A complex number is a combination of real and imaginary numbers. Its formula is a + bi, where a and b are real numbers and i is an imaginary number. Complex numbers leave the line that a real number occupies and fill a plane (also called the complex plane).

Complex numbers are useful because they allow us to express roots of polynomials, which cannot be expressed using only real numbers. The complex number concept also allows us to work very precisely in certain areas of physics, such as the measurement of electromagnetic waves and electric currents.

As complex numbers have been used to solve problems for over 250 years, the theory underlying their use has become well developed and is a very important part of mathematics. In fact, it has given rise to an entire branch of math called complex analysis.

While the topic is not as intuitive as it might appear at first glance, it has a long history and is a fundamental part of our mathematical tool kit. The development of complex numbers grew out of the need to represent the roots of polynomials, and this led to the fundamental theorem of algebra that gives every polynomial equation a solution.

This article provides a review of the basic operations of complex numbers, including multiplication and division. We also discuss the properties of complex conjugates and introduce a variety of algebraic and trigonometric forms of these numbers. Finally, we consider the polar representation of complex numbers and give some examples of vectors in the complex plane.