THE NUMORIX GUIDE
How to use the Complex Number Calculator
Last reviewed September 14, 2026
What this calculator does
Represent a=aReal+aImag*i and b=bReal+bImag*i.
Formula and method
Represent a=aReal+aImag*i and b=bReal+bImag*i. Add or subtract real and imaginary parts, multiply with i^2=-1, or divide by multiplying numerator and denominator by the conjugate. Report result magnitude sqrt(real^2+imag^2).
Variables and inputs
Operation defaults to Add; A defaults to 3+2i and B to 1+7i. Each complex number has separate real and imaginary fields.
Worked example
For the default addition: real part = 3+1=4, imaginary part = 2+7=9, so result = 4+9i; magnitude = sqrt(4^2+9^2) = sqrt(97) = 9.8489.
How to interpret the result
The real and imaginary components locate a point in the complex plane. Magnitude is its distance from the origin.
Common mistakes to avoid
Remember i^2=-1 when multiplying. For division, use bReal^2+bImag^2 as the denominator and do not divide real and imaginary parts independently.
Assumptions and limitations
The engine does not guard against division by the zero complex number and reports decimal components without a polar-angle result. It supports only binary arithmetic operations.
Practical use and checks
The Complex Number Calculator represents a number as a + bi and supports addition, subtraction, multiplication, and division. Enter a = 3 + 2i and b = 1 + 7i and choose addition as a check; the result should be 4 + 9i with magnitude sqrt(4^2 + 9^2), about 9.8489. Choosing multiplication should give (3 + 2i)(1 + 7i) = -11 + 23i, because i squared is -1. These checks help distinguish the real and imaginary components from the magnitude. Use complex arithmetic when a quadratic has no real roots, when an AC circuit uses phasors, or when a rotation is represented algebraically. The magnitude is always nonnegative, but the real and imaginary signs carry quadrant and phase information. Division requires a nonzero divisor magnitude: dividing by 0 + 0i is undefined and can produce nonfinite output. The engine rounds the displayed division steps, so use the returned components for a closer check. It does not attach units, choose a branch for complex logarithms or roots, or model conjugate conventions beyond the four basic operations. Keep the order of operands clear for multiplication and division, since complex multiplication is commutative but division is not.