Toán học - Chapter 0: Complex numbers

1 Math Dept, Faculty of Applied Science, HCM University of Technology ------------------------------------------------------------------------------------- Linear Algebra Chapter 0: Complex Numbers • Instructor Dr. Dang Van Vinh (10/2007) 2 A syllabus of Linear Algebra 1. Complex numbers 2. Matrix operations 3. Determinants 4. System of Linear equations 5. Vector spaces 6. Linear transformations 7. Eigenvalues, eigenvectors Quadratic form, orthogonality. 3 Student’s Tasks. Class attendance is expected. Do any homework and assignments. Be prepared for the next lesson. Tests, Final Exam. Midterm Test (20%) Assignments and homework (30%) Final Exam (50%) 4 Literature 1. V.A. Ilyin, Linear algebra; Mir Publishers Moscow, 1986 2. Strang G. Linear algebra and its applications, 1988. 3. David C. Lay, Linear Algebra and its applications, New York, 1993. 4. Meyer C.D. Matrix analysis and applied linear algebra, SIAM, 2000. 5. Kuttler K. Introduction to linear algebra for mathematicians, 6. Usmani R. Applied linear algebra, Marcel Dekker, 1987. 7. Kaufman L. Computational Methods of Linear Algebra ,2005. 8. Steven Leon, Linear Algebra with Applications, 2006 9. Muir T. Theory of determinants, Part I. Determinants in general 10. Golub G.H., van Loan C.F. Matrix computations. 3ed., JHU, 1996. 11. Nicholson W.K. Linear algebra with applications , PWS Boston, 1993. 12. Hefferon J. Linear algebra, 13. Proskuriyakov I.V. Problems in Linear algebra. 14. www.tanbachkhoa.edu.vn 5 CONTENTS --------------------------------------------------------------------------------------------------------------------------- 0.1 – The Standard form (algebraic form) 0.2 – The Trigonometric form 0.4 – The Power of complex numbers 0.5 – The Roots of complex numbers 0.6 – The Fundamental Theorem of Algebra 0.3 – The exponential form 6 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Definition of a complex number If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex number. The real number a is called the real part and the real number b is called the imaginary part of the complex number. The real numbers are a subset of the complex numbers. This can be observed by letting b = 0. Then a + bi = a + 0i, which a is a real number. 7 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Any number that can be written in the form 0 + bi, where b is a nonzero real number, is an imaginary number ( or a pure imaginary number). For example: i, -2i, 3i are all imaginary numbers. A complex number is in standard form when it is written in the form a + bi. 8 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Definition of addition and subtraction of complex numbers If a + bi and c + di are complex numbers, then Addition (a + bi) + (c + di) = (a + c) + (b + d) i Subtraction (a + bi) - (c + di) = (a - c) + (b - d) i Two complex numbers are said to be equal if they have the same real and imaginary parts. In other words, the complex numbers z1 = x1 + iy1 and z2 = x2 +iy2 are equal if and only if x1 = x2 and y1 = y2. Definition of Equality 9 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Definition of Multiplication of complex numbers If a + bi and c + di are complex numbers, then (a + bi) (c + di) = (ac – bd) + ( ad + bc)i Example Find an algebraic form of the number z = (2 + 5i).(3+ 2i) solution z = (2 + 5i)(3 + 2i) = 6 + 4i + 15i + 10 i2 Algebraic form of the given number is z = -4 + 19i. = 2.3 + 2.2i + 3.5i + 5i.2i = 6 + 19i + 10(-1) = -4 + 19i 10 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Addition, Subtraction, and Multiplication To add (subtract ) two complex numbers, simply add (subtract ) the corresponding real and imaginary parts. To multiply two complex numbers, use the distributive law and the fact that i2 = −1. 11 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- Definition of conjugates The complex numbers a + bi and a - bi are called complex conjugates or conjugates of each other. The conjugate of the complex number z is denoted by . z Example Find the conjugate of the number z = (2 + 3i) (4 - 2i). Solution. So conjugate of z is 14 8 . z i z = (2 + 3i) (4 - 2i) = 2.4 – 2.2i + 3i.4 – 3i.2i = 8 – 4i + 12i – 6i2 = 8 – 4i + 12i – 6(-1) = 14 + 8i. 12 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- If z and w are complex numbers, with conjugates and , then: z w 1. is a real number. z z 2. is a real number. z z 3. if and only if z is a real number. z z 4. z w z w   5. z w z w   6. z z 7. for all natural number n ( )n nz z Conjugate Theorems 13 0.1 The Standard form (algebraic form) ----------------------------------------------------------------- The trick for dividing two complex numbers is to multiply top and bottom by the complex conjugate of the denominator: Definition of the division 1 1 1 2 2 2 z a ib z a ib    1 1 1 2 2 2 2 2 2 2 ( )( ) ( )( ) z a ib a ib z a ib a ib      1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 z a a b b b a a b i z a b a b       14 0.1 The Standard form (algebraic form) --------------------------------------------------------------------------------------------------------------------------- Example. Find the quotient i i   5 23 Solution )5)(5( )5)(23( 5 23 ii ii i i      125 210315 2    iii i i 2 1 2 1 26 1313    Multiply numerator and denominator by 5 + i, which is the conjugate of the denominator Write in standard form 15 Remarks: Comparison with Real Analysis The concept of order in the real number system does not carry over to the complex number system. In other words, we cannot compare two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2. Statements such as z1 < z2 or z2 ≥ z1 have no meaning in C except in the special case when the two numbers z1 and z2 are real 0.1 The Standard form (algebraic form) --------------------------------------------------------------------------------------------------------------------------- 16 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- ( , )   M a b z a bi r b a o x y 2 2 mod( )  r a b z cos : sin          a r b r Real axis Imaginary axis 17 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- 2 2mod( ) | | .z z r a b    Definition of the Module The module of a complex number z = a + bi is a positive real number, denoted by Example Determine the module of the number z = 3 - 4i. Solution mod(z) = |z| = 2 2 2 23 ( 4) 5.    a ba = 3; b = -4. 18 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Let z = a + bi. Then 2 2| | z a b Remark. The distance between the point (a,b) and the origin. The distance between two points (a, b) and (c,d). Let z = a + bi and w = c + di. Then 2 2| | ( ) ( )z w a c b d     2 2( 0) ( 0)   a b 19 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Example Find all complex numbers z such that | 2 3 | 5  z i Solution | 2 3 | 5z i   | (2 3 ) | 5z i    This is a circle with a center at (2,-3) and a radius is 5. 20 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Example Find all complex numbers z such that | 2 | 1z i  Solution z i | 2 | 1 z i   | ( 2 ) | 1 This is a circle with the center at (0,-2) and the radius is 1. 21 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Example Find all the numbers z, such that | 2 | | 2 |z z   Solution | 2 | | 2 |  z z The set of all points in the plane such that the distance from these points to given two points (2,0) and (0,2) are equal. This is a vertical axis. 22 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Example Find all complex numbers z, such that | | | | 4z i z i    Solution | | | | 4   z i z i The set of all points, such that the sum of distances from these points to two given points (0,1) and (0,-1) is a constant. This is an ellipse. 23 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Definition of the argument The argument is the angle formed by the complex number on a polar graph with one real axis and one imaginary axis. arg( )z  Remember. 0 2   or      A formula for finding an argument. 2 2 2 2 cos sin              a a r a b b b r a b or tg  b a 24 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Solution Example Find an argument of 3 . z i 3; 1 a b . We find an angle such as  3 3 os = 23 1     a c r 1 1 sin = 23 1     b r It follows that 6    arg(z) = 6  25 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- 2 2; 0   z a bi a b (cos sin )  z r i The trigonometric form 2 2 2 2 2 2 ( )     a b z a b i a b a b (cos sin )z r i   26 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Solution Module: Write the number in trigonometric form Example 1 3  z i 1; 3.  a b 1 1 os = 23 1       a c r 3 3 sin = 23 1     b r Hence 2 3    trigonometric form: 2 2| | 2.   r z a b Argument: 2 2 1 3 2(cos sin ) 3 3       z i i 27 0.2 The Trigonometric Form ----------------------------------------------------------------------------------------------------- 1 1 1 1 2 2 2 2(cos sin ); (cos sin )z r i z r i       Equal numbers in trigonometric form 1 2 1 2 1 2 2 r r z z k         Product in trigonometric form 1 2 1 2 1 2 1 2(cos( ) sin( ))z z r r i        Quotient two numbers in trigonometric form 1 1 1 2 1 2 2 2 (cos( ) sin( )) z r i z r        28 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Solution (1 )(1 3)  z i i Trigonometric form: Example Find trigonometric, module and argument of the number (1 )(1 3).  z i i 2( os in ) 2( os in ) 4 4 3 3         z c is c is 2 2[ os( ) in( )] 4 3 4 3         z c is 2 2( os in ). 12 12     z c is 29 0.2 The Trigonometric Form --------------------------------------------------------------------------------------------------------------------------- Solution 2 2 3 3     i z i Trigonometric form: 7 7 2( os in ). 6 6     z c is Example Find a trigonometric, module and argument of the number 2 12 . 3     i z i - - 4(cos sin ) 3 3 5 5 2(cos sin ) 6 6        i i - 5 - 5 2[cos( - ) sin( - )] 3 6 3 6      z i 30 0.3 Exponential form of complex number --------------------------------------------------------------------------------------------------------------------------- cos sinie i    Euler’s Theorem z a bi  (cos sin )z r i   iz re  Algebraic form of z Trigonometric form of z Exponential form of z 31 0.3 Exponential form of complex number --------------------------------------------------------------------------------------------------------------------------- Example Find an exponential form of the number 3  z i Trigonometric: 5 5 2(cos sin ) 6 6 z i     Exponential: 5 62 i z e   32 0.3 Exponential form of complex number --------------------------------------------------------------------------------------------------------------------------- Example Represent following complex numbers in a complex plane. 2 ; iz e R   A module is unchanged. It follows that these numbers are in a circle with center at the origin and the radius is e2. 2(cos sin )z e i   33 0.3 Exponential form of complex number --------------------------------------------------------------------------------------------------------------------------- Example Represent following numbers in a complex plane. 3 ; a iz e a R  (cos3 sin3)az e i  Argument is unchanged, so these numbers are is in a half-line in the second - quarter plane. 34 0.3 The Power of complex numbers ----------------------------------------------------------------------------------------------------- Definition of Natural Power of a complex number z a bi  2 2 2( )( ) ( ) (2 )z z z a bi a bi a b ab i        3 3 3 2 2 3( ) 3 3 ( ) ( ) ...      z a bi a a bi a bi bi 0 1 1 2 2 2( ) ( ) ( ) ... ( )n n n n n n nn n n nz a bi C a C a bi C a bi C bi         nz A iB  35 0.3 The Power of complex numbers ----------------------------------------------------------------------------------------------------- Example. Let z = 2 + i. Find z5.  55 )2( iz  555 44 5 323 5 232 5 41 5 50 5 22222 iCiCiCiCiCC  iii 1.2.5).(4.10)1.(8.10.16.532 i4138 36 0.3 The Power of complex numbers -------------------------------------------------------------- The power of complex number i: ii 1 12 i iiiii  )1(23 1)1()1(224  iii iiiii  145 1)1(1246  iii iiiii  )(1347 111448  iii Powers of i If n is a positive integer, then in = ir, where r is the remainder of the division of n by 4. 37 0.3 The Power of complex numbers --------------------------------------------------------------------------------------------------------------------------- Example Find 1987z i 1987 4 496 3   1987z i 4 496 3 3i i i     38 0.3 The Power of complex numbers ----------------------------------------------------------------------------------------------------- Let z = 1 + i. a) Find z3; b) Find z100. Example 3 3) (1 )a z i  2 31 3 3i i i    1 3 3z i i    2 2z i   ) Similarly, but it's complicated so we try to do another way. b 39 [ (cos sin )] (cos sin )n nr i r n i n      The De Moivre formula Let r > 0 and n be a natural number. Then 0.3 The Power of complex numbers ----------------------------------------------------------------------------------------------------- z a bi  (cos sin )r i   2 2(cos2 sin2 )z z z r i     3 2 3(cos3 sin3 )z z z r i     1 (cos sin )n n nz z z r n i n     40 Example. Compute, using de Moivre’s formula : a) (1 + i)25 200)31( ib) 20 17 )212( )3( i i   c) Solution. a) Step 1. Write 1 + i in a trigonometric form ) 4 sin 4 (cos21  iiz  Step 2. Using the Moivre’s formula: ) 4 25 sin 4 25 (cos)2()] 4 sin 4 (cos2[ 252525  iiz  Step 3. Simplify a result ) 4 sin 4 (cos221225  iz  0.3 The Power of complex numbers ----------------------------------------------------------------------------------------------------- 41 0.4 The root of complex numbers ----------------------------------------------------------------------------------------------------- Definition of nth root of complex number A number w is an nth root of a nonzero complex number z if wn = z, where n is a positive integer. (cos sin )z a bi r i     2 2 (cos sin ) (cos sin )n nn k k k z r i z r i n n              Where k = 0, 1, 2, , n – 1. A complex number has exactly n n-th roots 42 0.4 The root of complex numbers ----------------------------------------------------------------------------------------------------- Example. Find the nth root of each case. Sketch the roots z1, z2, , zn-1on an appropriate circle centered at the origin. 3 8a) 5 1 ib) 8 16 1 i i c) 6 1 3 i i   d) 5 12ie) 1 2if) 43 0.4 The root of complex numbers ----------------------------------------------------------------------------------------------------- Solution a) Write in a trigonometric form 8 8(cos0 sin 0)i  Use the formula 3 0 2 0 28(cos0 sin 0) 2(cos sin ) 3 3 k k k i z i        0,1,2.k  44 0.4 The root of complex numbers ---------------------------------------------------------------------------------------------------- solution b) Write in a trigonometric form Use the formula 44 2 2 6 62(cos sin ) 2(cos sin ) 6 6 4 4            k k k i z i 0,1,2,3.k 3 2(cos sin ) 6 6     i i 0 z 1 z 2 z 3 z 45 0.5 The Fundamental Theorem of Algebra --------------------------------------------------------------------------------------------------------------------------- The German mathematician Carl Friedrich Gauss (1777-1855) was the first to prove that every polynomial has at least one complex zero. The Fundamental Theorem of Algebra If P(z) is a polynomial of degree with complex coefficients, then P(z) has at least one complex zero. 1n The Number of Zeros of a Polynomial If P(z) is a polynomial of degree with complex coefficients, then P(z) has exactly n complex zeros, provided each zero is counted according to its multiplicity. 1n 46 0.5 The Fundamental Theorem of Algebra --------------------------------------------------------------------------------------------------------------------------- Although the Fundamental Theorem and its corollary give information about the existence and the number of zeros of a polynomial, they do not provide a method of actually finding the zeros. The Conjugate Pair Theorem If a + bi is a complex zero of the polynomial P(z) with real coefficients, then the conjugate a – bi is also a complex zero of the polynomial. If a polynomial has real coefficients, then the following theorem can helps us determine the zeros of the polynomial. Proof. ... 47 0.5 The Fundamental Theorem of Algebra --------------------------------------------------------------------------------------------------------------------------- Example. (Use the conjugate Pair theorem to Find Zeros) Find all the zeros of given that 2 + i is a zero. 4536144)( 234  zzzzzP Solution. Because the coefficients are real numbers and 2 + i is a zero, the Conjugate Pair Theorem implies that 2 –i must also be a zero. P(z) has a factor (z – (2 + i))(z - (2 – i)) = = z2 – 4z + 5 P(z) may be written in the factored form P(z) = (z2 – 4z + 5)(z2 + 9) z2 + 9 has 3i and –3i as zeros. Therefore the four zeros of P(z) are 2 + i, 2 – i, 3i, -3i. 48 0.5 The Fundamental Theorem of Algebra --------------------------------------------------------------------------------------------------------------------------- Example. Solve following equations in C. 015  iza) 0122  izzd) 0224  zzc) 012  zzb) Solution. Solve the quadratic equation 02  cbzaz acb 42 Step 1. Compute Step 2. Find 2,1 2 4  acb Step 3. a b z a b z 2 ; 2 2 2 1 1     49 Conclusion --------------------------------------------------------------------------------------------------------------- 2. The trigonometric form )sin(cos  irz  3. The power of a complex number )sin(cos)]sin(cos[  ninrirz nnn  4. The n-th root of a complex number ) 2 sin 2 (cos)sin(cos n k i n k rzirz nkn n      .1,...,3,2,1  nk 1. The algebraic form biaz  50 Perform the following operation Exercise 1 )2( )32( 5 2 ii i z    51 Write the following number in the trigonometric form. Exercise 2 )3)(1( iiz  52 Write the given number in the standard form Exercise 3 5)32( iz  53 Construct the domains of point z, given the condition Exercise 4 1|21|  iz 54 Construct the domains of point z, given the condition Exercise 5 4 ||    Argz 55 Compute, using de Moivre’s formula Exercise 6 20 17 )3( )1( i i   56 Exercise 7 Express and in terms of functions of the angle . 5cos 5sin  Express and in terms of functions of the angle . ncos nsin  57 Exercise 8 Find , where 31 1 i i z   6 z 58 Exercise 9 Find , where 16z4 z 59 Exercise 10 Find 3 22 i 60 Exercise 11 Find i41 61 Exercise 12 Solve the equation 07  iz 62 Exercise 13 Solve the equation 012  izz 63 Exercise 14 Show that the complex number 2 + 3i is a root of the equation 05216174 234  zzzz and hence find the other three roots. 64 Exercise 15 Solve the equation 02)22()2( 23  iziziz given the equation has a pure imaginary root. 65 Exercise 16 Factor x3 + 8 as a product of linear factors. 66 Exercise 17 Find 100 100 98 100 4 100 2 100 0 100 CCCCCA  67 Exercise 18 Find  nA cos3cos2coscos  68 Exercise 19 Find 2           cos cos( ) cos( ) cos( )A b b b b n 69 Write in the standard form Exercise 20 3(1 2 ) 2 3 i z i    Find all the numbers z such that Exercise 21 | 1 2 | | 2 |z i z i     70 Let z be a complex number such that |z| = 2. Show that Exercise 22 6 8 13z i   Let z be a complex number such that |z| = 1. Show that Exercise 23 21 | 3| 4z   71 Find all numbers z such that Exercise 23 2z z i   Find all numbers z such that Exercise 24 2 1 12 6z i z   72 Let z be a nonzero complex number. Find the module of following number Exercise 25 2008z i z  Determine all complex numbers z such that Exercise 26 z k z i   where k is a positive real number. 73 Find all complex numbers z such that Exercise 28 4 1 z i z i       Find all numbers z satisfied following two conditions Exercise 27 1 1 z z i    and 3 1 z i z i    74 Find the real and imaginary part of the number Exercise 29 3 2 1 i i z i i      Solve the equation . Exercise 30 2 | | 0z z  75 Write following numbers in trigonometric form Exercise 31 2) sin 2sin 2 a i    ) cos (1 sin ) b i   Find the square root of the number z = - 8 + 6i. Exercise 32 76 Show that, if is an pure imaginary number, then |z| = 1. Exercise 34 1 1 z z   Find the real part of the number Exercise 33 1 1 z z   where |z| = 1 and 1.z  77