1.4. Complex numbers

The imaginary unit, \(i\) is defined as:

(1.18)\[i=\sqrt{-1}\]

A complex number is and expression of the form:

(1.19)\[z=x+iy\]

The distance from origo, the the point (x,y) in the complex plane is called the modulus and is defined as:

(1.20)\[|z|=\sqrt{x^2+y^2}\]

The norm of a complex number (equivalent of the vector length in the complex plane) is:

(1.21)\[|z|^2=x^2+y^2\]

The modulus of a product of two complex numbers \(z\) and \(w\) is:

(1.22)\[|zw|=|z||w|\]

The modulus of the sum of two complex numbers, also called “The triangle inequality” is defined:

(1.23)\[|z+w|<|z|+|w|\]

You can read more about complex numbers in Adams and Essex [AE18], Appendix 1.