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**Complex Numbers; Quadratic Equations in the Complex Number System**

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

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IMAGINARY NUMBERS Definition: The number i, called the imaginary unit, is the number such that i2 = −1. Definition: For any positive real number a,

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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.

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**COMPLEX NUMBER ARITHMETIC**

If a + bi and c + di are complex numbers, then Addition: Subtraction: Multiplication:

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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:

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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

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**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.

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**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.

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**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.

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THE QUADRATIC FORMULA In 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

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**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.

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