What are the applications of complex numbers in real life?
1 Complex Numbers I : Friendly Complex
Numbers
Complex numbers are widely used in physics. The solution of physical equations is often made simpler through the use of complex numbers and we will study examples of this
when solving differential equations later in this course. Another particularly important
application of complex numbers is in quantum mechanics where they play a central
role representing the state, or wave function, of a quantum system. In this course
I will give a straightforward introduction to complex numbers and to simple functions
of a complex variable. The first Section “Friendly Complex Numbers” is intended to
provide a presentation of basic definitions and properties of complex numbers suitable
for those who have not studied the subject.
Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians. Imaginary numbers were defined in 1572 by Rafael Bombelli, an Italian mathematician who wrote an influential algebra text and made free use of both negative numbers and complex numbers. Bombelli solved equations, and he introduced +i and -i and described how they both worked in Algebra, thus defining imaginary numbers.
One of the most familiar examples where we use imaginary numbers, or "complex numbers" as they are sometimes called, is from electrical engineering, where imaginary numbers are used to keep track of the amplitude and phase of an electrical oscillation, such as an audio signal, or the electrical voltage and current that power electrical appliances.
Closely related to the electrical engineering example is the use of complex numbers in signal processing, which has applications to telecommunications (cellular phone), radar (which assists the navigation of airplanes), and even biology (in the analysis of firing events from neurons in the brain)....