Complex Number Formulas: Essential Equations for Operations

Core Complex Number Formulas

Complex numbers are written as a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i² = -1. The formulas for basic operations are straightforward and follow from treating i like a variable, but with the rule that i² becomes -1.

Addition and Subtraction

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

(a + bi) - (c + di) = (a - c) + (b - d)i

These work just like combining like terms: real parts add together, imaginary parts add together. For example, (3 + 2i) + (1 - 4i) = 4 - 2i.

Multiplication

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

To multiply, use the distributive property (FOIL) and simplify i² to -1. The cross terms ad i and bc i combine, while bd i² becomes -bd. For instance, (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i² = 2 + 11i - 12 = -10 + 11i.

Division

(a + bi) / (c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²)

Division is trickier. Multiply numerator and denominator by the conjugate of the denominator (c - di) to make the denominator real. The result is then separated into real and imaginary parts. For example, (1 + i) / (1 - i) = [(1+ i)(1 + i)] / [(1 - i)(1 + i)] = (1 + 2i + i²) / (1 - i²) = (2i) / 2 = i.

Modulus and Argument

|z| = √(a² + b²) (modulus)

arg(z) = arctan(b / a) (argument, in radians or degrees)

The modulus is the distance from the origin to the point (a, b) in the complex plane—think of it like the length of a vector. The argument is the angle the vector makes with the positive real axis. When a = 0, the argument is π/2 (or 90°) if b > 0, and -π/2 if b < 0.

Polar and Exponential Forms

z = r(cos θ + i sin θ) (polar form)

z = r e^{iθ} (exponential form)

Here r = |z| and θ = arg(z). The polar form is especially useful for multiplication: to multiply two complex numbers, multiply their moduli and add their arguments. For example, 2(cos 30° + i sin 30°) * 3(cos 60° + i sin 60°) = 6(cos 90° + i sin 90°) = 6i. The exponential form uses Euler's formula e^{iθ} = cos θ + i sin θ and makes exponentiation and root extraction even simpler.

Understanding these formulas is the first step toward mastering complex numbers. For a step-by-step guide on performing these calculations by hand, see How to Calculate Complex Numbers: Step-by-Step Guide (2026).

Historical Origin and Intuition

The imaginary unit i was first introduced in the 16th century by Italian mathematician Gerolamo Cardano to solve cubic equations. At the time, even square roots of negative numbers were considered “sophisticated” or impossible. Later, Rafael Bombelli formalized the rules, and Euler and Gauss developed the full theory. The intuition behind complex number formulas comes from interpreting them as points on a 2D plane (the complex plane), where the real axis is horizontal and the imaginary axis vertical. Addition is vector addition—place the second vector at the tip of the first. Multiplication, particularly in polar form, corresponds to scaling and rotating: multiplying by i rotates a point 90° counterclockwise. This geometric view is why formulas like (a+bi)*(c+di) yield the same result as rotating and scaling vectors.

Practical Implications

Complex number formulas are not just abstract math—they are used in engineering, physics, and signal processing. In electrical engineering, AC circuit analysis relies heavily on complex numbers: voltages and currents are represented as phasors with magnitude and phase. The polar form simplifies calculations involving impedances and phase shifts. For a deeper look, see Complex Numbers in Electrical Engineering: Phasors and AC Circuits (2026). Additionally, complex numbers describe wave interference, quantum mechanics, and control systems. Mastering these formulas allows you to efficiently solve problems that would be much harder with real numbers alone.

Edge Cases and Special Considerations

When using these formulas, watch out for special cases:

• Division by zero: The denominator c + di must not be zero. If both c = 0 and d = 0, the division is undefined.
• Argument ambiguity: The arctan function returns values only between -π/2 and π/2, but the actual argument may need adjustment based on the quadrant. For example, -1 + i has argument 3π/4, not -π/4. Use the atan2 function (or check signs) to get the correct angle.
• Modulus is always non-negative: |z| ≥ 0. It is zero only for z = 0.
• Conjugate: The conjugate z̄ = a - bi is used in division and in finding the real and imaginary parts of a complex expression.

For answers to common questions about these edge cases, visit Complex Numbers FAQ: Common Questions Answered (2026). By understanding both the formulas and their limits, you can use the Complex Number Calculator confidently, knowing exactly what each operation does.