Complex Numbers

A complex number (also called an imaginary number) is a number that can be written in the form a + bi, where a and b are real numbers and i is a constant defined by the rule i² = −1. The quantity a is called the real part, and b is called the imaginary part. When b is zero, the number is an ordinary real number; when a is zero and b is nonzero, it is called a pure imaginary number.

Complex numbers were developed to solve equations that have no solution among real numbers, for example x² + 1 = 0. Addition, subtraction, multiplication, and division are performed in the same way as for real numbers, except that whenever i² appears it is replaced with −1. As a whole, complex numbers form a mathematical field, commonly denoted C.

Graphical representation

A complex number a + bi can be represented as a point on a two-dimensional plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This plane is called the complex plane or Argand diagram. Addition of complex numbers corresponds to vector addition, while multiplication corresponds to scaling lengths and adding angles.

Modulus and polar form

The modulus of a number z = a + bi is |z| = √(a² + b²), which is the distance from the point to the origin. The angle that z makes with the real axis is called the argument. Accordingly, the number can be written in polar form: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. This form simplifies multiplication and exponentiation.

Conjugate

The complex conjugate of z = a + bi is a − bi. The product of a number and its conjugate gives a real number equal to the square of its modulus, which is used in dividing complex numbers and simplifying fractions.

Significance and applications

Complex numbers are fundamental to many areas of mathematics and science. The fundamental theorem of algebra states that every polynomial equation of degree greater than zero has a solution among complex numbers. They are also used in electrical engineering and signal processing, such as in describing AC circuits, and in quantum physics, where wave functions take complex values.