Complex Roots Calculator: All Roots of a Complex Number

Find every n-th root of a complex number. The default solves z to the power 6 equals 1, which has six solutions.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Write the number in polar form with a size r and an angle. The n-th roots all have size r to the power one over n, and their angles are the original angle plus a whole turn for each k, all divided by n.

This gives n different roots, evenly spaced round a circle. For z to the power 6 equal to 1 the roots are at 0, 60, 120, 180, 240 and 300 degrees.

Formulas used

  • z_k = r^{1/n}\,e^{i(\theta + 2\pi k)/n}
  • k = 0, 1, \ldots, n-1

Worked example

With the default numbers in the calculator above:

  1. The number in polar form: 1.00 at 0.00 °
  2. Size of every root: 1.00
  3. Angle between neighbouring roots: 60.00 °
  4. Root k = 0: 1.00 + 0.00i (at 0.00 °)
  5. Root k = 1: 0.50 + 0.87i (at 60.00 °)
  6. Root k = 2: -0.50 + 0.87i (at 120.00 °)
  7. Root k = 3: -1.00 + 0.00i (at 180.00 °)
  8. Root k = 4: -0.50 - 0.87i (at 240.00 °)
  9. Root k = 5: 0.50 - 0.87i (at 300.00 °)
  10. Check: the roots add up to zero: 0.00 + 0.00i

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Real part of the number -8; Imaginary part of the number 0; Which root? (the power n) 3.

  1. The number in polar form: 8.00 at 180.00 °
  2. Size of every root: 2.00
  3. Angle between neighbouring roots: 120.00 °
  4. Root k = 0: 1.00 + 1.73i (at 60.00 °)
  5. Root k = 1: -2.00 + 0.00i (at 180.00 °)
  6. Root k = 2: 1.00 - 1.73i (at 300.00 °)
  7. Check: the roots add up to zero: 0.00 + 0.00i

Common questions

How many solutions does z^6 = 1 have?

Six. Two of them are real (1 and -1) and four are complex.

Why do the roots add up to zero?

They are spread evenly round a circle, so they cancel each other out.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Standard algebra and complex number texts (De Moivre theorem and roots of unity).

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