Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System IMAGINARY NUMBERS Definition: The number , called the imaginary
unit, is the number such that COMPLEX NUMBERS If and are real numbers and is the imaginary unit, then is called a complex number. The real number is called the real part and the real number is called the imaginary part.
COMPLEX NUMBER ARITHMETIC If and are complex numbers, then Addition: Subtraction:
Multiplication: COMPLEX CONJUGATES The complex numbers and are called complex conjugates or conjugates of each other. The conjugate of a complex number 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 , 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.
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.
POWERS OF i And so on. The powers of repeat with every fourth power. PRINCIPAL SQUARE ROOT
OF N Definition: If is a positive real number, we define the principle square root of N, denoted by , as where is the imaginary unit and . THE QUADRATIC FORMULA
In the complex number system, the solutions of the quadratic equation , where , , and are real numbers and , are given by the formula CHARACTER OF THE SOLUTIONS OF A QUADRATIC EQUATION In the complex number system, consider a
quadratic equation with real coefficients. 1. If , there are two unequal real solutions. 2. If , there is a repeated real solution, a double root. 3. If , the equation has two complex solutions that are not real. These solutions are conjugates of each other.
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