University Mathematics Handbook (2015)
X. Algebra
Chapter 1. Complex Numbers
1.1 Definition: Algebraic Operations in Complex Numbers
a. An ordered pair of real numbers is called a complex number.
b. , when , is the algebraic form of a complex number.
is called the real part of , and is denoted .
is called the imaginary part of , and is denoted .
c. is a conjugate to the number
d. Let and be two complex numbers:
1. Equality: if and only if and
2. Addition:
3. Multiplication:
4. Powers of : , ,
5. Division:
6.
1.2 Geometric Description, Modulus and Argument
a. Any complex number can be described as point in plane or a vector beginning in the origin and ending at point .
Plane is called complex plane, when -axis is Real Axis and -axis is Imaginary Axis.
b. In the complex plane, is described by a vector symmetrical to about -axis.
c. The length of vector s called the modulus or absolute value of complex number and is denoted .
d. Angle between the vector corresponding to complex number and the positive direction of the real axis is called the argument of and is denoted as . It is measured counterclockwise, in radians.
Notice: For every complex number, the modulus is specified definitely, but an argument has countless values different from each other by an integer product of .
e. is the polar form of a complex number. To have its polar form, we should just take an argument value of , or, alternately, , since an addition of for does not change the values of and .
f. The relation between the algebraic form and the polar form of a complex number is:
g. Properties of absolute value (modulus)
h. Properties of the argument
1.3 Powers and Roots of
a. De Moivre’s Formula: For every natural ,
b. The -th root of a complex number is a complex number holding . It is denoted .
c. For every complex number , there exist just different complex numbers for which , expressed by the formula