Complex Numbers, Explained

After reading this you can add, multiply and divide complex numbers by hand, convert them to polar and exponential form, and find all n roots of any complex number.

A complex number has the form z = a + bi, where a and b are real numbers and i satisfies i^2 = -1. The number a is the real part and b is the imaginary part. You can plot z as the point (a, b) on a plane, which is called the Argand diagram.

Here is the hook. The equation x^3 = 8i has no real solution, yet it has exactly three complex solutions spaced evenly around a circle: \sqrt{3} + i, -\sqrt{3} + i and -2i. This article shows how those three numbers appear and how to check them.

What complex numbers are for

Complex numbers solve equations that have no real answers. The quadratic x^2 + 1 = 0 has roots x = \pm i. More generally, every polynomial of degree n with complex coefficients has exactly n roots counted with multiplicity. That is the fundamental theorem of algebra, and it fails if you only allow real numbers.

Beyond algebra, complex numbers describe rotation and oscillation compactly. Multiplying by i rotates a point 90 degrees counterclockwise: i \cdot (3 + 4i) = -4 + 3i, and the point (3, 4) does move to (-4, 3). Engineers write alternating currents and signals as complex exponentials for the same reason.

Use this calculator when you need exact arithmetic, a conversion to polar form, or all roots of a number. If your task is really a matrix eigenvalue problem or a polynomial with many terms, the Matrix Calculator and the Polynomial Toolkit are better fits.

The arithmetic rules

Addition and subtraction work component by component:

(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i

Here a, b, c, d are the four real parts. So (3 + 4i) + (1 - 2i) = 4 + 2i.

Multiplication uses the distributive law together with i^2 = -1:

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

The term bi \cdot di = bd\,i^2 = -bd is why the real part gains a minus sign. For division, multiply top and bottom by the conjugate c - di of the denominator:

\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}

The denominator c^2 + d^2 is real and positive whenever c + di \ne 0, so the quotient is again a clean complex number.

The conjugate of z = a + bi is \bar{z} = a - bi. It reflects the point across the real axis, and z \cdot \bar{z} = a^2 + b^2 is always a real number. That identity is the trick behind every division.

Polar and exponential form

Every complex number has a distance from the origin and an angle. The distance is the modulus:

r = |z| = \sqrt{a^2 + b^2}

The angle from the positive real axis is the argument \theta, measured with the two-argument arctangent so the quadrant is correct. Then

z = r(\cos\theta + i\sin\theta) = r e^{i\theta}

The second equality is Euler's formula. For z = 3 + 4i you get r = \sqrt{9 + 16} = 5 and \theta = \arctan(4/3) \approx 0.9273 radians, about 53.13 degrees. So 3 + 4i \approx 5 e^{0.9273 i}.

Polar form makes multiplication easy. Multiply the moduli and add the arguments:

r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}

That is why multiplying by i = 1 \cdot e^{i\pi/2} adds 90 degrees and leaves the modulus unchanged.

The nth roots formula

To find all n roots of z = r e^{i\theta}, take the real nth root of the modulus and split the angle into n equal shares that wrap around the full circle:

w_k = r^{1/n}\, e^{i(\theta + 2\pi k)/n}, \quad k = 0, 1, \dots, n-1

Here r^{1/n} is the ordinary positive nth root of a positive number, and k counts the roots. Adding 2\pi k to the angle before dividing gives n distinct answers. They all have the same modulus r^{1/n}, so they sit on one circle, separated by equal angles of 2\pi/n.

Three roots equally spaced on the circle of radius 2, at 120 degrees apart, starting at 30 degrees.

Worked example: cube roots of 8i

Reproducing the demo: mode roots, z₁ = 8i, n = 3

Set the mode to n-th roots, enter 8i and n = 3, then press Calculate.

Write 8i in polar form. Its modulus is r = \sqrt{0^2 + 8^2} = 8. It lies on the positive imaginary axis, so \theta = \pi/2 (90 degrees).
The root modulus is 8^{1/3} = 2. All three roots have modulus 2.
The base angle is \theta/3 = \pi/6 (30 degrees). The other two angles add 2\pi/3 (120 degrees) each: 150 degrees and 270 degrees.
Convert back. At 30 degrees: 2(\cos 30^\circ + i \sin 30^\circ) = \sqrt{3} + i. At 150 degrees: -\sqrt{3} + i. At 270 degrees: -2i.

Check the first root by cubing it. (\sqrt{3} + i)^2 = 3 + 2\sqrt{3}\,i + i^2 = 2 + 2\sqrt{3}\,i. Multiply once more by \sqrt{3} + i: the real part is 2\sqrt{3} - 2\sqrt{3} = 0 and the imaginary part is 2 + 6 = 8. So the cube is 8i, as required.

Reading the results

The calculator reports a result in more than one form. Match each field to the math above.

Rectangular form: The a + bi view. Best for adding and for reading off coordinates.
Modulus r: Distance from the origin, always \ge 0. For 3 + 4i it is 5.
Argument \theta: Angle in radians or degrees, usually reported in the range (-\pi, \pi] or (-180^\circ, 180^\circ].
Exponential form: The compact r e^{i\theta}. Best for multiplication, division and powers.

When roots are requested, expect exactly n answers with identical modulus and arguments that differ by 360^\circ / n. If you see fewer, you either asked for n = 1 or two roots coincide, which happens only when the modulus is 0.

Common mistakes

The rule \sqrt{a}\sqrt{b} = \sqrt{ab} fails for negative numbers. Writing \sqrt{-1}\sqrt{-1} = \sqrt{1} = 1 is wrong; the correct value is i \cdot i = -1. Convert to i first, then multiply.

Three more traps to watch:

Argument quadrant. For -1 - i the naive \arctan(1) gives 45 degrees, but the point is in the third quadrant at 225 degrees, or equivalently -135^\circ. Always use the two-argument arctangent.
Only one root. The symbol \sqrt{4} means the single value 2 for real numbers, but "the square roots of 4" as complex numbers are 2 and -2. The nth roots mode returns the full set of n.
Degrees versus radians. Euler's formula uses radians: e^{i\pi} = -1, not e^{180 i}. Convert 180 degrees to \pi \approx 3.1416 before exponentiating.

Related calculators

Complex numbers show up across algebra. When a quadratic has a negative discriminant, its roots are complex conjugates; the Quadratic Equation Solver reports them directly. For higher-degree polynomials, the Equation Solver finds all real and complex roots.

To simplify a surd such as \sqrt{-72} into 6\sqrt{2}\,i, use the Radical Simplifier. For logarithms and nth roots of real numbers, see the Exponent & Logarithm Calculator. The Vector Calculator treats (a, b) as a 2D vector when you care about length and angle but not the multiplication rule.

Frequently asked questions

Should I write the imaginary unit as i or j?

Both are accepted. Mathematicians write i; electrical engineers write j to avoid clashing with current. So 3 + 4i and 3 + 4j are the same number.

What is the argument of 0?

Undefined. The number 0 has modulus 0 and no direction, so no single angle describes it. Every other complex number has a well-defined argument.

Why does i to the fourth power equal 1?

Powers of i cycle with period 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. In polar terms each power adds 90 degrees, so four steps return you to the start.

How many nth roots does a nonzero number have?

Exactly n, all distinct. They lie on a circle of radius |z|^{1/n} and are spaced 360^\circ/n apart. For n = 4 and z = 1 the four roots are 1, i, -1, -i.

Is a real number also a complex number?

Yes. A real number is a complex number with imaginary part 0. For example 5 equals 5 + 0i, with modulus 5 and argument 0.

Try it yourself in the tool