giving a range of (-π, π], whereas atan(b/a) has a range of (-π/2, π/2) and requires quadrant adjustments.

What if I enter 0 + 0i?

The only cube root of 0 is 0. The calculator will show this.

Related Tools and Internal Resources

• Complex Number Calculator: Perform basic arithmetic (addition, subtraction, multiplication, division) with complex numbers.
• Polar to Rectangular Calculator: Convert complex numbers from polar (r, θ) to rectangular (a + bi) form and vice-versa.
• De Moivre’s Theorem Explained: An article explaining the theorem for powers and roots of complex numbers.
• Roots of Unity: Learn about the nth roots of 1, a special case of complex number roots.
• Math Calculators: Explore other mathematical calculators.
• Algebra Calculators: A collection of calculators related to algebra.