The equation x² = −1 has no answer on the number line, so mathematicians invented a new number rather than admit defeat. In plain words, i is defined by i² = −1, and a complex number a + bi pairs a real part with an imaginary part. Its magnitude — the distance from zero on the complex plane — is √(a² + b²): for 3 + 4i that is 5. Multiply complex numbers with FOIL and replace every i² with −1 to land back in a + bi form. Electrical engineering students analyzing AC circuits meet complex numbers weekly, since i tracks phase shifts through capacitors and inductors. Algebra students meet them as the "two complex solutions" of x² + 1 = 0. They are called imaginary, but the power grid depends on them.

This calculator finds the magnitude. Type the real part a into the Variable A box, say 3, and the imaginary part b into the Variable B box, say 4. Read the magnitude from the Result box: 5. Verify on paper — √(9 + 16) = √25 = 5 — and remember the magnitude is always a positive real number, never imaginary.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Imaginary Number Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.