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

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