Complex Numbers Quadratic Equations in the Complex Number System

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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.
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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
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COMPLEX CONJUGATES
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PRODUCT OF CONJUGATES
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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 i 4 1 i 8 1
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 gt 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 lt 0, the equation has two
complex solutions that are not real. These
solutions are conjugates of each other.