A complex number is a two part structure that consist of the two "usual" numbers, real part and an imaginary part. A complex number represents a point in the complex plane. For this point, the real part is the horizontal (x) coordinate and the imaginary part is the vertical (y) coordinate. Ordinary real numbers are also complex numbers that have the zero imaginary part (hence they vertical coordinate on the complex plane is zero and they all lie on the horizontal (x) axis). In this way the complex number extends the ordinary real number in order to solve problems that would be impossible with only real numbers. Complex arithmetic is the compatible extension of the ordinary arithmetic that can be also applied to the real numbers (assuming zero imaginary part), obtaining same result. Complex numbers may appear and disappear on the way of solution also when both input values and results are ordinary real numbers.
A complex number is often written as
where a is the real part, b is the imaginary part i is a the imaginary unit. The square root operation for the complex number is defined so that
The functions to extract parts of the complex value separately are usually labelled as Re(z) (takes only real part) and Im(z) (takes only imaginary part). a and b directly represent x and y coordinates in the Cartesian plane.
Complex number can also can also be represented in polar coordinates as
is the the angle between the horizontal (x) axis and the line containing both the origin of coordinates (0,0) and the given point (Re(z), Im(z)). r is the distance from this point till the origin of coordinates and is always a non negative value.
Complex addition is the same as vector addition, adding real and imaginary parts independently:
Differently, complex multiplication is not the independent multiplication of real and imaginary parts. Complex multiplication is easier to explain using angular form: the modulus (r) parts are multiplied, but the angles are added:
Multiplication by a complex number of modulus 1 acts as a rotation.
The rule of complex multiplication is also consistent with the rule that multiplication of two negative real numbers results a positive value. The negative number has the 180 degree (or radians) angle. The sum of two such angles is equal to 360 degrees, or full circle. This is the same as zero, the angle of the positive scalar value.
The simplest way to understand complex multiplication for the numbers in Cartesian form seems to use :
The definitions of the complex root, division and other operations can be derived from the definitions of addition and multiplication.