Complex Numbers by lindseypsenter

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									Complex Numbers
     Definition of pure
    imaginary numbers:

Any positive real number b,
     2        2
 b  b  1  bi
where i is the imaginary unit
 and bi is called the pure
    imaginary number.
     Definition of pure
    imaginary numbers:

         i  1
          2
         i  1
       i is not a variable
it is a symbol for a specific
             number
  Simplify each expression.

1. 81        81 1  9i

2. 121x         121x 1 x
           5         4


                11 i x
                    2
                   x
3. 200  100 1 2x
       x
          10i 2x
  Simplify each expression.
4. 8i  3i  24i  24  1
                     2
                     2
           Remember i  1      24

5. 5 20  i 5 i 20
        Remember that 1  i

 i  100 110  10
    2
            2
 Remember i  1
             i
    Cycle of ""

i 1
0
            i 1
             4

 i i
  1
            i i
             5


i  1
2
            i  1
             6


i  i
 3
            i  i
             7
               Simplify.
     12    To figure out where we
 i        are in the cycle divide the
          exponent by 4 and look at
                the remainder.
12  4 = 3 with remainder 0
      So i  i  1
               12       0
               Simplify.
     17    Divide the exponent by 4
 i        and look at the remainder.


17  4 = 4 with remainder 1

      So i      17
                     i  i
                        1
              Simplify.
    26    Divide the exponent by 4
i        and look at the remainder.


26 4 = 6 with remainder 2

     So i     26
                    i  1
                       2
               Simplify.
     11    Divide the exponent by 4
 i        and look at the remainder.


11 4 = 2 with remainder 3

      So i  i  i
               11      3
Definition of Complex
      Numbers
  Any number in form
a+bi, where a and b are
 real numbers and i is
    imaginary unit.
  Definition of Equal
  Complex Numbers
Two complex numbers are
equal if their real parts are
equal and their imaginary
      parts are equal.
     If a + bi = c + di,
   then a = c and b = d
When adding or subtracting
complex numbers, combine like
terms.
Ex: 8  3i  2  5i 
    8  2  3i  5i

         10  2i
     Simplify.
8 7i 12 11i
8 12 7i  11i
    4  18i
    Simplify.
9 6i 12 2i 
9 12 6i  2i 
    3  8i
Multiplying
complex numbers.
 To multiply complex
 numbers, you use the
  same procedure as
multiplying polynomials.
    Simplify.
 8 5i2 3i
F    O    I     L
16 24i 10i 15i   2

  16 14i  15
     31 14i
     Simplify.
 6 2i 5 3i 
 F   O     I     L
3018i  10i  6i   2

  30  28i  6
   24  28i

								
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