Theory of complex numbers pdf

Definition 2 A complex number 3 is a number of the form a + bi where a and b are real numbers

If = a + bi then a is known as the real part of z and b as the imaginary part

Note that real numbers are complex – a real number is simply a complex number with zero imaginary part

Addition and subtraction of complex numbers has the same geometric interpretation as for vectors

The same holds for scalar multiplication of a complex number by a real number

(The geometric interpretation of multiplication by a complex number is di erent; we'll explain it soon

) Complex conjugation re ects a complex number in the real axis

Complex numbers have their origin in attempts to understand the roots of polynomials

By the sixteenth century, mathematicians had found formulas for the solutions of the equation p(x) = 0, whenpis a quadratic or cubic polynomial

For example, whenp(x) =ax2+bx+csuch solutionsx 0are given by the well-knownquadratic formula x 0= −b± √ b2−4ac 2a