Presentation on theme: "Complex Numbers; Quadratic Equations in the Complex Number System"— Presentation transcript:
1Complex Numbers; Quadratic Equations in the Complex Number System Section 1.3Complex Numbers; Quadratic Equations in the Complex Number System
2IMAGINARY NUMBERSDefinition: The number i, called the imaginary unit, is the number such thati2 = −1.Definition: For any positive real number a,
3COMPLEX 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.
4COMPLEX NUMBER ARITHMETIC If a + bi and c + di are complex numbers, thenAddition:Subtraction:Multiplication:
5COMPLEX 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:
6PRODUCT OF CONJUGATESTheorem: The product of a complex number and its conjugate is a nonnegative real number. That is, if z = a + bi, then
7DIVIDING 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.
8PROPERTIES 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.
9POWERS 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.
10THE 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
11CHARACTER 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.