Imaginary Unit and Standard Complex Form The Imaginary Unit is

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					Imaginary Unit and Standard Complex
Form
                             The Imaginary Unit is defined as

                                           i=        .

                 The reason for the name "imaginary" numbers is that when these
               numbers were first proposed several hundred years ago, people could
                                  not "imagine" such a number.

It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and
was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we
    find the imaginary unit being used in mathematics and science. Electrical engineers use the
                 imaginary unit (which they represent as j ) in the study of electricity.

                Imaginary numbers occur when a quadratic equation has
                          no roots in the set of real numbers.




                                                                          An imaginary
                                                                           number is a
                                                                          number whose
                                                                            square is
                                                                            negative.



                                 *i =         or -   i =-

               A pure imaginary number can be written in bi form where


                            b is a real number and i is           .

                         Examples:
                         pure imaginary
                         numbers
     A complex number is any number that can be written in the
        standard form a + bi, where a and b are real numbers and i is
                             the imaginary unit.
                         A complex number is a real number a,
                             or a pure imaginary number bi,
                                   or the sum of both.
                   Note these examples of complex numbers written
                        in standard a + bi form: 2 + 3i, -5 + 0i .

 Complex Number:
standard a + bi form                 a                           bi
       7 + 2i                        7                            2i
       1 - 5i                        1                           - 5i
         8i                          0                            8i




                       The set of real numbers
                  and the set of imaginary numbers
                  are subsets of the set of complex
                              numbers.
                               Practice with
Imaginary Unit and Standard Complex
                Form


                       Try the following problems:

      1.                            Solve the following       Answer
                                    quadratic equation:
                                       x2 + 4 = 0

2.    Solve the following expression for x:          Answer
                         2
                        x = -3

      3           Explain the circumstances under             Answer
                  which a square can be equal to a
                         negative number.



     4. State the values of the a and b                        Answer
        components of a + bi form for
          each complex number. State
          whether the value is the real part
          or the imaginary part of the
          number:

          a) 3 - 2i

          b) -2 - i

          c) 3i

          d) 0 + 0i

          e) -7 - 4i
Simplifying Square Roots with Negative
               Numbers


      Remember:                                   Consider:
      The Imaginary Unit is defined
      as

                 i=      .

        Whenever there is a negative under the radical sign, it comes out from
         underneath as i since the problem will contain a square root of -1.

        When simplifying these radical numbers in terms of i, follow the usual
        rules for simplifying radicals and treat the i with the rules for working
                                    with a variable.


Examples:
     Problem                          Solution                         bi Form
1.                                                                          6i
2.                                                                         14i
3.
4.
                            Practice with
Simplifying Square Roots with Negative
               Numbers


                  Try the following problems:

     1.                                                          Answer


                         Simplify:
     2.                                                          Answer

               Simplify in terms of i:




     3.                                                           Answer
                                     Simplify in terms of i:




    4.                                                         Answer
                   Simplify in terms of i:




                                                               Answer
          5.
                        Simplify:
Powers of i
 The powers of i repeat in a
      definite pattern:
        ( i, -1, -i, 1 )
     Powers of i                                                  ...
     Simplified
     form           i -1 -i 1              i -1 -i          1     ...
                                               Think about what happens
                                               when i is raised to a given
                                                         power:
  You need to remember that:




            LOOK OUT!!!
                     is not true when a and b are
                             both negative.

  False

    TRUE:




                   Whenever the exponent is greater than or equal to 5,
                                       you can
                     use the fact that    to simplify a power of i.
                           Another way to think of this process of
                           simplifying powers of i is to divide the
                                        exponent by 4,
   When raising i to        - if the remainder is 0, the answer is 1
                                             (i0).
 any integral power,        - if the remainder is 1, the answer is i
the answer is always                         (i1).
     i, -1, -i or 1.       -if the remainder is 2, the answer is -1
                                             (i2).
                            -if the remainder is 3, the answer is -i
                                             (i3).


  Let's examine two ways to simplify                          :
   Using the patterns                  Looking at remainders
shown in the robot table                when dividing by 4:
        above:
                                       with a remainder of 3,
                                      which means the answer is
                                                3
                                               i = -i.
       Practice with Powers of i
      Topic Index | Algebra2/Trig Index | Regents Exam Prep Center



     Solve the following problems and simplify the answers.

1.                                                         Choose:


            Simplify:                                         1
                                                              i
                                                              -i




2.                                                         Choose:

             Simplify:                                        0
                                                              1
                                                              2


              Explanation




3.                                                         Choose:

        Find the product of:                                   8
                                                               4
                                                               2i


              Explanation




4.                                                         Choose:

        Find the sum of:                                      6
                                                              6i
                                                              -6i
              Explanation




  5.                                             Choose:

                            Find the value of:     25
                                                   -10
                                                   -25


              Explanation




Adding and Subtracting Complex
           Numbers


        *Add like terms*
  Find the sum of the real components.           REMEMBER:
Find the sum of the imaginary components         Final answer must
  (the components with the i after them).          be in simplest

                                                         form.

                                Adding Rule:
Examples:
        1. Add: (7 + 5i) + (8 - 3i)



                       2. Add: (2 + 3i) + (-8 - 6i)




              3. Express the sum of                        and

                            in the form         .




                   4. Add                and              .




       *Subtract like terms*
  Find the difference of the real components.        REMEMBER:
      Find the difference of the imaginary          Final answer must be in
                   components                          simplest
    (the components with the i after them).                  form.
                                Subtracting Rule:
Examples:
     1. Subtract: (5 + 8i) - (2 + 2i)



            2. Subtract: (4 + 10i) - (-12 + 20i)




        3. Subtract               from             .




         4. Subtract            from               .
                       Practice with
      Arithmetic of Complex Numbers

      Practice makes perfect...



              Solve the following problems
           and express the result in    form.

1.
2.
3.
4.
5.
6.

7.

8.
9.
10.
11. Add:     ,     ,           and         .


12.    Subtract         from           .
Multiplying and Dividing Complex
Numbers

Multiplication:
Multiplying two complex numbers is accomplished in a manner similar to multiplying
     two binomials. You can use the FOIL process of multiplication, distributive
          multiplication, or your personal favorite means of multiplication.

                     Distributive Multiplication:
                     (2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 +
                     5i)
                     = 8 + 10i + 12i + 15i2
                     = 8 + 22i + 15(-1)
                     = 8 + 22i -15
                     = -7 + 22i Answer
                     Be sure to replace i2 with (-1) and
                     proceed with
                     the simplification. Answer should be in
                     a + bi
                     form.
             The product of two complex numbers is a complex
             number.

                           (a+bi)(c+di) = a(c+di) + bi(c+di)
                           = ac + adi + bci + bdi2
                           = ac + adi + bci + bd(-1)
                           = ac + adi + bci - bd
                           = (ac - bd) + (adi + bci)
                           = (ac -bd) + (ad + bc)i answer in
                           a+bi form

    The conjugate of a complex number a + bi is the complex number a - bi.
                For example, the conjugate of 4 + 2i is 4 - 2i.
             (Notice that only the sign of the bi term is changed.)

                    The product of a complex number and its
                                    conjugate
                     is a real number, and is always positive.
             (a + bi)(a - bi) = a2 + abi - abi - b2i2
             = a2 - b2 (-1) (the middle terms drop out)
             = a2 + b2 Answer
             This is a real number ( no i's ) and since
             both
             values are squared, the answer is
             positive.
Division:
     When dividing two complex numbers,
      1.write the problem in fractional form,
      2.rationalize the denominator by multiplying the numerator and the
        denominator by the conjugate of the denominator.
         (Remember that a complex number times its conjugate will give a real
         number.
         This process will remove the i from the denominator.)

                             Example:

                        Dividing using the conjugate:




                                               Answer
 Practice with Multiplying and Dividing
                Complex Numbers

Solve the following problems. Answers are to be in simplest a+bi form.
   You should be able to solve the problems both with, and without,
                       your graphing calculator.

    1.                                                Choose:

                        Multiply: (3 + 5i)(3 -              9 - 25i
                                                            25
                        5i)
                                                            34




    2.                                                Choose:

                                                            15 -
                                                      12i
                      Multiply: (8 + 9i)(7 - 3i)            29 -
                                                      39i
                                                            83 +
                                                      39i


                        Explanation




    3.                                                Choose:

                                                         25
                     Multiply: (4 - 3i)(3 - 4i)          -25i
                                                         12 - 12i


                         Explanation
4.                                            Choose:

                                                    21 + 20i
     Simplify: (2 +                                 -21 +
         5i)2                                 20i
                                                    29 + 20i


                            Explanation




5.                                             Choose:

                                                    7 + 8i
     Simplify: 8 + i(8 - i)                         8 + 8i
                                                    9 + 8i


              Explanation




6.                                             Choose:

                                                     5 - 2i
                                                     3 + 2i
                    Simplify:                        15 +
                                               10i


          Explanation




7.                                        Choose:

                                             35/37 +
                                          (12/37)i
                                             35 + 12i
                 Simplify:
                                             35/36 +
                                          (12/36)i


                  Explanation
8.                                    Choose:

                                         5 + 3i
                                         -5 - 3i
                       Simplify:
                                         5 - 3i


                Explanation




9.                                 Choose:

                                     2/15 + i/15
                                     2/15 - i/15
                       Simplify:
                                     1/45 + i/15


                Explanation




10.                                Choose:
           What is the
      multiplicative inverse          2/(1 + i)
                                      (1/2) -
                of                 (1/2)i
                                      (1 + i)/2


      Explanation
           Representing Complex Numbers
                     Graphically
                      (+ and -)

Due to their unique nature, complex numbers cannot be represented on a normal set of
                                  coordinate axes.

     In 1806, J. R. Argand developed a method for displaying complex numbers
 graphically as a point in a coordinate plane. His method, called the Argand diagram,
 establishes a relationship between the x-axis (real axis) with real numbers and the y-
                     axis (imaginary axis) with imaginary numbers.

                       In the Argand diagram, a complex number a + bi is
                        the point (a,b) or the vector from the origin to the
                                             point (a,b).
   Graph the
complex numbers:

1. 3 + 4i (3,4)

2. 2 - 3i (2,-3)

3. -4 + 2i (-4,2)

4. 3 (which is really 3
+ 0i) (3,0)

5. 4i (which is really 0
+ 4i) (0,4)

   The complex
    number is
represented by the
  point, or by the
  vector from the
origin to the point.
Absolute Value of Complex Numbers


                     The absolute value of a
                       complex number
                                      is written as .
               It is a nonnegative real number
                          defined as:

                                                          .

 Geometrically, the absolute value of a complex number is
the number's distance from the origin in the complex plane.

                                          In the diagram at the left, the complex
                                                           number
                                       8 + 6i is plotted in the complex plane on an
                                       Argand diagram (where the vertical axis is
                                        the imaginary axis). For this problem, the
                                       distance from the point 8 + 6i to the origin
                                       is 10 units. Distance is a positive measure.

                                      Notice the Pythagorean Theorem at work in
                                                     this problem.

           A complex number can be represented by a point, or by a
            vector from the origin to the point. When thinking of a
            complex number as a vector, the absolute value of the
         complex number is simply the length of the vector, called the
                                  magnitude.
       The formula for finding the absolute value of a complex number,


       can be derived from the Pythagorean theorem,
                               (see example 2 below).

                   In the Pythagorean Theorem, c is the hypotenuse and when
              represented in the coordinate plane, is always positive. This same idea
              holds true for the distance from the origin in the complex plane. Using
               the absolute value in the formula will always yield a positive result.

           To find the absolute value of a complex number a +      bi:
           1. Be sure your number is expressed in a + bi form
           2. Pick out the coefficients for a and b
           3. Substitute into the formula



Example 1:
Plot z = 8 + 6i on the complex plane, connect the graph of z to the origin (see graph
below), then find | z | by appropriate use of the definition of the absolute value of a
complex number.
Example 2:
Find the | z | by appropriate use of the Pythagorean Theorem when z = 2 - 3i.

                                           You can find the distance | z | by using the
                                         Pythagorean theorem. Consider the graph of
                                        2 - 3i shown at the left. The horizontal side of
                                          the triangle has length | a |, the vertical side
                                        has length | b |, and the hypotenuse has length
                                         | z |. By applying the Pythagorean Theorem,
                                                    you have, | z |2 = a2 + b2 .
                                            Notice: you can drop the absolute value
                                            symbols for a and b since | a |2 = a2 and
                                         | b |2 = b2. You must keep the absolute value
                                          symbol for z to insure that the final answer
                                                         will be positive.

                                        Solving this equation for | z |, you have just
                                       derived the formula for the absolute value of a
                                                      complex number:
                             Practice with
 Absolute Value of Complex Numbers


Solve the following problems dealing with the absolute value
                    of complex numbers.

       1.                                                        Answer
            Find the absolute value of the complex
                           number:



      2.                                                         Answer
                     Find the distance from the origin to this
                                      point.



       3.                                                    Answer
            If              , find |   z|.

      4.                                                         Answer
                            Find the absolute value of the
                                  complex number



      5.                                                         Answer
            Find the distance from the origin to
                         this point.

				
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