How the a+bi Form Calculator works
This calculator performs arithmetic directly on complex numbers written in a+bi (rectangular) form, where a is the real part and b is the coefficient of the imaginary unit i, defined by i² = −1. Enter the real and imaginary parts of two complex numbers, z₁ = a₁ + b₁i and z₂ = a₂ + b₂i, choose an operation, and the calculator returns the result in a+bi form along with its modulus and argument.
Formula and method
The four basic operations on complex numbers in a+bi form are:
- Addition: (a₁+b₁i) + (a₂+b₂i) = (a₁+a₂) + (b₁+b₂)i
- Subtraction: (a₁+b₁i) − (a₂+b₂i) = (a₁−a₂) + (b₁−b₂)i
- Multiplication: (a₁+b₁i)(a₂+b₂i) = (a₁a₂−b₁b₂) + (a₁b₂+b₁a₂)i
- Division: (a₁+b₁i) ÷ (a₂+b₂i) = [(a₁a₂+b₁b₂) + (b₁a₂−a₁b₂)i] ÷ (a₂²+b₂²), found by multiplying numerator and denominator by the conjugate of the denominator, a₂−b₂i
Once the result's real part (re) and imaginary part (im) are known, the calculator also reports the modulus |z| = √(re²+im²) — the distance of the point from the origin on the complex plane — and the argument θ = atan2(im, re), the angle measured from the positive real axis. Together, modulus and argument give the polar form r(cosθ + i·sinθ).
Common sources of error
- Sign errors on i²: remember i² = −1, so a term like b₁b₂i² becomes −b₁b₂, not +b₁b₂ — this trips up multiplication most often
- Dividing by zero: if z₂ = 0 + 0i, division is undefined because the denominator a₂²+b₂² is zero
- Degrees vs. radians: the argument here is reported in degrees; convert with θ(rad) = θ(deg) × π/180 if a later step expects radians
Checking your result
For addition and subtraction, the real parts and imaginary parts should each combine independently — nothing should "cross over" between them. For multiplication, verify with a known case: i × i should equal −1 (that is, 0+1i times 0+1i gives −1+0i). For the modulus, confirm it is never negative and matches √(re²+im²) using the result you were given.
Applications
Complex numbers in a+bi form appear throughout electrical engineering (AC circuit impedance), signal processing, control theory, and quantum mechanics, wherever a quantity has both a magnitude and a phase or direction that a single real number cannot capture. Converting to polar form (modulus and argument) is often the easier way to multiply, divide, or find powers and roots of complex numbers, since those operations become simple arithmetic on the moduli and angles.