Download - 009 chapter ii
Chapter II
This chapter centers on the complex numbers which is a number comprising a
real number and an imaginary number. Under this, we have the number i, the complex
plane where the points are plotted and the 4 arithmetic operations such as addition and
subtraction, multiplication and division of complex numbers. To round up the chapter,
simple equation involving complex numbers will be studied and solved.
TARGET SKILLS:
At the end of this chapter, students are expected to:
• identify complex numbers;
• differentiate the real pat and imaginary part of complex numbers; and
• explore solving of the 4 arithmetic operations on the complex numbers.
Lesson 2
Defining Complex NumbersOBJECTIVES:
At the end of this lesson, students are expected to:
identify complex numbers;
differentiate the real number and standard imaginary unit; and
extend the ordinary real number.
A complex number, in mathematics, is a number comprising a real number and
an imaginary number; it can be written in the form a + bi, where a and b are real
numbers, and i is the standard imaginary unit, having the property that i2 = −1.[1] The
complex numbers contain the ordinary real numbers, but extend them by adding in extra
numbers and correspondingly expanding the understanding of addition and
multiplication.
Equation 1: x2 - 1 = 0.
Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation
in x is equivalent to finding the x-intercepts of a graph; and, the graph of y = x 2 - 1
crosses the x-axis at (-1,0) and (1,0).
Equation 2: x2 + 1 = 0
Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.
Since the graph has no x-intercepts, the equation has no solutions. When we define
complex numbers, equation 2 will have two solutions.
Name: ___________________ Section: _______
Instructor: ________________ Date: _______ Rating: ____
Solve each equation and graph.
1. x² + 4 = 0
_____________________________________________
2. 2x² + 18 = 0
_____________________________________________
3. 2x² + 14 = 0
_____________________________________________
4. 3x² + 27 = 0
_____________________________________________
5. x² - 3 = 0
_____________________________________________
6. x² + 21 = 0
_____________________________________________
7. 3x² - 5 = 0
_____________________________________________
8. 5x² + 30 = 0
_____________________________________________
9. 2x² + 3 = 0
_____________________________________________
10. x² + 50 = 0
_____________________________________________
11. x² - 2 = 0
_____________________________________________
12. 3x² - 50 = 0
_____________________________________________
13. x² - 3 = 0
_____________________________________________
14. x² + 4 = 0
_____________________________________________
15. 2x² + 14 = 0
_____________________________________________
Lesson 3
The Number iOBJECTIVES:
At the end of this lesson, students are expected to:
recognize the property of the number i;
discuss the powers of i; and
solve the high powers of imaginary unit.
Consider Equations 1 and 2 again.
Equation 1 Equation 2
x2 - 1 = 0. x2 + 1 = 0.
x2 = 1. x2 = -1.
Equation 1 has solutions because the number 1 has two square roots, 1 and -1.
Equation 2 has no solutions because -1 does not have a square root. In other words,
there is no number such that if we multiply it by itself we get -1. If Equation 2 is to be
given solutions, then we must create a square root of -1.
The imaginary unit i is defined by
The definition of i tells us that i2 = -1. We can use this fact to find other powers of i.
Example
i3 = i2 * i = -1*i = -i.
i4 = i2 * i2 = (-1) * (-1) = 1.
Exercise:
Simplify i8 and i11.
We treat i like other numbers in that we can multiply it by numbers, we can add it
to other numbers, etc. The difference is that many of these quantities cannot be
simplified to a pure real number.
For example, 3i just means 3 times i, but we cannot rewrite this product in a
simpler form, because it is not a real number. The quantity 5 + 3i also cannot be
simplified to a real number.
However, (-i)2 can be simplified. (-i)2 = (-1*i)2 = (-1)2 * i2 = 1 * (-1) = -1.
Because i2 and (-i)2 are both equal to -1, they are both solutions for Equation 2 above.
Name: ___________________ Section: _______
Instructor: ________________ Date: _______ Rating: ____
Instruction: Express each number in terms of i and simplify.
1. √−64
______________________________________________________
2. √−94
_____________________________________________________
3. √−50
______________________________________________________
4. 2√−18 ______________________________________________________
5. 4√−45
______________________________________________________
6. √−1649
______________________________________________________
7. 3√−2516
______________________________________________________
8. −¿7√−4
______________________________________________________
9. 40 √−425
______________________________________________________
10. √−100
______________________________________________________
11. √−48
______________________________________________________
12. 5√−18100
______________________________________________________
13. 4 √−2527
______________________________________________________
14. √−649
______________________________________________________
15. −√−75
______________________________________________________
Lesson 4
The Complex PlaneOBJECTIVES:
At the end of this lesson, students are expected to:
distinguish the points on the plane;
differentiate the real and imaginary part; and
draw from memory the figure form by the plot points on the complex plane.
A complex number is one of the form a + bi, where a and b are real numbers. a is
called the real part of the complex number, and b is called the imaginary part.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal. I.e., a+bi = c+di if and only if a = c, and b = d.
Example.
2 - 5i.
6 + 4i.
0 + 2i = 2i.
4 + 0i = 4.
The last example above illustrates the fact that every real number is a complex
number (with imaginary part 0). Another example: the real number -3.87 is equal to the
complex number -3.87 + 0i.
It is often useful to think of real numbers as points on a number line. For
example, you can define the order relation c < d, where c and d are real numbers, by
saying that it means c is to the left of d on the number line.
We can visualize complex numbers by associating them with points in the plane.
We do this by letting the number a + bi correspond to the point (a,b), we use x for a and
y for b.
Exercises: Represent each of the following complex number by a point in the plane.
1. 3 + 2i
2. 1 – 4i
3. 4 + 3i
4. 2 – 5i
5. 4 – 3i
Name: ___________________ Section: _______
Instructor: ________________ Date: _______ Rating: ____
Instruction: Represent each of the following Complex Numbers by a point in the plane.
1. 4+3 i
______________________________________________________
2. 5 –7 i
______________________________________________________
3. 5 i
______________________________________________________
4. 0
______________________________________________________
5. 3
______________________________________________________
6.32−2i
_____________________________________________________
7.12
______________________________________________________
8. 3+2 i
______________________________________________________
9. 5 – 3i2
_____________________________________________________
10.1 – 4 i
______________________________________________________
11.32+4 i
_____________________________________________________
12.4+3 i
______________________________________________________
13.4−3 i
______________________________________________________
14.2−5 i
______________________________________________________
15.53+2 i
______________________________________________________
Lesson 5
Complex Arithmetic
OBJECTIVES:
At the end of this lesson, students are expected to:
define the four arithmetic operations on complex numbers;
comply with the steps in solving the different operations; and
solve the four arithmetic operations.
When a number system is extended the arithmetic operations must be defined
for the new numbers, and the important properties of the operations should still hold.
For example, addition of whole numbers is commutative. This means that we can
change the order in which two whole numbers are added and the sum is the same: 3 +
5 = 8 and 5 + 3 = 8.
We need to define the four arithmetic operations on complex numbers.
Addition and Subtraction
To add or subtract two complex numbers, you add or subtract the real parts and
the imaginary parts.
(a + bi) + (c + di) = (a + c) + (b + d)i.
(a + bi) - (c + di) = (a - c) + (b - d)i.
Example
(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.
(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.
Note
These operations are the same as combining similar terms in expressions that
have a variable. For example, if we were to simplify the expression (3 - 5x) + (6 + 7x) by
combining similar terms, then the constants 3 and 6 would be combined, and the terms
-5x and 7x would be combined to yield 9 + 2x.
The Complex Arithmetic applet below demonstrates complex addition in the
plane. You can also select the other arithmetic operations from the pull down list. The
applet displays two complex numbers U and V, and shows their sum. You can drag
either U or V to see the result of adding other complex numbers. As with other graphs in
these pages, dragging a point other than U or V changes the viewing rectangle.
Multiplication
The formula for multiplying two complex numbers is
(a + bi) * (c + di) = (ac - bd) + (ad + bc)i.
You do not have to memorize this formula, because you can arrive at the same
result by treating the complex numbers like expressions with a variable, multiply them
as usual, then simplify. The only difference is that powers of i do simplify, while powers
of x do not.
Example
(2 + 3i)(4 + 7i) = 2*4 + 2*7i + 4*3i + 3*7*i2
= 8 + 14i + 12i + 21*(-1)
= (8 - 21) + (14 + 12)i
= -13 + 26i.
Notice that in the second line of the example, the i2 has been replaced by -1.
Using the formula for multiplication, we would have gone directly to the third line.
Exercise
Perform the following operations.
(a) (-3 + 4i) + (2 - 5i)
(b) 3i - (2 - 4i)
(c) (2 - 7i)(3 + 4i)
(d) (1 + i)(2 - 3i)
Division
The conjugate (or complex conjugate) of the complex number a + bi is a - bi.
Conjugates are important because of the fact that a complex number times its
conjugate is real; i.e., its imaginary part is zero.
(a + bi)(a - bi) = (a2 + b2) + 0i = a2 + b2.
Example
Number Conjugate Product
2 + 3i 2 - 3i 4 + 9 = 13
3 - 5i 3 + 5i 9 + 25 = 34
4i -4i 16
Suppose we want to do the division problem (3 + 2i) ÷ (2 + 5i). First, we want to
rewrite this as a fractional expression .
Even though we have not defined division, it must satisfy the properties of
ordinary division. So, a number divided by itself will be 1, where 1 is the multiplicative
identity; i.e., 1 times any number is that number.
So, when we multiply by , we are multiplying by 1 and the number is
not changed.
Notice that the quotient on the right consists of the conjugate of the denominator
over itself. This choice was made so that when we multiply the two denominators, the
result is a real number. Here is the complete division problem, with the result written in
standard form.
Exercise:
Write (2 - i) ÷ (3 + 2i) in standard form.
We began this section by claiming that we were defining complex numbers so that
some equations would have solutions. So far we have shown only one equation that
has no real solutions but two complex solutions. In the next section we will see that
complex numbers provide solutions for many equations. In fact, all polynomial equations
have solutions in the set of complex numbers. This is an important fact that is used in
many mathematical applications. Unfortunately, most of these applications are beyond
the scope of this course. See your text (p. 195) for a discussion of the use of complex
numbers in fractal geometry.
Name: ___________________ Section: _______
Instructor: ________________ Date: _______ Rating: ____
Instruction: Perform the indicated operations and express the result in the form a+b i.
1. (8+2i)+(4+7 i)
_____________________________________________________
2. 3−i¿+(−4+2 i)
_____________________________________________________
3. (−7+4 i)+(5−6 i)
_____________________________________________________
4. (−8+6 i)−(−3+2 i)
_____________________________________________________
5. (1.2+3 i)+(3.88+1.6 i)
_____________________________________________________
6. (6−i)+(2−5 i)
_____________________________________________________
7. (5+3 i)−(3−2 i)
_____________________________________________________
8. 8−15 i¿+(−2+10 i)
_____________________________________________________
9. (8−2i)−(10−5 i)
_____________________________________________________
10. (6−5 i)−(6−4 i)
_____________________________________________________
11. (2+i)(3+i)
_____________________________________________________
12. (4−3i)(2+3 i)
_____________________________________________________
13. (1−2i)(5−2 i)
_____________________________________________________
14. (3+2 i)(−7+5 i)
_____________________________________________________
15. (8−7 i)(2 i+3 i)
_____________________________________________________
A. Define and/or describe each of the following terms.
1. Imaginary part
2. Real number
3. Complex number
4. Complex plane
5. Imaginary unit
6. Commutative property
7. Complex conjugate
B. 1. Simplify:
a. i15
b. i25
c. i106
d. i207
e. i21
2. Perform the indicated operation and express each answer.
a. √¯ 9 + √¯ 25
b. √¯ 16 + √¯ 49
c. √¯ 100 + √¯ 81
d. √¯ 169 + √¯ 225
e. √¯ 450 + √¯ 162
f. √¯ 147 + √¯ 48
3. Represent each complex numbers by a point in the plane.
a. 3 – i
b. -2 + 4i
c. -3 + 3i
d. 4 + 5i
e. -3 + 5i
4. Give the real part and the imaginary part of each complex numbers in #3.
5. Perform the indicated operations.
a. (3 – 2i) + (-7 + 3i)
b. (-4 + 7i) + (9 – 2i)
c. (14 – 9i) + (7 – 6i)
d. (5 + i) – (3 + 2i)
e. (7 – 2i) – (4 – 6i)
f. (8 + 3i) – (-4 – 2i)
g. (3 – 2i) (3 +2i)
h. (5 + 3i) (4 – i)
i. (11 + 2i)2 (5 – 2i)
j. (5 + 4i) / (3 – 2i)
k. (4 + i) (3 – 5i) / (2 – 3i)
l. (7 + 3i) / (3 – 3i / 4)