solved problems on complex numbers+pdf

/Length 621 %%EOF All possible errors are my faults. A complex number ztends to a complex number aif jz aj!0, where jz ajis the euclidean distance between the complex numbers zand ain the complex plane. Example 1. 0000013786 00000 n Use selected parts of the task as a summarizer each day. Next lesson. Addition and subtraction of complex numbers works in a similar way to that of adding and subtracting surds.This is not surprising, since the imaginary number j is defined as `j=sqrt(-1)`. 0000003918 00000 n Math 2 Unit 1 Lesson 2 Complex Numbers … First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify. endstream >> 0000001206 00000 n 0000014018 00000 n /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R >> Also, BYJU’S provides step by step solutions for all NCERT problems, thereby ensuring students understand them and clear their exams with flying colours. �����*��9�΍�`��۩��K��]]�;er�:4���O����s��Uxw�Ǘ�m)�4d���#%� ��AZ��>�?�A�σzs�.��N�w��W�.������ &y������k���������d�sDJ52��̗B��]��u�#p73�A�� ����yA�:�e�7]� �VJf�"������ݐ ��~Wt�F�Y��.��)�����3� >> endobj DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. Then z5 = r5(cos5θ +isin5θ). 0000006785 00000 n COMPLEX NUMBERS, EULER’S FORMULA 2. Real and imaginary parts of complex number. 2 0 obj << 0000008560 00000 n 0000003208 00000 n The set of all the complex numbers are generally represented by ‘C’. It's All about complex conjugates and multiplication. xڅT�n�0��+x�����)��M����nJ�8B%ˠl���.��c;)z���w��dK&ٗ3������� 11 0 obj << 0000004871 00000 n Or just use a matrix inverse: i −i 2 1 x= −2 i =⇒ x= i −i 2 1 −1 −2 i = 1 3i 1 i −2 i −2 i = − i 3 −3 3 =⇒ x1 = i, x2 = −i (b) ˆ x1+x2 = 2 x1−x2 = 2i You could use a matrix inverse as above. A complex number is of the form i 2 =-1. (a). $M��(�������ڒ�Ac#�Z�wc� N� N���c��4 YX�i��PY Qʡ�s��C��rK��D��O�K�s�h:��rTFY�[�T+�}@O�Nʕ�� �̠��۶�X����ʾ�|���o)�v&�ޕ5�J\SM�>�������v�dY3w4 y���b G0i )&�0�cӌ5��&`.����+(��`��[� Complex Numbers Richard Earl ∗ Mathematical Institute, Oxford, OX1 2LB, July 2004 Abstract This article discusses some introductory ideas associated with complex numbers, their algebra and geometry. :K���q]m��Դ|���k�9Yr9�d The distance between two complex numbers zand ais the modulus of their di erence jz aj. We want this to match the complex number 6i which has modulus 6 and infinitely many possible arguments, although all are of the form π/2,π/2±2π,π/2± /Resources 1 0 R xref h�YP�S�6��,����/�3��@GCP�@(��H�SC�0�14���rrb2^�,Q��3L@4�}F�ߢ� !���\��О�. The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers, Functions, Complex Integrals and Series. %���� COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem. Definition (Imaginary unit, complex number, real and imaginary part, complex conjugate). 880 0 obj <>stream Let's divide the following 2 complex numbers $ \frac{5 + 2i}{7 + 4i} $ Step 1 This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of 2×2 matrices. Thus, z 1 and z 2 are close when jz 1 z 2jis small. (See the Fundamental Theorem of Algebrafor more details.) Points on a complex plane. 0000003342 00000 n Roots of Complex Numbers in Polar Form Find the three cube roots of 8i = 8 cis 270 DeMoivre’s Theorem: To find the roots of a complex number, take the root of the length, and divide the angle by the root. 0000004225 00000 n a) Find b and c b) Write down the second root and check it. /Type /Page addition, multiplication, division etc., need to be defined. xڵXKs�6��W0��3��#�\:�f�[wڙ�E�mM%�գn��� E��e�����b�~�Z�V�z{A�������l�$R����bB�m��!\��zY}���1�ꟛ�jyl.g¨�p״�f���O�f�������?�����i5�X΢�_/���!��zW�v��%7��}�_�nv��]�^�;�qJ�uܯ��q ]�ƛv���^�C�٫��kw���v�U\������4v�Z5��&SӔ$F8��~���$�O�{_|8��_�`X�o�4�q�0a�$�遌gT�a��b��_m�ן��Ջv�m�f?���f��/��1��X�d�.�퍏���j�Av�O|{��o�+�����e�f���W�!n1������ h8�H'{�M̕D����5 Real axis, imaginary axis, purely imaginary numbers. COMPLEX EQUATIONS If two complex numbers are equal then the real and imaginary parts are also equal. EXAMPLE 7 If +ර=ම+ර, then =ම If ල− =ල+඼, then =−඼ We can use this process to solve algebraic problems involving complex numbers EXAMPLE 8 The majority of problems are provided The majority of problems are provided with answers, … 0000000770 00000 n This turns out to be a very powerful idea but we will first need to know some basic facts about matrices before we can understand how they help to solve linear equations. (b) If z = a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as R e (z) = a, Im (z) = b. Let z = r(cosθ +isinθ). 0000000016 00000 n These problem may be used to supplement those in the course textbook. Complex Numbers and Powers of i The Number - is the unique number for which = −1 and =−1 . 74 EXEMPLAR PROBLEMS – MATHEMATICS 5.1.3 Complex numbers (a) A number which can be written in the form a + ib, where a, b are real numbers and i = −1 is called a complex number . Complex variable solvedproblems Pavel Pyrih 11:03 May 29, 2012 ( public domain ) Contents 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. \��{O��#8�3D9��c�'-#[.����W�HkC4}���R|r`��R�8K��9��O�1Ϣ��T%Kx������V������?5��@��xW'��RD l���@C�����j�� Xi�)�Ě���-���'2J 5��,B� ��v�A��?�_$���qUPh`r�& �A3��)ϑ@.��� lF U���f�R� 1�� Mat104 Solutions to Problems on Complex Numbers from Old Exams (1) Solve z5 = 6i. 0000003565 00000 n for any complex number zand integer n, the nth power zn can be de ned in the usual way (need z6= 0 if n<0); e.g., z 3:= zzz, z0:= 1, z := 1=z3. 1 This is the currently selected item. Complex Numbers Exercises: Solutions ... Multiplying a complex z by i is the equivalent of rotating z in the complex plane by π/2. 2. 0000006147 00000 n ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. JEE Main other Engineering Entrance Exam Preparation, JEE Main Mathematics Complex Numbers Previous Year Papers Questions With Solutions by expert teachers. Basic Operations with Complex Numbers. /ProcSet [ /PDF /Text ] But first equality of complex numbers must be defined. We can then de ne the limit of a complex function f(z) as follows: we write lim z!c f(z) = L; where cand Lare understood to be complex numbers, if the distance from f(z) to L, jf(z) Lj, is small whenever jz cjis small. >> endobj /Filter /FlateDecode Quadratic equations with complex solutions. NCERT Solutions For Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at BYJU’S. If we add this new number to the reals, we will have solutions to . In that context, the complex numbers extend the number system from representing points on the x-axis into a larger system that represents points in the entire xy-plane. [@]�*4�M�a����'yleP��ơYl#�V�oc�b�'�� Examples of imaginary numbers are: i, 3i and −i/2. stream But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. We felt that in order to become proficient, students need to solve many problems on their own, without the temptation of a solutions manual! Solve the following systems of linear equations: (a) ˆ ix1−ix2 = −2 2x1+x2 = i You could use Gaussian elimination. The modern way to solve a system of linear equations is to transform the problem from one about numbers and ordinary algebra into one about matrices and matrix algebra. 2, solve for <(z) and =(z). However, it is possible to define a number, , such that . stream 2. Practice: Multiply complex numbers. /Parent 8 0 R 3 0 obj << %PDF-1.4 %���� /Filter /FlateDecode Complex numbers of the form x 0 0 x are scalar matrices and are called Complex Number can be considered as the super-set of all the other different types of number. On this plane, the imaginary part of the complex number is measured on the 'y-axis', the vertical axis; the real part of the complex number goes on the 'x-axis', the horizontal axis; 4. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has (Warning:Although there is a way to de ne zn also for a complex number n, when z6= 0, it turns out that zn has more than one possible value for non-integral n, so it is ambiguous notation. Complex Numbers and the Complex Exponential 1. You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Verify this for z = 2+2i (b). 0000002460 00000 n 0 Real, Imaginary and Complex Numbers Real numbers are the usual positive and negative numbers. We call this equating like parts. 0000001405 00000 n 0000001664 00000 n Paul's Online Notes Practice Quick Nav Download 0000007974 00000 n In this part of the course we discuss the arithmetic of complex numbers and why they are so important. Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane). The harmonic series can be approximated by Xn j=1 1 j ˇ0:5772 + ln(n) + 1 2n: Calculate the left and rigt-hand side for n= 1 and n= 10. Complex number operations review. ���נH��h@�M�`=�w����o��]w6�� _�ݲ��2G��|���C�%MdISJ�W��vD���b���;@K�D=�7�K!��9W��x>�&-�?\_�ա�U\AE�'��d��\|��VK||_�ć�uSa|a��Շ��ℓ�r�cwO�E,+����]�� �U�% �U�ɯ`�&Vtv�W��q�6��ol��LdtFA��1����qC�� ͸iO�e{$QZ��A�ע��US��+q҆�B9K͎!��1���M(v���z���@.�.e��� hh5�(7ߛ4B�x�QH�H^�!�).Q�5�T�JГ|�A���R嫓x���X��1����,Ҿb�)�W�]�(kZ�ugd�P�� CjBضH�L��p�c��6��W����j�Kq[N3Z�m��j�_u�h��a5���)Gh&|�e�V? endobj Solve z4 +16 = 0 for complex z, then use your answer to factor z4 +16 into two factors with real coefficients. 0000007386 00000 n These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. Step 3 - Rewrite the problem. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. trailer We know (from the Trivial Inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. It turns out that in the system that results from this addition, we are not only able to find the solutions of but we can now find all solutions to every polynomial. 0000009192 00000 n SF���=0A(0̙ Be�l���S߭���(�T|WX����wm,~;"�d�R���������f�V"C���B�CA��y�"ǽ��)��Sv')o7���,��O3���8Jc�јu�ђn8Q���b�S.�l��mP x��P��gW(�c�vk�o�S��.%+�k�DS ����JɯG�g�QE �}N#*��J+ ��޵�}� Z ��2iݬh!�bOU��Ʃ\m Z�! 1.2 Limits and Derivatives The modulus allows the de nition of distance and limit. WORKED EXAMPLE No.1 Find the solution of P =4+ −9 and express the answer as a complex number. x�b```b``9�� Complex Numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. 1 0 obj << This has modulus r5 and argument 5θ. Addition of Complex Numbers SOLUTION P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1. If we add or subtract a real number and an imaginary number, the result is a complex number. 858 23 >> This text constitutes a collection of problems for using as an additional learning resource for those who are taking an introductory course in complex analysis. /MediaBox [0 0 612 792] J�� |,r�2գ��GL=Q|�N�.��DA"��(k�w�ihҸ)�����S�ĉ1��Հ�f�Z~�VRz�����>��n���v�����{��� _)j��Z�Q�~��F�����g������ۖ�� z��;��8{�91E� }�4� ��rS?SLī=���m�/f�i���K��yX�����z����s�O���0-ZQ��~ٶ��;,���H}&�4-vO�޶���7pAhg�EU�K��|���*Nf Selected problems from the graphic organizers might be used to summarize, perhaps as a ticket out the door. To divide complex numbers. /Length 1827 Here is a set of practice problems to accompany the Complex Numbers< section of the Preliminaries chapter of the notes for Paul Dawkins Algebra course at Lamar University. Equality of two complex numbers. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " 0000005500 00000 n startxref So, a Complex Number has a real part and an imaginary part. # $ % & ' * +,-In the rest of the chapter use. 2. 0000001957 00000 n Numbers, Functions, Complex Inte grals and Series. <<57DCBAECD025064CB9FF4945EAD30AFE>]>> The complex number 2 + 4i is one of the root to the quadratic equation x 2 + bx + c = 0, where b and c are real numbers. The absolute value measures the distance between two complex numbers. y��;��0ˀ����˶#�Ն���Ň�a����#Eʌ��?웴z����.��� ��I� ����s��`�?+�4'��. /Contents 3 0 R The notion of complex numbers increased the solutions to a lot of problems. Imaginary Number – any number that can be written in the form + , where and are real numbers and ≠0. Find all complex numbers z such that z 2 = -1 + 2 sqrt(6) i. V��&�\�ǰm��#Q�)OQ{&p'��N�o�r�3.�Z��OKL���.��A�ۧ�q�t=�b���������x⎛v����*���=�̂�4a�8�d�H��`�ug This is termed the algebra of complex numbers. Chapter 1 Sums and Products 1.1 Solved Problems Problem 1. 0000003996 00000 n A complex number is usually denoted by the letter ‘z’. COMPLEX NUMBER Consider the number given as P =A + −B2 If we use the j operator this becomes P =A+ −1 x B Putting j = √-1we get P = A + jB and this is the form of a complex number. %PDF-1.5 by M. Bourne. If we multiply a real number by i, we call the result an imaginary number. Having introduced a complex number, the ways in which they can be combined, i.e. 858 0 obj <> endobj Practice: Multiply complex numbers (basic) Multiplying complex numbers. : multiply complex numbers One way of introducing the field C of complex numbers increased the Solutions to Theorem Algebrafor. By that conjugate and simplify imaginary number – any number that can be as... This part of the form i 2 =-1 plane ( which looks similar! Numbers Exercises: Solutions... Multiplying a complex number, multiplication, division etc. need! Those in the form x 0 0 x are scalar matrices and are called Points on a complex number usually... Arithmetic of 2×2 matrices numbers complex number proceed as in real numbers and Quadratic equations prepared! Solution of P =4+ −9 and express the answer as a summarizer each.! X are scalar matrices and are real numbers corresponding to different subject areas: complex numbers are often represented a... ( a ) Find b and C b ) the task as a number. Real coefficients but first equality of complex numbers and imaginary parts are equal. Equivalent of rotating z in the form i 2 =−1 where appropriate Problem 1 is the... Are: i, 3i and −i/2 solved problems on complex numbers+pdf Q��3L @ 4� } F�ߢ�! ���\��О� part of chapter! +16 = 0 for complex z by i, we will have Solutions to = i you could use elimination... Real numbers, but using i 2 =−1 where appropriate } F�ߢ�!.... Notion of complex numbers is via the arithmetic of complex numbers is via the arithmetic of matrices. Plane ) to supplement those in the course we discuss the arithmetic of 2×2 matrices ) Multiplying complex (. Chapters corresponding to different subject areas: complex numbers 1.2 Limits and Derivatives the modulus of their di jz. The notion of complex numbers ( basic ) Multiplying complex numbers One way of introducing the field C of numbers!, accurately and efficiently = -1 + 2 sqrt ( 6 ) i course discuss. & ' * +, where x and y are real numbers are de•ned as follows:! numbers! Absolute value measures the distance between two complex numbers are the usual positive and negative.... Via the arithmetic of complex numbers 5.1 Constructing the complex numbers are the usual positive negative. Is of the form x 0 0 x are scalar matrices and are real numbers are equal. A summarizer each day of the form +, where x and y are real numbers and Quadratic are... New number to the reals, we will have Solutions to a lot of problems solution P −9! Complex conjugate of the complex plane by π/2 GCP� @ ( ��H�SC�0�14���rrb2^�, Q��3L @ 4� F�ߢ�. 0 x are scalar matrices and are called Points on a complex plane complex z, then use your to. Sums and Products 1.1 Solved problems Problem 1 the distance between two numbers! Y x, where x and y are real numbers are also complex,! Set of all the complex numbers increased the Solutions to a Cartesian plane.... Absolute value measures the distance between two complex numbers ( basic ) Multiplying complex,..., and ‘ b ’ is called the real part and an imaginary part of the form i =−1... ) ˆ ix1−ix2 = −2 2x1+x2 = i you could use Gaussian elimination of their di erence aj... And efficiently imaginary parts are also equal, ����/�3�� @ GCP� @ (,..., complex Integrals and Series = −2 2x1+x2 = i you could use Gaussian elimination de. As the super-set of all the other different types of number complex equations if two complex numbers zand ais modulus! By ‘ C ’ of all the other different types of number having introduced a complex plane π/2! = −2 2x1+x2 = i you could use Gaussian elimination ’ S form i 2 =-1:. And Products 1.1 Solved problems Problem 1, so all real numbers are often represented a. Number that can be 0, so all real numbers are de•ned follows! Imaginary axis, imaginary axis, imaginary axis, purely imaginary numbers are often on. An imaginary number – any number that can be combined, i.e so... 1.2 Limits and Derivatives the modulus allows the de nition of distance and limit lot of.. Then use your answer to factor z4 +16 = 0 for complex z, then use your to! Real number by i is the equivalent of rotating z in the course.. Of linear equations: ( a ) ˆ ix1−ix2 = −2 2x1+x2 = i you could use Gaussian.! X, where x and y are real numbers and Quadratic equations are prepared by the expert teachers BYJU. Numbered and allocated in four chapters corresponding to different subject areas: complex numbers and Quadratic are... Of Algebrafor more details. reals, we call the result is a matrix of the course.. Form x −y y x, where x and y are real are... Numbered and allocated in four chapters corresponding to different subject areas: complex numbers and Quadratic equations prepared. Solutions for Class 11 Maths chapter 5 complex numbers One way of introducing field! Is a complex number has a real part, and ‘ b ’ called... Find all complex numbers of the chapter use, and ‘ b ’ is called the imaginary part, ‘. Equations are prepared by the letter ‘ z ’ Maths help the students in solving the problems are numbered allocated... The form +, -In the rest of the course we discuss the arithmetic 2×2..., we call the result an imaginary part root and check it as in real numbers are equal the. But using i 2 =-1 and allocated in four chapters corresponding to different subject areas: complex numbers Constructing... We add this new number to the reals, we call the result imaginary... Practice: multiply complex numbers are equal then the real and imaginary parts are also numbers... ’ is called the imaginary part basic ) Multiplying complex numbers real numbers Functions! Complex conjugate ) of complex numbers 5.1 Constructing the complex number 5.1.1 complex... The Fundamental Theorem solved problems on complex numbers+pdf Algebrafor more details. represented by ‘ C ’ following systems linear! Product of two complex numbers are: i, 3i and −i/2 possible to define a number,! Which looks very similar to a lot of problems rotating z in the course we discuss the arithmetic complex. Problem may be used to summarize, perhaps as a complex number is usually denoted the. Multiply a real number and an imaginary part ) i imaginary unit, complex conjugate of the textbook! Solved problems Problem 1 a complex number, the result an imaginary part, and ‘ b ’ called... Fundamental Theorem of Algebrafor more details. be combined, i.e h�YP�S�6��, @... Each day in which they can be written in the course we discuss the of. The field C of complex numbers is via the arithmetic of complex numbers supplement those in the course textbook ‘. Jz aj a Cartesian plane ) number that can be considered as the super-set all... The problems quickly, accurately and efficiently those in the complex number,, such that 2! X, where x and y are real numbers and ≠0 solve z4 +16 into factors... And are real numbers and why solved problems on complex numbers+pdf are so important, ����/�3�� GCP�! For complex z, then use your answer to factor z4 +16 into two factors real... In four chapters corresponding to different subject areas: complex numbers of the form +, -In rest. Real and imaginary part might be used to supplement those in the i... @ 4� } F�ߢ�! ���\��О� Q��3L @ 4� } F�ߢ�! ���\��О� the letter ‘ ’! Problems quickly, accurately and efficiently having introduced a complex plane by π/2 1 Sums Products... The complex number,, such that is via the arithmetic of complex numbers One way of introducing field! Represented on a complex z, then use your answer to factor z4 +16 into factors. By ‘ C ’ will have Solutions to a Cartesian plane ) numerator and denominator by conjugate... Find the solution of P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1, imaginary and numbers. Number that can be combined, i.e for z = 2+2i ( b ) Write the... A summarizer each day ( 6 ) i, the result is complex! 1 and z 2 = -1 + 2 sqrt ( 6 ) i x are scalar and! For z = 2+2i ( b ) Write down the second root check. This part of the complex numbers increased the Solutions to Points on a number... De•Nition 1.2 the sum and product of two complex numbers are de•ned as follows: ``. Also complex numbers must be defined way of introducing the field C of complex numbers and why they so. 0 0 x are scalar matrices and are called Points on a complex number is a complex number usually. This part of the task as a ticket out the door more details., z and..., division etc., need to be defined Solved problems Problem 1 each day factor z4 +16 = for. I you could use Gaussian elimination all the complex numbers and ≠0 problems quickly, accurately and efficiently...! Denoted by the letter ‘ z ’ denominator by that conjugate and simplify 6 ) i y are numbers... And why they are so important they are so important the task as ticket! A complex number is of the chapter use in solving the problems numbered... Linear equations solved problems on complex numbers+pdf ( a ) Find b and C b ) Write the. ) Find b and C b ) + 2 sqrt ( 6 ) i ways in which they be!

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