A complex number is a number that contains both a real and an imaginary part. The real part is a normal number, while the imaginary part is a real number multiplied by the imaginary unit, denoted by "i". The imaginary unit is defined as the square root of -1.
Complex numbers are typically written in the form a + bi, where a is the real part and bi is the imaginary part. The real and imaginary parts can also be represented graphically on a complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.
Some common operations on complex numbers include addition, subtraction, multiplication, and division. These operations follow similar rules to those of real numbers, but with the added complexity of dealing with the imaginary unit.
The conjugate of a complex number a + bi is the number a - bi. In other words, the conjugate has the same real part as the original complex number, but with the opposite sign on the imaginary part. The conjugate is useful in simplifying complex number expressions and finding the modulus of a complex number.
Complex numbers have a wide range of applications in fields such as engineering, physics, and mathematics. They are used to model and solve problems involving alternating currents, quantum mechanics, and signal processing, among others. They are also used in computer graphics and game development for representing and manipulating 3D objects.