Presentation on theme: "Complex Numbers; Quadratic Equations in the Complex Number System"— Presentation transcript:
1 Complex Numbers; Quadratic Equations in the Complex Number System Section 1.3Complex Numbers; Quadratic Equations in the Complex Number System
2 IMAGINARY NUMBERSDefinition: The number i, called the imaginary unit, is the number such thati2 = −1.Definition: For any positive real number a,
3 COMPLEX NUMBERSIf 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, thenAddition:Subtraction:Multiplication:
5 COMPLEX CONJUGATESThe 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 byEXAMPLES:
6 PRODUCT OF CONJUGATESTheorem: 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 divisionwe 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 FORMULAIn 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.