Complex Numbers, Polar Form, and the De Moivre Theorem

A complex number is nothing more exotic than a point in a plane, and the whole art of working with one is choosing the form that makes the operation you face easy. Rectangular form a + bi is built for adding and subtracting; polar form is built for multiplying, dividing, raising to powers, and extracting roots. The De Moivre theorem is the single rule that turns those last two operations into simple arithmetic on an angle, and phasors are just complex numbers wearing an electrical-engineering hat. This lesson carries every conversion and every rule all the way to a number so you can check your own habits against it.

The three forms of a complex number

A complex number has three interchangeable dresses. In rectangular (or standard) form it is z = a + bi, where a = Re(z) is the real part, b = Im(z) is the imaginary part, and i^2 = -1. Plot a as the horizontal coordinate and b as the vertical coordinate on the Argand plane and z becomes a single point, or the arrow from the origin to that point.

That arrow has a length and a direction, and naming them gives the polar (trigonometric) form. The length is the modulus r = |z| = sqrt(a^2 + b^2), and the direction is the argument theta, the angle the arrow makes with the positive real axis. Because a = r cos(theta) and b = r sin(theta), the number can be written

z = r(cos theta + i sin theta), abbreviated z = r cis theta

The Euler (exponential) form comes from Euler's formula, e^(i theta) = cos theta + i sin theta, which lets the same number be written as z = r e^(i theta) with theta in radians. All three describe the exact same point.

Form	How z is written	Best used for
Rectangular	a + bi	adding and subtracting
Polar (trigonometric)	r(cos theta + i sin theta), or r cis theta	multiplying, dividing, powers, roots
Euler (exponential)	r e^(i theta), theta in radians	calculus, phasors, compact algebra

Two habits keep the arithmetic honest. First, powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, then repeat, so i to any power reduces by taking the exponent modulo 4. Second, always simplify i^2 to -1 as the last algebraic step, never partway through, or sign errors creep in.

Converting between rectangular and polar

Going from rectangular to polar, compute r = sqrt(a^2 + b^2) and theta = arctan(b/a), then correct theta for the quadrant, because arctan on a calculator only reports angles between -90 and +90 degrees. If the point sits in the second or third quadrant, add 180 degrees to the raw arctan value. Going the other way, from polar to rectangular, read the components straight off the triangle: a = r cos(theta) and b = r sin(theta).

Worked example (rectangular to polar): Express z = 3 + 3i in polar form. The modulus is r = sqrt(3^2 + 3^2) = sqrt(18) = 4.24, and because the point is in the first quadrant the argument is theta = arctan(3/3) = arctan(1) = 45 degrees. So z = 4.24 cis 45 degrees.

Worked example (quadrant care): Express z = -1 + i sqrt(3) in polar form. Here r = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = 2. The raw arctan is arctan(sqrt(3) / -1) = -60 degrees, but the point lies in the second quadrant, so the true argument is -60 + 180 = 120 degrees. Thus z = 2 cis 120 degrees.

Worked example (polar to rectangular): Convert z = 10 cis 30 degrees. The real part is a = 10 cos(30 degrees) = 10(0.866) = 8.66 and the imaginary part is b = 10 sin(30 degrees) = 10(0.5) = 5. So z = 8.66 + 5i. Swapping sine and cosine, which would give 5 + 8.66i, is the standard slip.

Multiplication and division in polar form

Polar form pays for itself the moment you multiply or divide. To multiply, multiply the moduli and add the angles. To divide, divide the moduli and subtract the angles. The messy binomial expansion of rectangular multiplication disappears.

z1 z2 = r1 r2 cis(theta1 + theta2)

z1 / z2 = (r1 / r2) cis(theta1 - theta2)

Worked example (multiplication): Multiply (4 cis 20 degrees)(5 cis 40 degrees). Multiply the moduli, 4 x 5 = 20, and add the angles, 20 + 40 = 60 degrees, giving 20 cis 60 degrees. In rectangular form that is 20 cos(60 degrees) + i 20 sin(60 degrees) = 10 + 17.32i.

Worked example (division): Divide (12 cis 100 degrees) / (3 cis 30 degrees). Divide the moduli, 12 / 3 = 4, and subtract the angles, 100 - 30 = 70 degrees, giving 4 cis 70 degrees. Multiplying the moduli (36) or adding the angles (130 degrees) are the two classic errors.

Powers by the De Moivre theorem

The De Moivre theorem is what you get when you multiply a number by itself n times using the polar rule: the modulus is raised to the nth power and the angle is multiplied by n.

z^n = [r cis theta]^n = r^n cis(n theta) = r^n [cos(n theta) + i sin(n theta)]

The whole strategy for a power is therefore: convert to polar, apply De Moivre, convert back if a rectangular answer is wanted.

Worked example: Evaluate (1 + i)^8. First put the base in polar form: r = sqrt(1^2 + 1^2) = sqrt(2) and theta = 45 degrees, so 1 + i = sqrt(2) cis 45 degrees. Then z^8 = (sqrt(2))^8 cis(8 x 45 degrees) = 2^4 cis(360 degrees) = 16 cis 0 degrees = 16. The most common trap is using modulus 2 instead of sqrt(2), which gives the wrong 2^8 = 256.

Worked example: Evaluate (sqrt(3) + i)^6. The modulus is r = sqrt(3 + 1) = 2 and the argument is theta = arctan(1 / sqrt(3)) = 30 degrees, so the base is 2 cis 30 degrees. Then z^6 = 2^6 cis(6 x 30 degrees) = 64 cis(180 degrees) = 64(cos 180 degrees + i sin 180 degrees) = 64(-1 + 0i) = -64.

Roots by the De Moivre theorem

Run De Moivre in reverse and a surprise appears: a nonzero complex number has exactly n distinct nth roots, not one. To find them, take the nth root of the modulus and split the angle, remembering that the argument is only fixed up to full turns of 360 degrees:

w_k = r^(1/n) cis[(theta + 360k) / n], for k = 0, 1, 2, ..., n - 1

Every root has the same modulus r^(1/n), so all n of them sit on one circle, equally spaced by 360/n degrees. Sketching that circle is the fastest way to see them and to avoid missing any.

Worked example: Find the three cube roots of 8. Write 8 in polar form as 8 cis 0 degrees, so r = 8, theta = 0, n = 3. The common modulus is r^(1/3) = 8^(1/3) = 2, and the angles are (0 + 360k)/3 = 0, 120, and 240 degrees. The roots are w0 = 2 cis 0 degrees = 2, w1 = 2 cis 120 degrees = -1 + i(1.732), and w2 = 2 cis 240 degrees = -1 - i(1.732). Check: they add to zero, 2 + (-1) + (-1) = 0 real and 0 + 1.732 - 1.732 = 0 imaginary, as the roots of any pure power must.

Worked example: Find the two square roots of i. Write i = 1 cis 90 degrees, so r = 1, n = 2. The modulus of each root is 1^(1/2) = 1 and the angles are (90 + 360k)/2 = 45 and 225 degrees. The roots are cis 45 degrees = 0.707 + 0.707i and cis 225 degrees = -0.707 - 0.707i, which are negatives of each other, exactly as two square roots should be.

Phasor addition

In AC analysis a sinusoidal voltage or current is represented by a phasor, a complex number whose modulus is the amplitude and whose angle is the phase. Multiplying and dividing phasors (for impedance work) stays in polar form, but adding them does not: you cannot add the amplitudes unless the phases already match. The reliable method is to convert each phasor to rectangular form, add the real parts and the imaginary parts separately, then convert the sum back to polar.

Worked example: Add V1 = 10 cis 0 degrees and V2 = 10 cis 90 degrees. In rectangular form V1 = 10 + 0i and V2 = 0 + 10i, so the sum is 10 + 10i. Its modulus is sqrt(10^2 + 10^2) = sqrt(200) = 14.14 and its angle is arctan(10/10) = 45 degrees, giving 14.14 cis 45 degrees. Adding the amplitudes to get 20 is the mistake the figure warns against.

Worked example: Add the currents I1 = 3 cis 0 degrees A and I2 = 4 cis 90 degrees A. In rectangular form I1 = 3 + 0i and I2 = 0 + 4i, so the sum is 3 + 4i A. The magnitude is sqrt(3^2 + 4^2) = sqrt(25) = 5 A and the angle is arctan(4/3) = 53.13 degrees, so the total current is 5 cis 53.13 degrees A, the familiar 3-4-5 triangle in phasor clothing.

Operation	Rule in polar form	In words
Multiply z1 z2	r1 r2 cis(theta1 + theta2)	multiply moduli, add angles
Divide z1 / z2	(r1 / r2) cis(theta1 - theta2)	divide moduli, subtract angles
Power z^n	r^n cis(n theta)	raise modulus to n, multiply angle by n
Roots (n of them)	r^(1/n) cis[(theta + 360k)/n]	take nth root of modulus, split the angle
Add or subtract	convert to a + bi first	combine real and imaginary parts separately

Exam-day strategy

• Pick the form that fits the job: keep a + bi for sums and differences, but switch to polar the instant you see a product, a quotient, a power, or a root.
• After any arctan, look at the signs of a and b to place the point in its quadrant, and add 180 degrees when it lands in the second or third quadrant, since the calculator only returns -90 to +90 degrees.
• For a power, raise the modulus to the nth power and multiply the angle by n; the frequent slip is forgetting to raise the modulus, or using the rectangular modulus instead of the true one such as sqrt(2) for 1 + i.
• Expect n answers from an nth root, all on one circle of radius r^(1/n) and spaced 360/n degrees apart; find one root, then step around the circle to get the rest.
• Never add phasor amplitudes unless the phases are identical; convert each to a + bi, add the parts, then return to polar, and sanity-check that the resultant of two unequal-phase phasors is less than their amplitudes summed.
• Keep exact values like sqrt(2), sqrt(3), and clean angles as long as you can, converting to decimals only at the final step to avoid rounding drift.