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De Moivre's Theorem & Roots of Unity cheat sheet - grade 10-12

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Math Grade 10-12

De Moivre's Theorem & Roots of Unity Cheat Sheet

A printable reference covering De Moivre's theorem, polar form, powers of complex numbers, $n$th roots, and roots of unity for grades 10-12.

TopicsTrigonometryAlgebraFunctionsExponentsMathematicsStudy Guide
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De Moivre's theorem connects complex numbers, trigonometry, and powers in a simple formula. This cheat sheet helps students rewrite complex numbers in polar form, raise them to powers, and find roots efficiently. It is especially useful for precalculus, advanced algebra, and early calculus topics involving complex numbers.

The main goal is to make multiplication, powers, and roots easier by using angles and magnitudes instead of rectangular coordinates.

The core idea is that a complex number can be written as z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta), where rr is its modulus and θ\theta is its argument. De Moivre's theorem says [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta) for integer nn. The nnth roots of a complex number are found by taking the nnth root of the modulus and adding angle increments of 2πn\frac{2\pi}{n}.

Roots of unity are the special solutions to zn=1z^n = 1, evenly spaced around the unit circle.

Key Facts

  • The polar form of a complex number is z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta), where r=∣z∣=a2+b2r = |z| = \sqrt{a^2 + b^2} for z=a+biz = a + bi.
  • The argument θ\theta of z=a+biz = a + bi satisfies tan⁡θ=ba\tan \theta = \frac{b}{a}, but the quadrant must be chosen from the signs of aa and bb.
  • De Moivre's theorem states that [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta) for any integer nn.
  • Using cis notation, cis⁡θ=cos⁡θ+isin⁡θ\operatorname{cis}\theta = \cos \theta + i\sin \theta, so De Moivre's theorem becomes (rcis⁡θ)n=rncis⁡(nθ)(r\operatorname{cis}\theta)^n = r^n\operatorname{cis}(n\theta).
  • The nnth roots of z=rcis⁡θz = r\operatorname{cis}\theta are wk=rncis⁡(θ+2πkn)w_k = \sqrt[n]{r}\operatorname{cis}\left(\frac{\theta + 2\pi k}{n}\right) for k=0,1,2,…,n−1k = 0, 1, 2, \ldots, n - 1.
  • The nnth roots of unity are zk=cis⁡(2πkn)z_k = \operatorname{cis}\left(\frac{2\pi k}{n}\right) for k=0,1,2,…,n−1k = 0, 1, 2, \ldots, n - 1.
  • All nnth roots of unity lie on the unit circle because each has modulus 11.
  • The roots of zn=1z^n = 1 are evenly spaced by an angle of 2πn\frac{2\pi}{n} around the complex plane.

Vocabulary

Complex number
A number of the form a+bia + bi, where aa is the real part, bb is the imaginary part, and i2=−1i^2 = -1.
Modulus
The distance of a complex number z=a+biz = a + bi from the origin, given by ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.
Argument
The angle θ\theta that a complex number makes with the positive real axis in the complex plane.
Polar form
A way to write a complex number as z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta) using its modulus and argument.
De Moivre's theorem
The rule [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta) for powers of complex numbers in polar form.
Root of unity
A complex number that satisfies zn=1z^n = 1 for a positive integer nn.

Common Mistakes to Avoid

  • Using the wrong quadrant for θ\theta is wrong because tan⁡θ=ba\tan \theta = \frac{b}{a} can give the same reference angle for different complex numbers.
  • Multiplying the modulus by nn in De Moivre's theorem is wrong because the modulus must be raised to the power, so rr becomes rnr^n, not nrnr.
  • Forgetting the 2πk2\pi k term when finding roots is wrong because it gives only one root instead of all nn distinct roots.
  • Using degrees and radians together is wrong because formulas like 2πn\frac{2\pi}{n} assume radian measure unless angles are consistently converted.
  • Listing more or fewer than nn roots for zn=az^n = a is wrong because a nonzero complex number has exactly nn distinct nnth roots.

Practice Questions

  1. 1 Write z=1+i3z = 1 + i\sqrt{3} in polar form r(cos⁡θ+isin⁡θ)r(\cos \theta + i\sin \theta).
  2. 2 Use De Moivre's theorem to find (2cis⁡π6)4(2\operatorname{cis}\frac{\pi}{6})^4 in polar form.
  3. 3 Find all cube roots of unity, which are the solutions to z3=1z^3 = 1.
  4. 4 Explain why the nnth roots of unity form a regular polygon on the unit circle in the complex plane.

Understanding De Moivre's Theorem & Roots of Unity

Complex multiplication has a geometric meaning. Multiplying by a complex number changes two things at once. It stretches or shrinks a point by its distance from the origin, then turns it by its angle from the positive horizontal axis.

Repeated multiplication repeats the same stretch and the same turn. This is why powers become manageable in polar form.

Instead of expanding long products and collecting many real and imaginary terms, you can track one length and one rotation. This viewpoint is useful whenever a problem involves repeated rotations, spirals, waves, or points moving around a center.

Finding roots reverses that process. A root must have a length that becomes the original length after repeated multiplication. Its angle must become the original angle after repeated rotation.

There is not just one possible starting angle because a full turn can be shared among the steps in several ways. For example, fourth roots come in four directions, separated by one quarter turn. After all of them have been listed, adding another full turn gives a repeat rather than a new root.

Students often miss roots by using only the calculator's displayed angle. That angle is one representative of an angle family, not the only angle connected to the complex number.

Roots of unity show the symmetry especially clearly. They form the corners of a regular polygon centered at the origin. The third roots make an equilateral triangle.

The fourth roots make a square. The fifth roots make a regular pentagon. This geometry explains a useful result.

When there is more than one root, all the roots of unity add to zero. Their arrows balance because the polygon is symmetric in every direction. This idea later appears in factoring polynomial expressions, solving recurrence patterns, and the discrete Fourier transform.

The Fourier transform breaks a complicated signal into repeating circular motions. It is used in sound processing, image compression, and communication systems.

Good work with these problems depends on careful angle handling. First locate the point in the correct quadrant before choosing its angle. An inverse tangent calculation alone can point to the wrong quadrant.

Keep degree mode and radian mode separate because mixing them changes every angle calculation. Radians are usually the natural choice in higher mathematics because a full turn has a value of two pi radians. When checking an answer, raise one proposed root back to the required power.

Its length should match the original length, and its final direction should match after allowing for complete turns. A sketch of the complex plane catches many errors before they spread through the calculation.