What is the polar form of a complex number?

The polar form represents a complex number as r * e^(i*theta) or equivalently r(cos(theta) + i*sin(theta)), where r is the magnitude (distance from origin) and theta is the argument (angle from the positive real axis). r = sqrt(a^2 + b^2) and theta = atan2(b, a).

What is the conjugate of a complex number?

The conjugate of a + bi is a - bi. It is obtained by negating the imaginary part. The product of a complex number and its conjugate always gives a real number: (a + bi)(a - bi) = a^2 + b^2.

Complex Number Calculator - Add, Multiply, Convert |

Q: What is a complex number?

A complex number is a number of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i^2 = -1. Complex numbers extend the real number system and are essential in engineering, physics, and mathematics.

Q: How do you multiply complex numbers?

To multiply (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i. The real part is ac - bd and the imaginary part is ad + bc.

Q: How do you divide complex numbers?

To divide (a + bi) by (c + di), multiply numerator and denominator by the conjugate of the denominator: ((a + bi)(c - di)) / (c^2 + d^2). The result is ((ac + bd) + (bc - ad)i) / (c^2 + d^2).

Complex Number Calculator

Perform arithmetic on complex numbers and convert between rectangular (a + bi) and polar (r∠θ) forms. Results update in real time as you type.

Complex Number Operations

Operation

z1 = a + bi

Real (a)

Imaginary (b)

z2 = c + di

Real (c)

Imaginary (d)

Result

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How to Use the Complex Number Calculator

Select an operation from the dropdown, enter the real and imaginary parts of your complex numbers, and see the result instantly. For polar-to-rectangular conversion, enter the magnitude r and angle θ in degrees. All arithmetic follows the standard rules for complex numbers.

For example, multiplying (3 + 4i) by (1 + 2i): real part = 3×1 − 4×2 = −5, imaginary part = 3×2 + 4×1 = 10, giving −5 + 10i.

Complex Number Arithmetic

Complex numbers extend the real numbers by including the imaginary unit i, where i² = −1. Every complex number can be written as a + bi where a is the real part and b is the imaginary part. The four basic arithmetic operations follow naturally from the distributive law and the rule i² = −1.

Polar and Rectangular Forms

The rectangular form a + bi locates a complex number on the complex plane using Cartesian coordinates. The polar form r∠θ uses the distance from the origin (magnitude r = √(a² + b²)) and the angle from the positive real axis (argument θ = atan2(b, a)). Multiplication and division are often simpler in polar form because magnitudes multiply and angles add.

Applications

Complex numbers appear in electrical engineering (AC circuit analysis, impedance, phasors), signal processing (Fourier transforms), quantum mechanics (wave functions), control theory (transfer functions), and fractal geometry. Understanding how to manipulate them is essential for advanced STEM work.

Frequently Asked Questions

What is a complex number?

A number of the form a + bi where i² = −1. It extends the real number system with an imaginary axis.

How do you multiply complex numbers?

Use FOIL: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.

What is the polar form?

r∠θ where r = √(a² + b²) is the magnitude and θ = atan2(b, a) is the angle.

How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator to get a real denominator.

What is the conjugate?

The conjugate of a + bi is a − bi. Their product is always a² + b², a real number.

Disclaimer: This calculator is for informational and educational purposes only. Results are estimates and should not be considered professional expert advice. Consult a qualified professional before making decisions based on these calculations. See our full Disclaimer.