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# 009 chapter ii

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### 009 chapter ii

1. 1. Chapter II<br />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.<br /> TARGET SKILLS: <br />At the end of this chapter, students are expected to:<br />• identify complex numbers;<br />• differentiate the real pat and imaginary part of complex numbers; and<br />• explore solving of the 4 arithmetic operations on the complex numbers.<br />Lesson 2<br />Defining Complex Numbers<br />OBJECTIVES:<br />At the end of this lesson, students are expected to:<br /><ul><li>identify complex numbers;
2. 2. differentiate the real number and standard imaginary unit; and
3. 3. extend the ordinary real number.</li></ul>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.<br />Equation 1:  x2 - 1 = 0.<br />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 = x2 - 1 crosses the x-axis at (-1,0) and (1,0).<br />Equation 2:  x2 + 1 = 0<br />Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.<br />Since the graph has no x-intercepts, the equation has no solutions. When we define complex numbers, equation 2 will have two solutions.<br />-438785-684530Name: ___________________ Section: _______<br />Instructor: ________________ Date: _______ Rating: ____<br />Solve each equation and graph.<br />x² + 4 = 0<br /> _____________________________________________<br />2x² + 18 = 0<br /> _____________________________________________<br />2x² + 14 = 0<br /> _____________________________________________<br />3x² + 27 = 0<br /> _____________________________________________<br />x² - 3 = 0 <br /> _____________________________________________<br />x² + 21 = 0<br />-466725-741103 _____________________________________________<br />3x² - 5 = 0<br /> _____________________________________________<br />5x² + 30 = 0<br /> _____________________________________________<br />2x² + 3 = 0<br /> _____________________________________________<br /> x² + 50 = 0<br /> _____________________________________________<br /> x² - 2 = 0<br /> _____________________________________________<br /> 3x² - 50 = 0<br /> _____________________________________________<br /> x² - 3 = 0<br /> _____________________________________________<br />-514350-683895 x² + 4 = 0<br /> _____________________________________________<br /> 2x² + 14 = 0<br /> _____________________________________________<br />Lesson 3<br />The Number i<br />OBJECTIVES:<br />At the end of this lesson, students are expected to:<br /><ul><li>recognize the property of the number i;
4. 4. discuss the powers of i; and
5. 5. solve the high powers of imaginary unit.</li></ul>Consider Equations 1 and 2 again.<br />Equation 1Equation 2 x2 - 1 = 0.x2 + 1 = 0. x2 = 1.x2 = -1. <br />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.<br />The imaginary unit i is defined by<br />The definition of i tells us that i2 = -1. We can use this fact to find other powers of i.<br />Example<br />i3 = i2 * i = -1*i = -i.<br />i4 = i2 * i2 = (-1) * (-1) = 1.<br />Exercise:<br />Simplify i8 and i11. <br />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.<br />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.<br />However, (-i)2 can be simplified. (-i)2 = (-1*i)2 = (-1)2 * i2 = 1 * (-1) = -1.<br />Because i2 and (-i)2 are both equal to -1, they are both solutions for Equation 2 above.<br />-485775-683953Name: ___________________ Section: _______<br />Instructor: ________________ Date: _______ Rating: ____<br />Instruction: Express each number in terms of i and simplify.<br /><ul><li>-64</li></ul> ______________________________________________________<br /><ul><li>-9 4</li></ul> _____________________________________________________<br /><ul><li>-50</li></ul> ______________________________________________________<br /><ul><li>2-18</li></ul> ______________________________________________________<br /><ul><li>4-45</li></ul> ______________________________________________________<br /><ul><li>-514350-693420-16 49</li></ul> ______________________________________________________<br /><ul><li>3-25 16</li></ul> ______________________________________________________<br /><ul><li>-7-4</li></ul> ______________________________________________________<br /><ul><li>40-425</li></ul> ______________________________________________________<br /><ul><li> -100</li></ul> ______________________________________________________<br /><ul><li> -48</li></ul> ______________________________________________________<br /><ul><li> 5-18100</li></ul> ______________________________________________________<br /><ul><li>-534789-636213 4-2527</li></ul> ______________________________________________________<br /><ul><li> -649</li></ul> ______________________________________________________<br /><ul><li> --75</li></ul> ______________________________________________________<br />Lesson 4<br />The Complex Plane<br />OBJECTIVES:<br />At the end of this lesson, students are expected to:<br /><ul><li>distinguish the points on the plane;
6. 6. differentiate the real and imaginary part; and
7. 7. draw from memory the figure form by the plot points on the complex plane.</li></ul>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.<br />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.<br />Example.<br />2 - 5i.<br />6 + 4i.<br />0 + 2i = 2i.<br />4 + 0i = 4.<br />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.<br />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.<br />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.<br />Exercises: Represent each of the following complex number by a point in the plane. <br /><ul><li>3 + 2i
8. 8. 1 – 4i
9. 9. 4 + 3i
10. 10. 2 – 5i
11. 11. 4 – 3i</li></ul>-485775-683953Name: ___________________ Section: _______<br />Instructor: ________________ Date: _______ Rating: ____<br />Instruction: Represent each of the following Complex Numbers by a point in the plane.<br /><ul><li>4 + 3i</li></ul> ______________________________________________________<br /><ul><li>5 – 7i</li></ul> ______________________________________________________<br /><ul><li>5i</li></ul> ______________________________________________________<br /><ul><li>0</li></ul> ______________________________________________________<br /><ul><li>3</li></ul> ______________________________________________________<br /><ul><li>32-2i
12. 12. _____________________________________________________
13. 13. 12</li></ul> ______________________________________________________<br /><ul><li>3 + 2i</li></ul> ______________________________________________________<br /><ul><li>-504826-6838955 – 3i2</li></ul> _____________________________________________________<br /><ul><li>1 – 4i</li></ul> ______________________________________________________<br /><ul><li>32+ 4i</li></ul> _____________________________________________________<br /><ul><li>4+3i</li></ul> ______________________________________________________ <br /><ul><li>4-3i</li></ul> ______________________________________________________<br /><ul><li>2-5i</li></ul> ______________________________________________________<br /><ul><li>53+2i</li></ul> ______________________________________________________<br />Lesson 5<br />Complex Arithmetic<br />OBJECTIVES:<br />At the end of this lesson, students are expected to:<br /><ul><li>define the four arithmetic operations on complex numbers;
14. 14. comply with the steps in solving the different operations; and
15. 15. solve the four arithmetic operations.</li></ul>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.<br />We need to define the four arithmetic operations on complex numbers.<br />Addition and Subtraction<br />To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.<br />(a + bi) + (c + di) = (a + c) + (b + d)i.(a + bi) - (c + di) = (a - c) + (b - d)i.<br />Example <br />(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.<br />(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.<br />Note<br />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.<br />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.<br />Multiplication<br />The formula for multiplying two complex numbers is<br />(a + bi) * (c + di) = (ac - bd) + (ad + bc)i.<br />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.<br />Example <br />(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.<br />Notice that in the second line of the example, the i2 has been replaced by -1.<br />Using the formula for multiplication, we would have gone directly to the third line.<br />Exercise <br />Perform the following operations.<br />(a) (-3 + 4i) + (2 - 5i)<br />(b) 3i - (2 - 4i)<br />(c) (2 - 7i)(3 + 4i)<br />(d) (1 + i)(2 - 3i)<br />Division<br />The conjugate (or complex conjugate) of the complex number a + bi is a - bi.<br />Conjugates are important because of the fact that a complex number times its conjugate is real; i.e., its imaginary part is zero.<br />(a + bi)(a - bi) = (a2 + b2) + 0i = a2 + b2.<br />Example <br />NumberConjugateProduct2 + 3i2 - 3i4 + 9 = 133 - 5i3 + 5i9 + 25 = 344i-4i16<br />Suppose we want to do the division problem (3 + 2i) ÷ (2 + 5i). First, we want to rewrite this as a fractional expression .<br />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.<br />So, when we multiply by , we are multiplying by 1 and the number is not changed.<br />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.<br />Exercise:<br />Write (2 - i) ÷ (3 + 2i) in standard form. <br />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.<br />-485775-683953Name: ___________________ Section: _______<br />Instructor: ________________ Date: _______ Rating: ____<br />Instruction: Perform the indicated operations and express the result in the form a +bi.<br /><ul><li>(8+2i)+(4+7i)</li></ul>_____________________________________________________<br /><ul><li>3-i)+(-4+2i)</li></ul>_____________________________________________________<br /><ul><li>(-7+4i)+(5-6i)</li></ul>_____________________________________________________<br /><ul><li>(-8+6i)-(-3+2i)</li></ul>_____________________________________________________<br /><ul><li>(1.2+3i)+(3.88+1.6i)
16. 16. _____________________________________________________
17. 17. (6-i)+(2-5i)</li></ul>_____________________________________________________<br /><ul><li>(5+3i)-(3-2i)</li></ul>_____________________________________________________<br /><ul><li>-533399-6266888-15i)+(-2+10i)</li></ul>_____________________________________________________<br /><ul><li>(8-2i)-(10-5i)</li></ul>_____________________________________________________<br /><ul><li> (6-5i)-(6-4i)</li></ul>_____________________________________________________<br /><ul><li> (2+i)(3+i)</li></ul>_____________________________________________________<br /><ul><li> (4-3i)(2+3i)</li></ul>_____________________________________________________<br /><ul><li> (1-2i)(5-2i)</li></ul>_____________________________________________________<br /><ul><li> (3+2i)(-7+5i)</li></ul>_____________________________________________________<br /><ul><li> (8-7i)(2i+3i)</li></ul>_____________________________________________________<br />Define and/or describe each of the following terms.<br />Imaginary part<br />Real number<br />Complex number<br />Complex plane<br />Imaginary unit<br />Commutative property<br />Complex conjugate<br />1. Simplify: <br />i15 <br />i25<br />i106<br />i207<br />i21<br />2. Perform the indicated operation and express each answer.<br /><ul><li>a. ¯9 + ¯25
18. 18. b. ¯16 + ¯49
19. 19. c. ¯100 + ¯81
20. 20. d. ¯169 + ¯225
21. 21. e. ¯450 + ¯162</li></ul>f. ¯147 + ¯48<br />3. Represent each complex numbers by a point in the plane.<br />a. 3 – i<br />b. -2 + 4i<br />c. -3 + 3i<br />d. 4 + 5i<br />e. -3 + 5i<br />4. Give the real part and the imaginary part of each complex numbers in #3.<br />5. Perform the indicated operations.<br />a. (3 – 2i) + (-7 + 3i)<br />b. (-4 + 7i) + (9 – 2i)<br />c. (14 – 9i) + (7 – 6i)<br />d. (5 + i) – (3 + 2i)<br />e. (7 – 2i) – (4 – 6i)<br />f. (8 + 3i) – (-4 – 2i)<br />g. (3 – 2i) (3 +2i)<br />h. (5 + 3i) (4 – i)<br />i. (11 + 2i)2 (5 – 2i)<br />j. (5 + 4i) / (3 – 2i)<br />k. (4 + i) (3 – 5i) / (2 – 3i)<br />l. (7 + 3i) / (3 – 3i / 4)<br />