5.3Complex Numbers; Quadratic

Equations with a Negative Discriminant

Complex numbers are numbers of theform a + bi, where a and b are realnumbers. The real number a is called thereal part of the number a + bi; the realnumber b is called the imaginary part ofa + bi.

(a + bi) + (c + di) = (a + c) + (b + d)i

(2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i

= 1 + 10i

Sum of Complex Numbers

(a + bi) - (c + di) = (a - c) + (b - d)i

(3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i

= 2 + 3i

Difference of Complex Numbers

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.