Complex Numbers

Complex Number

A number in the form x + iy where x, y are real numbers and i² = -1.

Imaginary Number

A number that can be written in the form n i, where n is a real number and i = √(-1).

Conjugate of a Complex Number

For a complex number of the form a + ib, its conjugate is a - ib.

Modulus of a Complex Number

The distance from the origin to the endpoint of a complex number in the complex plane, denoted |z|.

Argument of a Complex Number

The angle θ formed with the positive real axis, often denoted as arg(z).

i

The imaginary unit, defined as the square root of -1, satisfying the equation i² = -1.

Addition of Complex Numbers

Combine real and imaginary parts separately to sum complex numbers.

Multiplication of Complex Numbers

Multiply complex numbers using distributive properties while applying the rule i² = -1.

Powers of i

Powers can be simplified, e.g., i^2 = -1, i^3 = -i, and i^4 = 1.

Equal Complex Numbers

Two complex numbers are equal if their real parts and imaginary parts are separately equal.