Let me start with the basic definition of complex numbers. In simple words it can be defined as:
DEFINITION
Complex numbers are those numbers that can be separated into both a real component and an imaginary component. Complex numbers are generally expressed in the form a + bi, where a represents any real number (rational or irrational) and b represents the real coefficient (rational or irrational) of the imaginary number bi.
PROPERTIES
ADDITION
- Addition and with complex numbers are similar to addition and subtraction with real numbers, with the sums (or differences) of real components handled independently of imaginary components. For example:
$ (a + bi) + (c + di) = (a + c) + (b + d)i $
MULTIPLICATION
- Multiplication of complex numbers is similar to multiplying two first-order polynomials. Expressed generally, the product of two complex numbers is
$ (a + bi)(c + di) = ac + adi + bci + bdi2 = (ac - bd) + (ac + bd)i. $
Rules and Identities
$ i^2 = (i)(i) = -1 $ $ i^3 = (i^2)(i) = (-1)(i) = -i $ $ i^4 = (i^2)(i^2) = (-1)(-1) = +1 $
Eulers Formula
While dealing with complex nos. the most frequent identity which we shall come across is the Euler's Foemula
$ e^{ix} = cox(x) + isin(x) $
- This formula links together the exponential function and the trigonometric functions.
You can also represent cos and sin in terms of e in a way very similar to hyperbolic trig functions, which is why they hyperbolics are named sinh and cosh.
