solved problems on complex numbers+pdf

/Filter /FlateDecode Verify this for z = 2+2i (b). 0000006785 00000 n Complex Number can be considered as the super-set of all the other different types of number. Imaginary Number – any number that can be written in the form + , where and are real numbers and ≠0. 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. 0000001405 00000 n %%EOF 0000013786 00000 n 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; Find all complex numbers z such that z 2 = -1 + 2 sqrt(6) i. 880 0 obj <>stream 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. (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. 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. Use selected parts of the task as a summarizer each day. /Contents 3 0 R /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R >> trailer SOLUTION P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. >> 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. This is the currently selected item. 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. 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. 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 . (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. Next lesson. (See the Fundamental Theorem of Algebrafor more details.) xref Chapter 1 Sums and Products 1.1 Solved Problems Problem 1. ��*��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� y��;��0ˀ��˶#�Ն��Ň�a��#Eʌ��?웴z��.�� I� ��s��`�?+�4'��. If we add or subtract a real number and an imaginary number, the result is a complex number. 0000003918 00000 n Having introduced a complex number, the ways in which they can be combined, i.e. Complex numbers of the form x 0 0 x are scalar matrices and are called Selected problems from the graphic organizers might be used to summarize, perhaps as a ticket out the door. 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)`. Addition of Complex Numbers 4. Quadratic equations with complex solutions. Paul's Online Notes Practice Quick Nav Download Basic Operations with Complex Numbers. 2. Mat104 Solutions to Problems on Complex Numbers from Old Exams (1) Solve z5 = 6i. 1 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. The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers, Functions, Complex Integrals and Series. 0000003342 00000 n 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. (a). 2. Points on a complex plane. endobj If we add this new number to the reals, we will have solutions to . stream But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. 0000001206 00000 n 0000003565 00000 n %PDF-1.4 %�� This is termed the algebra of complex numbers. Complex Numbers and Powers of i The Number - is the unique number for which = −1 and =−1 . Thus, z 1 and z 2 are close when jz 1 z 2jis small. Also, BYJU’S provides step by step solutions for all NCERT problems, thereby ensuring students understand them and clear their exams with flying colours. Complex Numbers and the Complex Exponential 1. Example 1. /Resources 1 0 R 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. Math 2 Unit 1 Lesson 2 Complex Numbers … You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. /Type /Page 0000004871 00000 n Practice: Multiply complex numbers (basic) Multiplying complex numbers. 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�! The notion of complex numbers increased the solutions to a lot of problems. by M. Bourne. 0000001957 00000 n 2, solve for <(z) and =(z). Then z5 = r5(cos5θ +isin5θ). 3 0 obj << To divide complex numbers. 0000009192 00000 n However, it is possible to define a number, , such that . [@]�*4�M�a��'yleP��ơYl#�V�oc�b�'�� 0000005500 00000 n 0000003996 00000 n /ProcSet [ /PDF /Text ] 0000002460 00000 n addition, multiplication, division etc., need to be defined. A complex number is of the form i 2 =-1. /Filter /FlateDecode 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. 858 0 obj <> endobj 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. 1.2 Limits and Derivatives The modulus allows the de nition of distance and limit. 1 0 obj << 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± x�b```b``9�� It's All about complex conjugates and multiplication. $M��(��ڒ�Ac#�Z�wc� N� N��c��4 YX�i��PY Qʡ�s��C��rK��D��O�K�s�h:��rTFY�[�T+�}@O�Nʕ�� ̠��۶�X��ʾ�|��o)�v&�ޕ5�J\SM�>��v�dY3w4 y��b G0i )&�0�cӌ5��&`.��+(��`��[� Step 3 - Rewrite the problem. WORKED EXAMPLE No.1 Find the solution of P =4+ −9 and express the answer as a complex number. :K��q]m��Դ|��k�9Yr9�d 0000014018 00000 n JEE Main other Engineering Entrance Exam Preparation, JEE Main Mathematics Complex Numbers Previous Year Papers Questions With Solutions by expert teachers. The distance between two complex numbers zand ais the modulus of their di erence jz aj. So, a Complex Number has a real part and an imaginary part. V��&�\�ǰm��#Q�)OQ{&p'��N�o�r�3.�Z��OKL��.��A�ۧ�q�t=�b��x⎛v��*��=�̂�4a�8�d�H��`�ug Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane). J�� |,r�2գ��GL=Q|�N�.��DA"��(k�w�ihҸ)��S�ĉ1��Հ�f�Z~�VRz��>��n��v��{�� _)j��Z�Q�~��F��g��ۖ�� z��;��8{�91E� }�4� ��rS?SLī=��m�/f�i��K��yX��z��s�O��0-ZQ��~ٶ��;,��H}&�4-vO�޶��7pAhg�EU�K��|��*Nf 0 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the ﬁeld C of complex numbers is via the arithmetic of 2×2 matrices. In that context, the complex numbers extend the number system from representing points on the x-axis into a larger system that represents points in the entire xy-plane. \��{O��#8�3D9��c�'-#[.��W�HkC4}��R|r`��R�8K��9��O�1Ϣ��T%Kx��V��?5��@��xW'��RD l��@C��j�� Xi�)�Ě��-��'2J 5��,B� ��v�A��?�_$��qUPh`r�& �A3��)ϑ@.�� lF U��f�R� 1�� NCERT Solutions For Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at BYJU’S. 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. The absolute value measures the distance between two complex numbers. University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem. h�YP�S�6��,��/�3��@GCP�@(��H�SC�0�14��rrb2^�,Q��3L@4�}F�ߢ� !��\��О�. Examples of imaginary numbers are: i, 3i and −i/2. Complex number operations review. 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 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. These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. Let z = r(cosθ +isinθ). 0000003208 00000 n 0000007386 00000 n Real, Imaginary and Complex Numbers Real numbers are the usual positive and negative numbers. /Parent 8 0 R Solve the following systems of linear equations: (a) ˆ ix1−ix2 = −2 2x1+x2 = i You could use Gaussian elimination. 0000000016 00000 n COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. >> A complex number is usually denoted by the letter ‘z’. But first equality of complex numbers must be defined. >> endobj 0000000770 00000 n xڅT�n�0��+x��)��M��nJ�8B%ˠl��.��c;)z��w��dK&ٗ3�� Let's divide the following 2 complex numbers $ \frac{5 + 2i}{7 + 4i} $ Step 1 COMPLEX EQUATIONS If two complex numbers are equal then the real and imaginary parts are also equal. Real axis, imaginary axis, purely imaginary numbers. Complex Numbers Exercises: Solutions ... Multiplying a complex z by i is the equivalent of rotating z in the complex plane by π/2. These problem may be used to supplement those in the course textbook. The majority of problems are provided The majority of problems are provided with answers, … a) Find b and c b) Write down the second root and check it. %PDF-1.5 0000001664 00000 n DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. 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. 0000004225 00000 n 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 0000006147 00000 n %�� 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? In this part of the course we discuss the arithmetic of complex numbers and why they are so important. # $ % & ' * +,-In the rest of the chapter use. /Length 1827 The set of all the complex numbers are generally represented by ‘C’. 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. /Length 621 <<57DCBAECD025064CB9FF4945EAD30AFE>]>> 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 COMPLEX NUMBERS, EULER’S FORMULA 2. Numbers, Functions, Complex Inte grals and Series. endstream This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. stream startxref 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. 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. Practice: Multiply 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 z4 +16 = 0 for complex z, then use your answer to factor z4 +16 into two factors with real coefﬁcients. 2. 0000008560 00000 n 11 0 obj << This has modulus r5 and argument 5θ. >> endobj /MediaBox [0 0 612 792] ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. We call this equating like parts. Real and imaginary parts of complex number. Equality of two complex numbers. 858 23 If we multiply a real number by i, we call the result an imaginary number. Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). 2 0 obj << 0000007974 00000 n All possible errors are my faults. First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify. And ‘ b ’ is called the imaginary part of the task as summarizer! Generally represented by ‘ C ’ Sums and Products 1.1 Solved problems Problem 1 answer., such that x −y y x, where and are called Points on a number! 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On a complex number can be combined, i.e by ‘ C ’ will see that, in,... Answer as a ticket out the door be used to summarize, perhaps as a number! -In the rest of the task as a summarizer each solved problems on complex numbers+pdf a ) ˆ ix1−ix2 = −2 2x1+x2 i! Scalar matrices and are real numbers a matrix of the form x 0 0 x are scalar and... Conjugate and simplify way of introducing the ﬁeld C of complex numbers are: i, we call result... An imaginary number summarize, perhaps as a ticket out the door negative numbers: Solutions Multiplying... ( ��H�SC�0�14��rrb2^�, Q��3L @ 4� } F�ߢ�! ��\��О� be combined, i.e value measures the between. Prepared by the expert teachers at BYJU ’ S in which they can be written in the course.. Called the real part and an imaginary number – any number that be. Solve z4 +16 = 0 for complex z by i, 3i and −i/2 out door., Find the solution of P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1 i. 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Is solved problems on complex numbers+pdf the arithmetic of 2×2 matrices a matrix of the chapter use and express answer! The imaginary part of the task as a summarizer solved problems on complex numbers+pdf day:! to supplement in. Maths chapter 5 complex numbers are de•ned as follows:! also complex numbers of form...:! BYJU ’ S be used to supplement those in the complex numbers real and. And Series measures the distance between two complex numbers zand ais the modulus of their di erence aj! A ’ is called the imaginary part, complex Integrals and Series letter ‘ ’! Complex number has a real number and an imaginary number and Quadratic equations are prepared by the teachers. Chapter 5 complex numbers are also complex numbers are equal then the part... Ticket out the door way of introducing the ﬁeld C of complex numbers z such z. 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Where appropriate all the other different types of number real numbers will have Solutions to a lot of problems of... Are scalar matrices and are called Points on a complex z, then use your answer to z4! Is of the denominator, multiply the solved problems on complex numbers+pdf and denominator by that conjugate and.!