009 chapter ii

31
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

Upload: aleli-ariola

Post on 24-May-2015

765 views

Category:

Technology


1 download

TRANSCRIPT

Page 1: 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.

Page 2: 009 chapter ii

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).

Page 3: 009 chapter ii

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.

Page 4: 009 chapter ii

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

Page 5: 009 chapter ii

_____________________________________________

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

_____________________________________________

Page 6: 009 chapter ii

14. x² + 4 = 0

_____________________________________________

15. 2x² + 14 = 0

_____________________________________________

Lesson 3

Page 7: 009 chapter ii

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.

Page 8: 009 chapter ii

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: ____

Page 9: 009 chapter ii

Instruction: Express each number in terms of i and simplify.

1. √−64

______________________________________________________

2. √−94

_____________________________________________________

3. √−50

______________________________________________________

4. 2√−18 ______________________________________________________

5. 4√−45

______________________________________________________

6. √−1649

Page 10: 009 chapter ii

______________________________________________________

7. 3√−2516

______________________________________________________

8. −¿7√−4

______________________________________________________

9. 40 √−425

______________________________________________________

10. √−100

______________________________________________________

11. √−48

______________________________________________________

12. 5√−18100

______________________________________________________

13. 4 √−2527

Page 11: 009 chapter ii

______________________________________________________

14. √−649

______________________________________________________

15. −√−75

______________________________________________________

Lesson 4

Page 12: 009 chapter ii

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.

Page 13: 009 chapter ii

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: ____

Page 14: 009 chapter ii

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

_____________________________________________________

Page 15: 009 chapter ii

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

Page 16: 009 chapter ii

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

Page 17: 009 chapter ii

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.

Page 18: 009 chapter ii

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

Page 19: 009 chapter ii

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

Page 20: 009 chapter ii

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: ____

Page 21: 009 chapter ii

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)

_____________________________________________________

Page 22: 009 chapter ii

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)

_____________________________________________________

Page 23: 009 chapter ii

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

Page 24: 009 chapter ii

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)