Complex Numbers

A complex number has a real part and an imaginary part, and is written a + bi. The imaginary unit i is defined by $i^2=-1$i² = −1. Such a number does not fit on a line, so complex numbers live on a plane, where adding is a shift and multiplying is a turn.

What is the square root of a negative number?

No real number has a negative square, because a positive squared is positive and a negative squared is also positive. So $x^2=-1$x² = −1 has no solution on the number line.

The way out is to define a new number. Let i be a number with $i^2=-1$i² = −1. Every other square root of a negative then follows: $\sqrt{-9}=\sqrt{9}\times \sqrt{-1}=3i$√(−9) = √9 × √(−1) = 3i. Algebra still works with i in an expression: expand brackets as before, and replace $i^2$i² with −1 wherever it appears. See The Imaginary Unit.

The powers of i repeat in a cycle of four: i, −1, −i, 1. To find i²⁰²⁷, divide the exponent by 4 and keep the remainder. Since 2027 = 4 × 506 + 3, i²⁰²⁷ $=i^3=-i$= i³ = −i. See Powers of i.

How do you add and multiply complex numbers?

To add, add the real parts and the imaginary parts separately: (3 + 2i) + (1 − 5i) = 4 − 3i. The two parts never mix. See Adding Complex Numbers.

To multiply, expand the brackets as if i were a letter, then replace $i^2$i² with −1:

$\left(3+2i\right)\left(1-5i\right)=3-15i+2i-10i^2=3-13i+10=13-13i$(3 + 2i)(1 − 5i) = 3 − 15i + 2i − 10i² = 3 − 13i + 10 = 13 − 13i.

The last step is where mistakes happen: $-10i^2$−10i² is −10 × (−1) = +10, not −10. See Multiplying Complex Numbers.

Dividing with the conjugate

The conjugate of a + bi is a − bi. A number times its conjugate is always real, because the middle terms cancel and $-b^2{i}^2$−b²i² becomes $+b^2$+b²: $\left(a+\text{bi}\right)\left(a-\text{bi}\right)=a^2+b^2$(a + bi)(a − bi) = a² + b². See Complex Conjugates.

To divide, multiply the numerator and the denominator by the conjugate of the denominator. The denominator becomes real, and the division is then ordinary. It is the same move as rationalizing a denominator with a surd in it. See Dividing Complex Numbers.

(4 + i) / (2 − 3i) = (4 + i)(2 + 3i) / ((2 − 3i)(2 + 3i)) = (5 + 14i) / 13.

Where do complex numbers come from in an equation?

From a quadratic whose discriminant is negative. The graph misses the x-axis, but the quadratic formula still applies. For $x^2-4x+13=0$x² − 4x + 13 = 0, the discriminant is 16 − 52 = −36, so $x=\left(4\pm 6i\right)$x = (4 ± 6i) / $2=2\pm 3i$2 = 2 ± 3i. See Complex Roots of a Quadratic.

The two roots are conjugates of each other. When the coefficients are real, non-real roots always come in conjugate pairs.

So a cubic with real coefficients has either three real roots or one real root and a conjugate pair, and its graph always crosses the x-axis. Given one non-real root, its conjugate is a second root. Multiply the two matching factors to get a real quadratic, then divide the cubic by it to find the real root. See Complex Roots of a Cubic and the quadratics and polynomials guide.

What does a complex number look like?

Like a point on a plane. Plot the real part along the horizontal axis and the imaginary part up the vertical axis. That plane is the Argand diagram. See The Argand Diagram.

Two measurements fix a point. The modulus |z| is its distance from the origin. The arrow to a + bi is the hypotenuse of a right-angled triangle with legs a and b, so $|z|=\sqrt{a^2+b^2}$|z| = √(a² + b²). For 3 + 4i the modulus is $\sqrt{9+16}=5$√(9 + 16) = 5. See The Modulus of a Complex Number.

The argument is the angle the arrow makes with the positive real axis, measured counterclockwise, and in the same triangle $\tan \theta =\frac{b}{a}$tan θ = b/a. A calculator's inverse tangent only returns angles between −90° and 90°, so for a point in the second or third quadrant, sketch it and adjust by 180°. See The Argument of a Complex Number.

Writing a number by its modulus r and argument $\theta$θ is polar form: $z=r\left(\cos \theta +i\sin \theta \right)$z = r(cos θ + i sin θ), because the real part is $r\cos \theta$r cos θ and the imaginary part is $r\sin \theta$r sin θ. See Polar Form and The Form $r\left(\cos \theta +i\sin \theta \right)$r(cos θ + i sin θ).

Division reverses both steps: divide the moduli and subtract the arguments. See Dividing in Polar Form.

What is De Moivre's theorem?

It is the rule for powers in polar form: $\left[r\left(\cos \theta +i\sin \theta \right)\right]$[r(cos θ + i sin θ)]ⁿ = rⁿ$\left(\cos n\theta +i\sin n\theta \right)$(cos nθ + i sin nθ). Multiplying z by itself n times multiplies the argument by n and raises the modulus to the power n. See De Moivre's Theorem.

One use is to derive trigonometric identities. Expand ${\left(\cos \theta +i\sin \theta \right)}^3$(cos θ + i sin θ)³ with the binomial theorem and match the real part against $\cos 3\theta$cos 3θ. Replace ${\sin }^2\theta$sin²θ with $1-{\cos }^2\theta$1 − cos²θ to get $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$cos 3θ = 4cos³θ − 3cos θ. See Multiple Angles by De Moivre.

The reverse direction turns a power of $\cos \theta$cos θ into cosines of multiple angles, the form needed before integrating. If |z| = 1, then $\frac{1}{z}$1/z is the conjugate of z, so $z+\frac{1}{z}=2\cos \theta$z + 1/z = 2 cos θ and zⁿ $+\frac{1}{z}$+ 1/zⁿ $=2\cos n\theta$= 2 cos nθ. See The Identity $z+\frac{1}{z}=2\cos \theta$z + 1/z = 2 cos θ.

De Moivre also sums series of cosines: the sum of $\cos k\theta$cos kθ is the real part of the sum of z^k, which is a geometric series with ratio z. See Summing a Series with De Moivre and the sequences and series guide.

The roots of unity

The equation zⁿ = 1 has n solutions, each of modulus 1, with arguments 0°, $\frac{360^{\circ }}{n}$360°/n, $2\times \frac{360^{\circ }}{n}$2 × 360°/n and so on, because each of those angles times n is a whole number of turns. See The Roots of Unity.

Joined in order, they are the vertices of a regular polygon: the fifth roots of 1 form a regular pentagon. See Regular Polygons from the Roots of Unity.

The same method finds the nth roots of any complex number. Take the real nth root of the modulus, divide the argument by n, then step around by $\frac{360^{\circ }}{n}$360°/n to collect the rest. The cube roots of 8i all have modulus 2, at 30°, 150° and 270°. See The nth Roots of a Complex Number.

Behind all of this is the exponential form. Substitute $x=i\theta$x = iθ into the series for e^x and the terms sort into the series for cosine and sine, so e^$i\theta$iθ $=\cos \theta +i\sin \theta$= cos θ + i sin θ and every complex number is re^$i\theta$iθ. De Moivre is then the index law (e^$i\theta$iθ)ⁿ = e^in$\theta$θ. See The Exponential Form of a Complex Number.

How do you read a complex equation as a picture?

Read every modulus as a distance and every argument as a direction. |z − a| is the distance from z to the point a, so |z − a| = r is the circle with center a and radius r. For example, |z − 3 + 2i| = 4 is the circle of radius 4 centered at 3 − 2i.

|z − a| = |z − b| says z is the same distance from a as from b, so z lies on the perpendicular bisector of the segment from a to b. See Circles and Bisectors on the Argand Diagram.

Fixing an argument gives a half-line: arg(z − a) = 45° is the ray from a pointing at 45°. Changing an equals sign to an inequality shades a region: $|z|\le 3$|z| ≤ 3 is the whole disc of radius 3, and a strict inequality leaves out the boundary. See Rays and Regions on the Argand Diagram.

The mistakes worth naming

• Losing the sign on $i^2$i². $-10i^2$−10i² is +10, because $i^2=-1$i² = −1.
• Trusting the calculator's inverse tangent for the argument. It returns an angle between −90° and 90°. Sketch the point, and add or subtract 180° in the second or third quadrant.
• Writing $\sqrt{-4}\times \sqrt{-9}=\sqrt{36}=6$√(−4) × √(−9) = √36 = 6. The rule $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$√a × √b = √(ab) holds only for non-negative a and b. Convert first: $2i\times 3i=6i^2=-6$2i × 3i = 6i² = −6.
• Forgetting that non-real roots come in pairs. If 2 + 3i is a root and the coefficients are real, then 2 − 3i is also a root.
• Giving only one nth root. A complex number has three cube roots, evenly spaced around a circle.

Where this leads next

Rotation is multiplication by a complex number here and a 2 × 2 matrix in the matrices guide. The identity e^$i\theta$iθ $=\cos \theta +i\sin \theta$= cos θ + i sin θ comes from the Maclaurin series in the sequences and series guide, and every polynomial in the quadratics and polynomials guide has its full set of roots once complex numbers are allowed.

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