## Complex numbers are numbers that consist of a real part and an imaginary part. They are an extension of the real numbers and are denoted by the symbol "i" for the imaginary unit.

In a complex number, the real part is a real number, and the imaginary part is a real number multiplied by "i". A complex number is typically written in the form "a + bi", where "a" is the real part, "b" is the imaginary part, and "i" is the imaginary unit.

Complex numbers have several important properties and operations, including:

Addition: The addition of two complex numbers is done by adding their real parts separately and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Subtraction: The subtraction of two complex numbers is done by subtracting their real parts separately and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

Multiplication: The multiplication of two complex numbers is done using the distributive property and the fact that "i^2" is equal to -1. For example, (a + bi) * (c + di) = (ac - bd) + (ad + bc)i.

Division: The division of two complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator, which is the complex number with the same real part but with the opposite sign for the imaginary part. For example, (a + bi) / (c + di) = (ac + bd)/(c^2 + d^2) + (bc - ad)i/(c^2 + d^2).

Conjugate: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of (a + bi) is (a - bi).

Modulus: The modulus of a complex number, denoted as |a + bi|, is the absolute value of the complex number and is calculated as the square root of the sum of the squares of its real and imaginary parts. For example, |a + bi| = √(a^2 + b^2).

Polar form: Complex numbers can also be represented in polar form, which involves expressing them in terms of their modulus and argument. The modulus is the distance from the origin to the complex number in the complex plane, and the argument is the angle between the positive real axis and the line connecting the origin to the complex number. The polar form of a complex number is given as r(cosθ + i sinθ), where "r" is the modulus and "θ" is the argument.

Complex numbers are used in various fields of mathematics, science, engineering, and technology, including electrical engineering, physics, signal processing, and computer graphics. They have applications in solving equations, analyzing alternating current circuits, representing waveforms, and many other areas.