Operation
z₁
z₂
—
Polar form—
Modulus |z|—
Argument arg(z)—
Conjugate z̄—
z₁—
z₂—
Complex Number Arithmetic Rules
All operations are performed in rectangular form z = a + bi using these rules:
Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
Subtraction: (a+bi) − (c+di) = (a−c) + (b−d)i
Multiply: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Division: (a+bi)/(c+di) = [(ac+bd) + (bc−ad)i] / (c²+d²)
Conjugate: conj(a+bi) = a − bi
Modulus: |a+bi| = √(a² + b²)
Argument: arg(a+bi) = atan2(b, a) [in radians]
Multiplication and division are often easier in polar form: if z₁ = r₁∠θ₁ and z₂ = r₂∠θ₂, then z₁·z₂ = r₁r₂∠(θ₁+θ₂) and z₁/z₂ = (r₁/r₂)∠(θ₁−θ₂).
Polar Form and De Moivre's Theorem
Every complex number can be written in polar form as:
z = r(cosθ + i·sinθ) = r∠θ
where r = |z| = √(a²+b²) (modulus)
θ = arg(z) = atan2(b,a) (argument, in radians)
De Moivre's Theorem:
[r(cosθ + i·sinθ)]ⁿ = rⁿ(cos(nθ) + i·sin(nθ))
Euler's Formula:
e^(iθ) = cosθ + i·sinθ
De Moivre's theorem is the key tool for computing integer and fractional powers of complex numbers. For example, the n-th roots of a complex number z = r∠θ are rˡ/ⁿ∠(θ + 2πk)/n for k = 0, 1, ..., n−1.
Key properties of complex numbers:
- |z₁·z₂| = |z₁|·|z₂| — modulus is multiplicative
- arg(z₁·z₂) = arg(z₁) + arg(z₂) — arguments add
- z·z̄ = |z|² — a number times its conjugate gives a real number
- The complex conjugate reflects across the real axis
A complex number is a number of the form a + bi, where a is the real part and b is the imaginary part. The symbol i represents the imaginary unit, defined as the square root of −1 (i² = −1). Complex numbers extend the real number line into a two-dimensional plane called the complex plane or Argand diagram, where the x-axis represents the real part and the y-axis the imaginary part.
The modulus (or absolute value) of a complex number z = a + bi is its distance from the origin in the complex plane. It is calculated as |z| = √(a² + b²) — exactly like the Pythagorean theorem. The modulus is always a non-negative real number. It represents the magnitude (r) in polar form r∠θ and satisfies |z₁·z₂| = |z₁|·|z₂|.
To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. For (a+bi)/(c+di), multiply top and bottom by (c−di) to get [(ac+bd) + (bc−ad)i] / (c²+d²). This technique removes the imaginary part from the denominator (since (c+di)(c−di) = c²+d² which is always real), yielding a standard form complex number.
De Moivre's theorem states that for any complex number in polar form r(cosθ + i·sinθ) and any real number n: [r(cosθ + i·sinθ)]ⁿ = rⁿ(cos(nθ) + i·sin(nθ)). This theorem makes it easy to compute powers and roots of complex numbers by working in polar form — multiply the modulus by the power n and multiply the argument by n. It's especially useful for finding nth roots: the equation zⁿ = w has n distinct complex solutions.