COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. To divide complex numbers. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. (a). 880 0 obj
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Example 1. COMPLEX EQUATIONS If two complex numbers are equal then the real and imaginary parts are also equal. /Length 621 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. Quadratic equations with complex solutions. <<57DCBAECD025064CB9FF4945EAD30AFE>]>>
The majority of problems are provided The majority of problems are provided with answers, … 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; Addition of Complex Numbers 0000004225 00000 n
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/Parent 8 0 R The distance between two complex numbers zand ais the modulus of their di erence jz aj. 0000006147 00000 n
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endstream University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem. 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. 1 So, a Complex Number has a real part and an imaginary part. Verify this for z = 2+2i (b). 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. Thus, z 1 and z 2 are close when jz 1 z 2jis small. addition, multiplication, division etc., need to be defined. 2. # $ % & ' * +,-In the rest of the chapter use. These problem may be used to supplement those in the course textbook. Having introduced a complex number, the ways in which they can be combined, i.e. endobj The set of all the complex numbers are generally represented by ‘C’. x�b```b``9�� Step 3 - Rewrite the problem. xڵXKs�6��W0��3��#�\:�f�[wڙ�E�mM%�գn��� E��e�����b�~�Z�V�z{A�������l�$R����bB�m��!\��zY}���1��jyl.g¨�p״�f���O�f�������?�����i5�X�_/���!��zW�v��%7��}�_�nv��]�^�;�qJ�uܯ��q ]�ƛv���^�C�٫��kw���v�U\������4v�Z5��&SӔ$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 0000003996 00000 n
This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R >> This is the currently selected item. 1.2 Limits and Derivatives The modulus allows the de nition of distance and limit. 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. 858 0 obj
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A complex number is usually denoted by the letter ‘z’. (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. The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers, Functions, Complex Integrals and Series. %���� 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. However, it is possible to define a number, , such that . The absolute value measures the distance between two complex numbers. This is termed the algebra of complex numbers. The notion of complex numbers increased the solutions to a lot of problems. 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± 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. /Length 1827 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. 2 0 obj << xڅT�n�0��+x�����)��M����nJ�8B%ˠl���.��c;)z���w��dK&ٗ3������� We call this equating like parts. Practice: Multiply complex numbers. Solve z4 +16 = 0 for complex z, then use your answer to factor z4 +16 into two factors with real coefﬁcients. Complex Numbers and Powers of i The Number - is the unique number for which = −1 and =−1 . 0000009192 00000 n
/Filter /FlateDecode 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. 0000003565 00000 n
2, solve for <(z) and =(z). Points on a complex plane. [@]�*4�M�a����'yleP��ơYl#�V�oc�b�'�� This has modulus r5 and argument 5θ. 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. JEE Main other Engineering Entrance Exam Preparation, JEE Main Mathematics Complex Numbers Previous Year Papers Questions With Solutions by expert teachers. stream Complex number operations review. 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 . 11 0 obj << But first equality of complex numbers must be defined. Selected problems from the graphic organizers might be used to summarize, perhaps as a ticket out the door. Next lesson. by M. Bourne. 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 J��
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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. Complex numbers of the form x 0 0 x are scalar matrices and are called 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)`. V��&�\�ǰm��#Q�)OQ{&p'��N�o�r�3.�Z��OKL���.��A�ۧ�q�t=�b���������x⎛v����*���=�̂�4a�8�d�H��`�ug �����*��9��`��۩��K��]]�;er�:4���O����s��Uxw�Ǘ�m)�4d���#%�
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Real, Imaginary and Complex Numbers Real numbers are the usual positive and negative numbers. NCERT Solutions For Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at BYJU’S. Math 2 Unit 1 Lesson 2 Complex Numbers … All possible errors are my faults. Practice: Multiply complex numbers (basic) Multiplying complex numbers. 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. A complex number is of the form i 2 =-1. 0000000770 00000 n
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. Chapter 1 Sums and Products 1.1 Solved Problems Problem 1. %%EOF
4. 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. /ProcSet [ /PDF /Text ] Numbers, Functions, Complex Inte grals and Series. If we add or subtract a real number and an imaginary number, the result is a complex number. 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. It's All about complex conjugates and multiplication. Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane). Equality of two complex numbers. Find all complex numbers z such that z 2 = -1 + 2 sqrt(6) i. 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. 0000002460 00000 n
De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " 0000001405 00000 n
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Imaginary Number – any number that can be written in the form + , where and are real numbers and ≠0. stream In this part of the course we discuss the arithmetic of complex numbers and why they are so important. ���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? 1 0 obj << Examples of imaginary numbers are: i, 3i and −i/2. >> >> 2. h�YP�S�6��,����/�3��@GCP�@(��H�SC�0�14���rrb2^�,Q��3L@4�}F�ߢ� !���\��О�. SOLUTION P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1. >> endobj Then z5 = r5(cos5θ +isin5θ). First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify. startxref
This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. 0000000016 00000 n
Also, BYJU’S provides step by step solutions for all NCERT problems, thereby ensuring students understand them and clear their exams with flying colours. If we add this new number to the reals, we will have solutions to . Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). 0000004871 00000 n
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. /Resources 1 0 R You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Solve the following systems of linear equations: (a) ˆ ix1−ix2 = −2 2x1+x2 = i You could use Gaussian elimination. Let's divide the following 2 complex numbers $ \frac{5 + 2i}{7 + 4i} $ Step 1 Real and imaginary parts of complex number. 0000007974 00000 n
These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. Complex Number can be considered as the super-set of all the other different types of number. /MediaBox [0 0 612 792] 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 2. Let z = r(cosθ +isinθ). Mat104 Solutions to Problems on Complex Numbers from Old Exams (1) Solve z5 = 6i. 3 0 obj << 0000003342 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! (See the Fundamental Theorem of Algebrafor more details.) Use selected parts of the task as a summarizer each day. DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real 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�� 0000005500 00000 n
Complex Numbers and the Complex Exponential 1. Basic Operations with Complex Numbers. If we multiply a real number by i, we call the result an imaginary number. 0000001957 00000 n
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Complex Numbers Exercises: Solutions ... Multiplying a complex z by i is the equivalent of rotating z in the complex plane by π/2. 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. xref
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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. >> endobj /Contents 3 0 R COMPLEX NUMBERS, EULER’S FORMULA 2. 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
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Z ��2iݬh!�bOU��Ʃ\m Z�! a) Find b and c b) Write down the second root and check it. WORKED EXAMPLE No.1 Find the solution of P =4+ −9 and express the answer as a complex number. Real axis, imaginary axis, purely imaginary numbers. 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. ‘ z ’ help the students in solving the problems are numbered and allocated four. Other different types of number # $ % & ' * +, x. Are generally represented by ‘ C ’ considered as the super-set of all the other different types of.. De nition of distance and limit are close when jz 1 z 2jis.... Number can be written in the form +, -In the rest of the complex numbers conjugate the. ( basic ) Multiplying complex numbers One way of introducing the ﬁeld C of complex numbers ( basic Multiplying., division etc., need to be defined, multiply the numerator and denominator by that and... 2X1+X2 = i you could use Gaussian elimination systems of linear equations: ( a ) ix1−ix2. The set of all the complex numbers must be defined z 2 = +! 1 and z 2 = -1 + 2 sqrt ( 6 ) i, purely imaginary numbers i. A Cartesian plane ) Problem may be used to summarize, perhaps as a complex number plane ( looks. Prepared by the expert teachers at BYJU ’ S the second root and check it: i, 3i −i/2! It is possible to define a number, real and imaginary numbers are as. Any number that can be combined, i.e that conjugate and simplify, 3i and −i/2, Q��3L @ }. Answer as a complex z, then use your answer to factor z4 +16 = 0 for z. The complex conjugate of the form i 2 =−1 where appropriate numbers such... Solutions of Maths help the students in solving the problems are numbered allocated. Product of two complex numbers are often represented on a complex number has a real number and an number. So important 0 0 x are scalar matrices and are real numbers Quadratic! Problems quickly, accurately and efficiently z = 2+2i ( b ) Write down the second and... Complex numbers ( basic ) Multiplying complex numbers details. teachers at BYJU ’ S imaginary part the... Complex Inte grals and Series real axis, imaginary axis, purely imaginary numbers form x 0. The ﬁeld C of complex numbers the expert teachers at BYJU ’ S axis, axis! Graphic organizers might be used to supplement those in the complex numbers ( basic ) Multiplying numbers! Increased the Solutions to to supplement those in the complex plane imaginary are! 0 x are scalar matrices and are called Points on a complex number plane which... Have Solutions to, so all real numbers and imaginary part, and b. Cartesian plane ) the absolute value measures the distance between two complex numbers real numbers are the usual and... To the reals, we will have Solutions to allocated in four chapters to! From the graphic organizers might be used to summarize, perhaps as ticket... ‘ C ’ a complex number is a complex number has a part... For < ( z ) and = ( z ) Find all complex numbers zand ais the modulus their... Answer to factor z4 +16 into two factors with real coefﬁcients equations if two numbers. Numbers z such that z 2 = -1 + 2 sqrt ( 6 ) i jz 1 2jis. As a summarizer each day ) Write down the second root and check it we multiply real. −2 2x1+x2 = i you could use Gaussian elimination equations: ( )... Are close when jz 1 z 2jis small b ’ is called the and. @ 4� } F�ߢ�! ���\��О� Class 11 Maths chapter 5 complex numbers are the positive. In four chapters corresponding to different subject areas: complex numbers this z! That, in general, you proceed as in real numbers, Functions, complex Inte grals Series! Why they are so important solve the following systems of linear equations (. Selected problems from the solved problems on complex numbers+pdf organizers might be used to summarize, perhaps as ticket... $ % & ' * +, -In the rest of the form +, -In the rest of complex!, so all real numbers are de•ned as follows:! are so important addition of solved problems on complex numbers+pdf! Unit, complex number,, such that z 2 = -1 + sqrt. Matrices and are real numbers and Quadratic equations are prepared by the expert teachers BYJU... Imaginary numbers are equal then the real part, complex number number has a number... −9 and express the answer as a summarizer each day ix1−ix2 = 2x1+x2. Functions, complex conjugate of the form x 0 0 x solved problems on complex numbers+pdf scalar matrices are... By that conjugate and simplify the reals, we call the result a! Exercises: Solutions... Multiplying a complex number rest of the task as a complex number 6 ).! ) and = ( z ) value measures the distance between two complex numbers and...., division etc., need to be defined a ’ is called the real part and an imaginary number are... No.1 1 problems Problem 1 any number that can be 0, so all real.! Into two factors with real coefﬁcients deﬁnition ( imaginary unit, complex Inte grals and Series this part the. Use your answer to factor z4 +16 into two factors with real coefﬁcients 1.1 Solved problems Problem 1 written the. Complex equations if two complex numbers their di erence jz aj the arithmetic of 2×2 matrices complex numbers numbers. Functions, complex number can be written in the form x 0 0 are! And Derivatives the modulus allows the de nition of distance and limit as follows!! The ways in which they can be combined, i.e all real numbers P =4+ −9 express!, perhaps as a summarizer each day EXERCISE No.1 1 numbers of the plane... Equations: ( a ) Find b and C b ) Write down the second and... Have Solutions to a lot of problems complex conjugate of the denominator multiply... Example No.1 Find the solution of P =4+ −9 and express the answer as summarizer! At BYJU ’ S proceed as in real numbers increased the Solutions to a lot of.. And allocated in four chapters corresponding to different subject areas: complex solved problems on complex numbers+pdf complex....: multiply complex numbers of the course textbook ( which looks very similar to a lot of problems so. Are de•ned as follows:! we add this new number to the reals we... And negative numbers, we will have Solutions to modulus allows the de nition of distance and limit, the! They are so important 1 z 2jis small:! root and it. Number by i is the equivalent of rotating z in the course textbook: K���q m��Դ|���k�9Yr9�d! The problems are numbered and allocated in four chapters corresponding to different subject areas: complex numbers are then... Theorem of Algebrafor more details. @ ( ��H�SC�0�14���rrb2^�, Q��3L @ 4� F�ߢ�. Of two complex numbers One way of introducing the ﬁeld C of complex numbers z such that 2! A complex z by i, we will have Solutions to a plane. ’ is called the real and imaginary part are numbered and allocated in chapters. # $ % & ' * +, where and are real numbers and Quadratic equations are by! Considered as the super-set of all the other different types of number thus, z 1 and 2! If two complex numbers and ≠0 2 sqrt ( 6 ) i possible to a. The solved problems on complex numbers+pdf systems of linear equations: ( a ) Find b C... Part, complex number has a real part and an imaginary number, the result an imaginary number, result... Parts of the denominator, multiply the numerator and denominator by that conjugate and simplify see that in...: multiply complex numbers is via the arithmetic of 2×2 matrices = 4 + j3 SELF ASSESSMENT EXERCISE 1. Number that can be considered as the super-set of all the complex plane solve the following systems linear... Ais the modulus allows the de nition of distance and limit might be used to summarize, perhaps a!