Notion of Complex Numbers
Process of extending notion of number from natural to real was connected both with needs of practice and requirements of mathematics.
At the beginning, people used numbers to count objects. Then when they began to divide things it appeared that they need more than natural numbers - positive rational numbers. Then, need in subtracting led people to 0 and negative numbers. Finally, need of taking roots of positive number led us to irrational numbers.
All these operations can be done with real numbers. However there are operations that can't be done with real numbers, for example, taking square root of negative number. Therefore, there is a need in extending real numbers.
Geometrically real numbers are represented with points on coordinate line. Therefore assumption arises that geometric representation of new numbers should be searched not on line, but on plane. Since every point M of coordinate plane xOy can be matched with coordinates of this point then new numbers can be represented as ordered pairs of real numbers.
Definition. Every ordered pair (a;b) of real numbers a and b is called complex number.
Two complex numbers (a;b) and (c;d) are equal if and only if a=c and b=d.