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What is a complex number?

A complex number is expressed in the standard form z = a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit defined such that i² = -1.

How do you multiply two complex numbers?

Using FOIL expansion: (a + bi)(c + di) = ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i.

How do you divide two complex numbers?

To divide (a + bi) / (c + di), multiply numerator and denominator by the complex conjugate of the denominator (c - di): [(ac + bd) + (bc - ad)i] / (c² + d²).

What is the modulus and argument of a complex number?

The modulus |z| = √(a² + b²) represents the distance from the origin on the complex plane. The argument θ = atan2(b, a) represents the counter-clockwise angle from the positive real axis.

Imaginary & Polar Algebra

Complex Numbers (a + bi) Calculator

Compute complex addition, multiplication, division, modulus, conjugates, and polar Euler forms with complete analytical proofs.

Complex Number Z₁ (a + bi)

+ i

Complex Number Z₂ (c + di)

+ i

Addition: Z₁ + Z₂

4 + 2i

Subtraction: Z₁ - Z₂

2 + 6i

Multiplication: Z₁ × Z₂

11 - 2i

Division: Z₁ / Z₂

-1 + 2i

Properties of Z₁:

• Modulus |Z₁|: 5.0000

• Argument θ: 53.13° (0.9273 rad)

• Polar Form: 5.00(cos 53.13° + i sin 53.13°)

• Euler Form: 5.00 e^(0.93i)

• Complex Conjugate: 3 - 4i

Properties of Z₂:

• Modulus |Z₂|: 2.2361

• Argument θ: -63.43° (-1.1071 rad)

• Polar Form: 2.24(cos -63.43° + i sin -63.43°)

• Euler Form: 2.24 e^(-1.11i)

• Complex Conjugate: 1 + 2i

Understanding Complex Numbers

Complex numbers extend the 1-dimensional real number line into a 2-dimensional plane known as the Argand plane. The horizontal axis plots real values, while the vertical axis plots imaginary values. They are vital in electrical engineering (AC circuits and impedances), quantum mechanics, and signal processing.

Frequently Asked Questions

Why do we multiply by the conjugate during division?

Multiplying a complex number (c + di) by its conjugate (c - di) yields a purely real number: c² + d². This clears imaginary units from the denominator.

What is Euler's formula?

Euler's famous formula states that e^(iθ) = cos(θ) + i sin(θ), establishing an equivalence between trigonometric functions and complex exponentials.