Complex number

 A complex number is a mathematical construct that extends the concept of real numbers by introducing an imaginary unit denoted as "i." Complex numbers are expressed in the form "a + bi," where "a" and "b" are real numbers, and "i" represents the imaginary unit. In this expression:

"a" is called the real part of the complex number.

"b" is called the imaginary part of the complex number.

The imaginary unit "i" is defined as the square root of -1, which means that "i^2" equals -1.

Some key concepts related to complex numbers include:

1. Addition and Subtraction**: Complex numbers can be added and subtracted by combining their real and imaginary parts separately.

2. Multiplication**: When multiplying complex numbers, you use the distributive property and the fact that "i^2" is -1 to simplify the calculations.

3. Division: Dividing complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number "a + bi" is "a - bi."

4. Complex Plane**: Complex numbers can be represented graphically on a complex plane, which is a two-dimensional plane where the horizontal axis represents the real part, and the vertical axis represents the imaginary part.

5. Modulus (Magnitude)**: The modulus or magnitude of a complex number "a + bi" is given by the square root of the sum of the squares of its real and imaginary parts: |a + bi| = √ (a^2 + b^2). It represents the distance of the complex number from the origin on the complex plane.

6. Argument (Angle)**: The argument of a complex number is the angle it forms with the positive real axis on the complex plane. It is usually denoted by θ (theta).

Complex numbers have wide applications in various fields, including mathematics, physics, engineering, and signal processing. They are used to solve equations that don't have solutions in the real number system, and they play a crucial role in understanding phenomena involving oscillations, waves, and more. The study of complex analysis deals with properties and functions of complex numbers.