Review of Complex Numbers
Complex numbers are the numbers that consist of a real part and an imaginary part, and it usually can be illustrated on a co-ordinate system.
Definition
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i denotes a number satisfying i$ ^2 $ = − 1. Of course, since the square of any real number is nonnegative, i cannot be a real number. At first glance, it is not even clear whether such an object exists and can be reasonably called a number; for example, can we sensibly associate with i natural operations such as addition and multiplication? As it happens, we can define mathematical operations for these "complex numbers" in a consistent and sensible way and, perhaps more importantly, using complex numbers provides mathematicians, physicists, and engineers with an extremely powerful approach to expressing parts of these sciences in a convenient and natural-feeling way.
Properties
- Addition: $ \,(a + bi) + (c + di) = (a + c) + (b + d)i $
- Subtraction: $ \,(a + bi) - (c + di) = (a - c) + (b - d)i $
- Multiplication: $ \,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i $
- Division: $ \,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,, $
where c and d are not both zero.
