Add, subtract, multiply, or divide complex numbers in a+bi form — with step-by-step work for each operation.
i = √(−1) → i² = −1
Addition / Subtraction: Combine real parts, combine imaginary parts.
(a+bi) ± (c+di) = (a±c) + (b±d)i
Multiplication: Use FOIL, then replace i²=−1 to simplify.
(a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i
Division: Multiply numerator and denominator by the conjugate of the denominator. The denominator becomes real: c²+d².
(a+bi)(a−bi) = a² + b²
The conjugate of a complex number a+bi is a−bi — just flip the sign of the imaginary part.
Multiplying a complex number by its conjugate always gives a real number: (a+bi)(a−bi) = a²−(bi)² = a²+b².
This property is what makes division work: multiply top and bottom by the conjugate of the denominator to make the denominator real, then divide.
One-on-one Algebra 2 tutoring makes complex numbers concrete — we work through your actual problems and build the intuition that carries into polynomials, trigonometry, and beyond.