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

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Presentation on theme: "Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System."— Presentation transcript:

1 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 i 2 = 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 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

9 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.

10 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

11 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.


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