Perform arithmetic on complex numbers in rectangular (a + bi) or polar (r∠θ) form. Get step-by-step solutions and an Argand plane visualization.
A complex number has the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as √(−1). Complex numbers extend the real number line into a two-dimensional plane called the complex plane or Argand plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Every complex number can also be expressed in polar form as z = r(cos θ + i sin θ), where r = |z| is the modulus (distance from the origin) and θ is the argument (angle from the positive real axis). De Moivre's Theorem states that zⁿ = rⁿ(cos nθ + i sin nθ), making it easy to raise complex numbers to integer powers.
i² = −1. This is the defining property of the imaginary unit. It follows that i³ = −i, i⁴ = 1, and the pattern repeats with period 4. This property is what makes complex multiplication produce the cross term: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i.
The modulus |z| is the distance from the origin to the point (a, b) in the complex plane. It's calculated as |z| = √(a² + b²) — the same as the Pythagorean theorem. For example, |3 + 4i| = √(9 + 16) = √25 = 5.
The conjugate of z = a + bi is z̄ = a − bi — you flip the sign of the imaginary part. Multiplying a complex number by its conjugate gives a real number: z × z̄ = a² + b² = |z|². This property is used in complex division to eliminate the imaginary part from the denominator.
To divide (a+bi)÷(c+di), multiply both numerator and denominator by the conjugate of the denominator (c−di). This makes the denominator real: (a+bi)(c−di) ÷ (c²+d²). Then separate the real and imaginary parts to get the result in a+bi form.
Complex numbers are fundamental in electrical engineering (impedance in AC circuits), signal processing (Fourier transforms), quantum mechanics (wave functions), control theory (stability analysis), and fluid dynamics. They also appear in fractal geometry — the famous Mandelbrot set is defined using complex number iteration.