1. Complex Numbers; Quadratic Equations in the Complex Number System
Section 1.3
Complex Numbers; Quadratic Equations in the Complex Number System

2. IMAGINARY NUMBERS
Definition: The number i, called the imaginary unit, is the number such that
i2 = −1.
Definition: For any positive real number a,

3. COMPLEX NUMBERS
If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex number. The real number a is called the real part and the real number b is called the imaginary part.

4. COMPLEX NUMBER ARITHMETIC
If a + bi and c + di are complex numbers, then
Addition:
Subtraction:
Multiplication:

5. COMPLEX CONJUGATES
The complex numbers a + bi and a − bi are called complex conjugates or conjugates of each other. The conjugate of a complex number z is denoted by
EXAMPLES:

6. PRODUCT OF CONJUGATES
Theorem: The product of a complex number and its conjugate is a nonnegative real number. That is, if z = a + bi, then

7. DIVIDING COMPLEX NUMBERS
To perform the division
we multiply the numerator and denominator by the conjugate of the denominator. Then simplify the complex number into standard form.

8. PROPERTIES OF CONJUGATES
The conjugate of the conjugate is the number itself.
The conjugate of a sum is the sum of the conjugates.
The conjugate of a product is the product of the conjugates.

9. POWERS OF i i 1 = i i 5 = i i 2 = −1 i 6 = −1 i 3 = −i i 7 = −i
And so on. The powers of i repeat with every fourth power.

10. THE QUADRATIC FORMULA
In the complex number system, the solutions of the quadratic equation ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0, are given by the formula

11. CHARACTER OF THE SOLUTIONS OF A QUADRATIC EQUATION
In the complex number system, consider a quadratic equation ax2 + bx + c = 0 with real coefficients.
1. If b2 − 4ac > 0, there are two unequal real solutions.
2. If b2 − 4ac = 0, there is a repeated real solution, a double root.
3. If b2 − 4ac < 0, the equation has two complex solutions that are not real. These solutions are conjugates of each other.