5.3 Complex Numbers; Quadratic Equations with a Negative Discriminant

5.3Complex Numbers; Quadratic Equations with a Negative Discriminant

C o m p l e x n u m b e r s a r e n u m b e r s o f t h e + a b i f o r m , w h e r e a a n d b a r e r e a l n u m b e r s . T h e r e a l n u m b e r a i s c a l l e d t h e r e a l p a r t o f t h e n u m b e r a + b i ; t h e r e a l i m a g i n a r y p a r t n u m b e r b i s c a l l e d t h e o f a + b i .

Sum of Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i (2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i = 1 + 10i

Difference of Complex Numbers (a + bi) - (c + di) = (a - c) + (b - d)i (3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i = 2 + 3i

Product of Complex Numbers

If z=a +bi is a complex number, then its conjugate, denoted by

Theorem The product of a complex number and its conjugate is a nonnegative real number. Thus if z=a +bi, then

Theorem

If N is a positive real number, we define the principal square root of -N as

In the complex number system, the solution of the quadratic equation where a, b, and c are real numbers and are given by the formula

Solve:

Discriminant of a Quadratic Equation is called a discriminant >0, there are 2 unequal real solutions. =0, there is a repeated real solution. <0, there are two complex solutions. The solutions are conjugates of each other.

5.3 Complex Numbers; Quadratic Equations with a Negative Discriminant