Complex numbers were first employed when the general formula for solutions of cubic and quartic equations gave rise to the definition of the square root of negative numbers. This had bewildered many prominent mathematicians, even Euler.
Now, we coin the number i such that i^2=-1. Also, this naturally led to the invention of -i. Note that however, we do not say that i>0 or -i<0. It is senseless to talk about their "positiveness", as proven by the following fact: If a>0, then a^2>0*a=0. However, i^2=-1<0, a contradiction.
Right then, people began investigating the property of such numbers. These numbers behave in a similar manner to surds (i.e. square root of real positive numbers), in a sense that they can be added and multiplied in an algebraic manner. They are demonstrated below:
Let a, b, c, d be real numbers. Then, the following properties of complex numbers hold true:
1) (a+bi)+(c+di)=(a+c)+(b+d)i
2) (a+bi)(c+di)=(ac-bd)+(ad+bc)i
3) (a+bi)c=ac+bci
For division, we first rationalize the denominator, then carry out multiplication in the above manner. Refer to the page on Surds.
Problems:
