Presentation on theme: "Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System."— Presentation transcript:
Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System
IMAGINARY NUMBERS Definition: The number i, called the imaginary unit, is the number such that i 2 = 1. Definition: For any positive real number a,
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.
COMPLEX NUMBER ARITHMETIC If a + bi and c + di are complex numbers, then Addition: Subtraction: Multiplication:
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:
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
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.
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. PROPERTIES OF CONJUGATES
POWERS OF i i 1 = ii 5 = i i 2 = 1i 6 = 1 i 3 = ii 7 = i i 4 = 1i 8 = 1 And so on. The powers of i repeat with every fourth power.
THE QUADRATIC FORMULA In the complex number system, the solutions of the quadratic equation ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0, are given by the formula
CHARACTER OF THE SOLUTIONS OF A QUADRATIC EQUATION In the complex number system, consider a quadratic equation ax 2 + bx + c = 0 with real coefficients. 1.If b 2 4ac > 0, there are two unequal real solutions. 2.If b 2 4ac = 0, there is a repeated real solution, a double root. 3.If b 2 4ac < 0, the equation has two complex solutions that are not real. These solutions are conjugates of each other.