Guides / Complex numbers

Complex numbers in polar form: find the angle from the quadrant, multiply the lengths and add the angles

A complex number a + bi is a point in the plane, so it can also be given by its distance r from the origin and its angle θ from the positive real axis: z = r(cos θ + i sin θ) = re^iθ. In that form, multiplying, dividing, taking powers and taking roots all become arithmetic on r and θ. The algebra of i² = −1 is covered in the complex numbers guide; this one is about the polar form, where most wrong answers come from an angle in the wrong quadrant or from treating r and θ the wrong way in a power or a root.

Why it works

Euler’s formula, e^iθ = cos θ + i sin θ, puts the angle into an exponent, and exponents add when you multiply: e^iαe^iβ = e^{i(α + β)}. Multiplying out (cos α + i sin α)(cos β + i sin β) by hand gives the same thing, because the real part is the cosine addition formula and the imaginary part is the sine one. So multiplying by re^iθ stretches by r and rotates by θ.

A power is repeated multiplication, so the length is raised to the power and the angle is multiplied by it. Roots run that backward, and here the angle is the subtle part: θ, θ + 2π, θ + 4π and so on all name the same point, but after dividing by n they name different points. That is where the n different roots come from, spaced 2π/n apart on a circle of radius r^1/n.

Example 1: a point in the third quadrant

Write z = −√3 − i in polar form, with −π < θ ≤ π.

1. r = √(3 + 1) = 2The distance to (−√3, −1).
2. cos θ = −√3/2Divide the real part by r. Negative: the point is left of the imaginary axis.
3. sin θ = −1/2Negative too, so the point is below the real axis: third quadrant.
4. reference angle π/6The angle with cosine √3/2 and sine 1/2.
5. θ = −π + π/6 = −5π/6Go clockwise to the negative real axis, then back π/6.
6. z = 2e^−5πi/6Check: 2 cos(−5π/6) = −√3 and 2 sin(−5π/6) = −1. The angle 7π/6 names the same point, outside the range asked for.

Example 2: a product and a quotient

Let z = 2e^iπ/3 and w = 4e^iπ/6. Find zw and z/w in both forms.

1. zw = 8e^{i(π/3 + π/6)}Multiply the lengths, 2 · 4, and add the angles.
2. zw = 8e^iπ/2 = 8iπ/3 + π/6 = π/2, straight up.
3. z/w = (1/2)e^iπ/6Divide the lengths and subtract the angles: π/3 − π/6.
4. z/w = √3/4 + i/4(1/2) cos(π/6) = √3/4 and (1/2) sin(π/6) = 1/4.
5. zw = 8i, z/w = √3/4 + i/4Check in rectangular form: z = 1 + √3 i and w = 2√3 + 2i, and (1 + √3 i)(2√3 + 2i) = 2√3 + 2i + 6i − 2√3 = 8i.

Example 3: a fifth power

Compute (√3 + i)⁵.

1. √3 + i = 2e^iπ/6r = √(3 + 1) = 2, first quadrant, arctan(1/√3) = π/6.
2. (2e^iπ/6)⁵ = 32e^5πi/6De Moivre: 2⁵ = 32, and the angle is multiplied by 5.
3. cos(5π/6) = −√3/2Second quadrant, reference angle π/6.
4. sin(5π/6) = 1/2Sine is positive in the second quadrant.
5. (√3 + i)⁵ = −16√3 + 16iMultiplying √3 + i by itself five times in rectangular form gives the same, with far more arithmetic.

Example 4: the fourth roots of −16

Solve z⁴ = −16.

1. −16 = 16e^iπOn the negative real axis.
2. z_k = 2e^{i(π + 2πk)/4}The real fourth root of the length, 16^1/4 = 2, and the angle divided by 4 after adding 2πk, for k = 0, 1, 2, 3.
3. π/4, 3π/4, 5π/4, 7π/4The four angles, a quarter turn apart.
4. 2e^iπ/4 = √2 + √2 i2 cos(π/4) = √2; the others follow by symmetry.
5. z = ±√2 ± √2 iAll four sign choices. Check one: (√2 + √2 i)² = 4i, and (4i)² = −16.

Common mistakes, and the exact line they happen on

1. Taking arctan(b/a) when a is negative

The point of Example 1.

1. r = 2Right so far.
2. θ = arctan(−1/−√3) = π/6This is the mistake. The two minus signs cancel inside the fraction, so arctan cannot tell −√3 − i from √3 + i. Its answers always lie between −π/2 and π/2, on the right half of the plane. Here 2e^iπ/6 = √3 + i, the point exactly opposite. The right angle is π/6 − π = −5π/6.

2. Raising the length but not multiplying the angle

The power of Example 3.

1. √3 + i = 2e^iπ/6Right so far.
2. (√3 + i)⁵ = 32e^iπ/6This is the mistake. Five factors of e^iπ/6 rotate by π/6 five times, so the angle is 5π/6. This version gives 16√3 + 16i, the right length pointing in the wrong direction; the right answer is −16√3 + 16i.

3. Dividing the length by n instead of taking its root

The roots of Example 4.

1. −16 = 16e^iπRight so far.
2. z₀ = 4e^iπ/4This is the mistake. The angle is divided by 4, but the length is the fourth root, not a quarter: 16^1/4 = 2, not 16/4 = 4. Checking catches it at once: (4e^iπ/4)⁴ = 256e^iπ = −256.

Practice

1. Write −5 in polar form.

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   5e^iπ.

2. Write 1 − i in polar form, with −π < θ ≤ π.

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   √2 e^−iπ/4: fourth quadrant, reference angle π/4.

3. Find 3e^iπ/2 · 2e^iπ/3 in rectangular form.

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   6e^5πi/6 = −3√3 + 3i.

4. Compute (1 − i)⁴.

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   (√2)⁴e^−iπ = −4. Check: (1 − i)² = −2i and (−2i)² = −4.

5. Find both square roots of i.

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   i = e^iπ/2, so the roots are e^iπ/4 and e^5πi/4, that is ±(√2/2)(1 + i).