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# Imaginary Unit and Standard Complex Form The Imaginary Unit is by ajizai

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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:

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:

Simplify:

Simplify in terms of i:

Simplify in terms of i:

Simplify in terms of i:

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     ...
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
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,
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

Numbers

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.

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

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

3. Express the sum of                        and

in the form         .

*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
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)
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)
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:

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,

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.

Find the absolute value of the complex
number:

Find the distance from the origin to this
point.

If              , find |   z|.