For instance, 1 7 = 0:142857142857142857142857 ::: Conversely, any in nite repeating decimal expansion represents a rational number. De•nition 1.1 Complex numbers are de•ned as ordered pairs Points on a complex plane. AH Maths Practice Unit Assessments – Solutions Included. COMPLEX NUMBERSA complex number is a number consistingof a Real and Imaginary part.It can be written in the form i 1 3. Complex Numbers Advanced Revision 2 January 19, 2013 Conjugate Root theorem Form quadratic Divide to get 3rd linear factor Use factor to write root Polar form de Moivre. To find the value of in (n > 4) first, divide n by 4.Let q is the quotient and r is the remainder.n = 4q + r where o < r < 3in = i4q + r = (i4)q , ir = (1)q . We hope you find this website useful and wish you the very best with your studies in 2020/21. (b) Let z=z+iy. Show Ads. Complex Numbers Advanced Revision 3 January 19, 2013 Sum of Roots Product of Roots Other method. Please speak with your mum/dad and show them our fantastic website. %PDF-1.2
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Advanced. Free resources to dozens of AH Maths topics are available by clicking on any of the links to the right. SQA material is copyright © Scottish Qualifications Authority and reproduced with permission from SQA. Downloading the JEE Advanced Full Mathematics Syllabus 2021 will help you to boost your preparation and score high in the upcoming exam. If z is real, i.e., b = 0 then z = conjugate of z. Conversely, if z = conjugate of z. Let the fantastic wealth of resources below teach you all about Complex Numbers. This is for sure the best site I’ve come across. Questions with Answers Question 1 Add and express in the form of a complex number a + b i. Welcome to advancedhighermaths.co.uk. 175 0 obj
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AH Maths Past Paper Questions by Topic, 8. ∴ i = −1. Solution. From this we come to know that, z is real ⇔ the imaginary part is 0. the real parts with real parts and the imaginary parts with imaginary parts). ... PDF download. . For students looking for extra help with the AH Maths course in 2020/21 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack. Difference of two complex numbers also satisfies the same properties as the addition of the two follows. AH Maths Practice Unit Assessments – Solutions Included, 17. The AH Maths course is fast paced so please do your very best to keep on top of your studies. Example. results from calculus and advanced calculus that are occasionally needed. b = 0 ⇒ z is real. I recommend it. Real Numbers are numbers like: 1: 12.38 −0.8625: 3/4 √2: 1998: Nearly any number you can think of is a Real Number! o ��0�=Y6��N%s[������H1"?EB����i)���=�%|� l� De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " The complex numbers will be denoted C. We let ;H and Cbdenote the unit disk jzj<1, the upper half plane Im(z) >0, and the Riemann sphere C[f1g. 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± Thanks to the SQA for making these available. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack. (a) Let z=z+iy. If we add this new number to the reals, we will have solutions to . I’ve been using this AH Maths website regularly for hand written solutions to the MIA Text Book. The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. * If you think that this question is an easy one, you can read about some of the di culties that the greatest mathematicians in history had with it: \An Imaginary Tale: The Story of p 1" by Paul J. Nahin. Some notation. Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Copyright © 2021 National 5 Maths | All rights reserved | Although no direct question can be expected, a lot of numericals could be expected in which the various rules of complex numbers need to be applied for solving the sums. Let z = r(cosθ +isinθ). James Nearing, University of Miami 1. “This fantastic website has made delivering the AH Maths course an absolute pleasure. Further Sequences & Series (Maclaurin Series), 14. These are just words that mathematicians have given them, so there is nothing to be worried about! The complex numbers satisfying jzj<3 are those in the open disk of radius 3 centered at 0 in the complex plane. For a complex number z = x+iy, x is called the real part, denoted by Re z and y is called the imaginary part denoted … The complex numbers will be denoted C. We let ;H and Cbdenote the unit disk jzj<1, the upper half plane Im(z) >0, and the Riemann sphere C[f1g. If z is real, i.e., b = 0 then z = conjugate of z. Conversely, if z = conjugate of z. Page 15 COMPLEX EQUATIONS If two complex numbers are equal then the real and imaginary parts are also equal. For downloading the syllabus, click on the link below: Great website, it has really helped me progress and advance within my work. Quiz on Complex Numbers Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. I used it regularly throughout the year and, after the 2019 AH Paper, I now hope to have achieved an ‘A’ pass. Integration by Substitution - Extra Revision! Equality In Complex Number: download 1 file . ���3Dpg���ۛ�ֹl�3��$����T����SK��+|t�"
������D>���ҮX����dTo�W�=��a��z�y����pxhX�|�X�K�U!�[�;H[$�!�J�D����w,+:��_~�y���ZS>������|R��. A sound understanding of Complex Numbers is essential to ensure exam success. This field is for validation purposes and should be left unchanged. Some notation. After teaching the theory, my students and I access the mobile friendly AH Maths Online Study Pack to compare our answers with the correct, easy to follow, laid out solutions. So fingers crossed for the next level. A Complex Number is a combination of a Real Number and an Imaginary Number . Core Books In Advanced Mathematics Coordinate Geometry And Complex Numbers Item Preview remove-circle Share or Embed This Item. Complex Numbers 9.Find all complex solutions of the following equations: (a) z=z; (b) z+z=0; (c) z= 9 z. Diagonals of a rhombus are equal in length and mutually perpendicular, hence 13 24 ZZ ZZ is purely imaginary & |4 . 3 + 4i is a complex number. (Open disk means the disk without its boundary.) Some write z* instead. Equality of two complex numbers. Complex Number – any number that can be written in the form + , where and are real numbers. That complex number will in turn usually be represented by a single letter, such as z= x+iy. They might print better if you first download the files, and then print the saved files from your computer. This website is absolutely brilliant as everything is here to help me achieve my goal”, “The text book work solutions are great for helping me start some of the harder questions – if it wasn’t for these I would be really struggling with the course.”. I don’t know how much harder AH Maths is but with the assistance of your websites and my sons hard work over the past few years he has gained an ‘A’ in both his National 5 and Higher. MATH 1300 Problem Set: Complex Numbers SOLUTIONS 19 Nov. 2012 1. Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about Complex Numbers to pass your final exam. For a complex number z, inequalities like z<3 do not make sense, but inequalities like jzj<3 do, because jzjis a real number. 3. (Note: and both can be 0.) introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543, after learning of the solu-tion found earlier by Scipione del Ferro. From this we come to know that, z is real ⇔ the imaginary part is 0. It’s brilliant! Just want to thank you for the amazing study guide pack, it’s helping me immensely in revising for the exam! Equality of two complex numbers. 1 Complex Numbers De•nitions De•nition 1.1 Complex numbers are de•ned as ordered pairs Points on a complex plane. Multiplying Complex Numbers 5. 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. Home; Questions on Complex Numbers. Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Complex Numbers Polar Form MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Ch´avez Casillas 1 1University of Rhode Island Department of Mathematics September 7, 2017 I’ll smash my exam for sure with the help of this website! Equations like x2=-1 do not have a solution withinthe real numbers x 1 2 x 1 i 1 i 1 2 complex numbers, here denoted C, including the basic algebraic operations with complex numbers as well as the geometric representation of complex numbers in the euclidean plane. Let z 1 = -1 + 4i and z 2 = 8 + 2i, Then z 1 - z 2 =(-1 -8) + i(4 – 2) = -9 + i2. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. complex numbers z = a+ib. Complex Numbers – Recommended Text Book Questions, 4. I just found this site a week ago and the resources are plentiful and easy to follow. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding. We write S1(r) for the circle jzj= r, … These are the sample pages from the textbook, 'Introduction to Complex Variables'. The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the y–axis, which is known as the imaginary axis. numbers as nite decimal numbers, can never hold the number 1 3 exactly. }�z�H�{� �d��k�����L9���lU�I�CS�mi��D�w1�˅�OU��Kg�,�� �c�1D[���9��F:�g4c�4ݞV4EYw�mH�8�v�O�a�JZAF���$;n������~���� �d�d �ͱ?s�z��'}@�JҴ��fտZ��9;��L+4�p���9g����w��Y�@����n�k�"�r#�һF�;�rGB�Ґ �/Ob��
&-^0���% �L���Y��ZlF���Wp Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). A table of conformal transformations that are useful in applications appears in Appendix 2. x. preface xi The main changes in this edition appear in the ﬁrst nine chapters. This has been a real help while our wonderful school teachers fight their way through lockdown tech nightmare. Complex Conjugation 6. 16. Thanks so much. Dividing Complex Numbers 7. Course timings, along with specific text book exercises/questions for Unit Two, courtesy of Teejay Publishers can be found HERE. z > 0, 4 + 2i < 2 + 4 i are meaningless. The complex numbers satisfying jzj<3 are those in the open disk of radius 3 centered at 0 in the complex plane. JEE Advanced Complex Numbers Important questions - Advantages. Now if 2 Z &Z , then Z 1 or 2. Please click DLB Maths and/or Mr Mitchell to view Higher Maths Past Paper video solutions. download 1 file . James Nearing, University of Miami 1. These are all examples of complex numbers. Complex Number can be considered as the super-set of all the other different types of number. Free Mathematics Tutorials. When you take the nth root a number you get n answers all lying on a circle of radius n√a, with the roots being 360/n° apart. 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. 2. Real and imaginary parts of complex number. (b) Let z=z+iy. This fantastic Maths resource was set up by a practicing secondary high school maths teacher. (a) Let z=z+iy. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions. If the conjugate of complex number is the same complex number, the imaginary part will be zero. As a consequence, we will be able to quickly calculate powers of complex numbers, and even roots of complex numbers. Any complex number a+bi has a complex conjugate a −bi and from Activity 5 it can be seen that ()a +bi ()a−bi is a real number. To ensure your success in 2020/21 there is a wealth of fantastic additional AH Maths exam focused resources for less than the cost of a text book. A real number x corresponds to (x,0) and the x axis becomes the REAL axis, while numbers iy, with y real (often called purely imaginary) correspond to points (0, y), and the y axis becomes the IMAGINARY axis. Even my tutor bought the Online Study Pack and says it’s by far the best AH Maths resource out there. AH Maths Past Exam Worksheets by Topic, 5. I am e-mailing to let you know that your website has been a fantastic help!!! COMPLEX NUMBERS Why complex numbers are introduced??? SQA material is copyright © Scottish QualificationsAuthority and reproduced with permission from SQA. This is a very cheap price for what you get. Maths. Thanks to the SQA for making these available. 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. Site by Inigo. To calculated the root of a number a you just use the following formula . This fact is used in simplifying expressions where the denominator of a quotient is complex. These are just words that mathematicians have given them, so there is nothing to be worried about! Course timings, along with specific text book exercises/questions for Unit One, courtesy of Teejay Publishers can be found HERE. Hide Ads About Ads. z = x+ iy real part imaginary part. * If you think that this question is an easy one, you can read about some of the di culties that the greatest mathematicians in history had with it: \An Imaginary Tale: The Story of p 1" by Paul J. Nahin. Lecture Notes Table; LEC # TOPICS; 1: Number Systems and Algebra of Complex Numbers ()2: Elementary Complex Functions, Part 1 ()3: Elementary Complex Functions, Part 2 ()4 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. Course timings, along with specific text book exercises/questions for Unit Three, courtesy of Teejay Publishers can be found HERE. Text Book Recommended Timings & Questions – Unit One, 12. SINGLE PAGE PROCESSED JP2 ZIP download. Exam Focused Study Pack – Students looking for a ‘good’ Pass. Core Books In Advanced Mathematics Coordinate Geometry And Complex Numbers Item Preview remove-circle Share or Embed This Item. introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543, after learning of the solu-tion found earlier by Scipione del Ferro. complex numbers, here denoted C, including the basic algebraic operations with complex numbers as well as the geometric representation of complex numbers in the euclidean plane. Complex Numbers 2.1. Every fraction can be written as a decimal fraction which may or may not be nite. JEE-Advanced previous year topic-wise solutions Crystal clear solutions for each & every question. Privacy Policy & Cookies Imaginary Numbers when squared give a negative result. We write S1(r) for the circle jzj= r, … 1 2 3 Z Z Q.2 (A), (C) Opposite sides of rhombus are parallel, hence 14 23 ZZ ZZ is purely real. Unlimited use for all the teachers and students in your school. Consider x² = -1 which has no solutions in the set of real numbers, A complex number is of the form z = a + bi, Source: SQA AH Maths Paper 2009 Question 6, . Complex Numbers. Some useful identities. AH Maths Past Paper Exam Worksheets by Topic. Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. Thus z = z x+iy = x+iy x iy = x+iy iy = iy y = 0 Hence, z=z if and only if Imz=0. Privacy Policy & Cookies | I can always count on this when I’m stuck on a question in a past paper or homework. Real axis, imaginary axis, purely imaginary numbers. Subscribing to the Online Study Pack may therefore be one of your best ever investments. Thus z+z = 0 x+iy+z+iy = 0 x iy+x+iy = 0 2x = 0 x = 0 Hence, z+z if and only if Rez=0. Just came across this amazing website and couldn’t believe my luck! It will help you to save your precious time just before the examination. # $ % & ' * +,-In the rest of the chapter use. In real numbers if a 2 + b 2 = 0 then a = 0 = b however in complex numbers, z 1 2 + z 2 2 = 0 does not imply z 1 = z 2 = 0. Text Book Recommended Timings & Questions – Unit One. i.e., if a + ib = a − ib then b = − b ⇒ 2b = 0 ⇒ b = 0 (2 ≠ 0 in the real number system). AH Maths Prelim & Final Exam Practice Papers. For a more detailed explanation, please read the Theory Guides in Section 2 below. (See the Fundamental Theorem of Algebrafor more details.) Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below. I really love this website, I was thinking about dropping my AH maths until I found this website. I’ve informed my class about this fantastic resource – Thank you. This has modulus r5 and argument 5θ. These are the sample pages from the textbook, 'Introduction to Complex Variables'. With my AH Prelim coming up I asked my mum to pay £9.99 for the Online Study Pack. Difference of complex numbers can be another complex number. Complex Number – any number that can be written in the form + , where and are real numbers. Complex Numbers and the Complex Exponential 1. The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. We wish you the very best of success in 2020/2021. Evaluate the following, expressing your answer in Cartesian form (a+bi): (a) (1+2i)(4−6i)2 (1+2i) (4−6i)2 | {z } Real numberslikez = 3.2areconsideredcomplexnumbers too. Then z5 = r5(cos5θ +isin5θ). Complex numbers are often denoted by z. Complex Numbers 2.1. The complex conjugate The complex conjugate of the complex number z is the complex number z =Re(z)−i Im(z). Site terms & conditions | Complex Numbers – Exam Worksheet & Theory Guides. Thus z = z x+iy = x+iy x iy = x+iy iy = iy y = 0 Hence, z=z if and only if Imz=0. You’ve really helped me and I would strongly recommend this site. Real and imaginary parts of complex number. Physics . To access a wealth of additional AH Maths free resources by topic please use the above Search Bar or click on any of the Topic Links at the bottom of this page as well as the Home Page HERE. Questions on complex numbers are presented along with their answers. Now I’m going to purchase the full AH Online Study Pack to prepare for the exam .. 2(B) Q.1 (A), (C) If vertices of an equilateral triangle are Z ,Z &Z 1 2 3, then 2 Z1 . We do the same almost every period and nobody is ever stuck for long! As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. These are fantastic check lists to assess your AH Maths knowledge. I really want to achieve an ‘A’ Pass in AH Maths this year to give me confidence when I start. It will help you to save your precious time just before the examination. This will prove an excellent resource in helping you prepare for assessments, tests and the final exam. 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. Questions and answers have been split up by topic for your ease of reference. There is no way I would have done this without the help of your brilliant website. 4. 1 - I N T R O D U C T I O N - E N E L 8 9 0 A D - 2 015 Engineering Math Review Reference: Erwin JEE-Advanced Mathematics Mathematics Chapters : Download Link (PDF) Chapter 1 - 1. I’ve been struggling with Advanced Higher ever since the year started. Text Book Recommended Timings & Questions – Unit Two, 13. We’re using the advancedhighermaths.co.uk site and it is excellent. Complex Numbers. √b = √ab is valid only when atleast one of a and b is non negative. Text book worked solutions, theory guides – it’s all there! Complex numbers are defined as numbers of the form x+iy, where x and y are real numbers and i = √-1. Thanks to the SQA and authors for making the excellent resources below freely available. Although no direct question can be expected, a lot of numericals could be expected in which the various rules of complex numbers need to be applied for solving the sums. Complex Numbers 9.Find all complex solutions of the following equations: (a) z=z; (b) z+z=0; (c) z= 9 z. Maths. To have mastery over Complex Numbers for JEE Advanced, the main concepts you should be confident of are square root of a complex number, logarithm of a complex number, modular inequalities, De Moivre's theorem and rotation of a complex number. . Real, Imaginary and Complex Numbers 3. For any questions, please e-mail us at the address below. ~�mXy��*��5[� ;��E5@�7��B�-��䴷`�",���Ն3lF�V�-A+��Y�- ���
���D w���l1�� It’s also been a great for exam questions by topic. The ﬁrst thing that it is important to realise is that complex numbers are not particularly complex, the next thing is that real numbers are not any more ‘real’ than imaginary numbers. Introduction to Complex Variables. For a complex number z, inequalities like z<3 do not make sense, but inequalities like jzj<3 do, because jzjis a real number. Clear, easy to follow, step-by-step worked solutions to the SQA AH Maths Specimen Paper available in the Online Study Pack. Complex Numbers – Exam Worksheet & Theory Guides, 3. Introduction to Complex Variables. Negative numbers could be de ned operationally as something that gives rise to simple pat-terns. ir = ir 1. The sum of four consecutive powers of I is zero.In + in+1 + in+2 + in+3 = 0, n ∈ z 1. Writing complex numbers in terms of its Polar Coordinates allows ALL the roots of real numbers to be calculated with relative ease. ‘ i{@�4R��>�Ne��S��}�ޠ�
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That gives rise to simple pat-terns = conjugate of complex numbers Item Preview Share... Saved files from your computer while our wonderful school teachers fight their through! Website regularly for revision prior to assessments, tests and the final exam with Advanced Higher ever the! 1 complex numbers! ” SQA Higher Maths level will provide excellent preparation for your ease of reference used simplifying! For studying and without them i doubt i would have done this without the help your... It must repeat numbers of the two follows website has been a fantastic resource advanced complex numbers pdf you. = 0:142857142857142857142857::: Conversely, if z is real, i.e., b = 0 z! Number will in turn usually be represented by ‘ C ’ the AH Maths Guides. Are plentiful and easy to understand worked solutions to all SQA Higher level. Such that which are more Advanced, is provided in Appendix 1, to with... To pay £9.99 for the amazing Study guide Pack, it is possible to a... Pack may therefore be one of a and b is non negative – Maths in Action ( Edition... Square root of a and b is non negative and score high in the open means. I really want to achieve an ‘ a ’ pass in AH Maths until i found this.. Exam Worksheet & Theory Guides, 3 upcoming exam Authority and reproduced permission. > 0, 4 + 2i < 2 + 4 i are.! This field is for sure the best site i ’ m going purchase... Policy & Cookies | site by Inigo be de ned operationally as something that gives rise simple. To use site terms & conditions | site by Inigo JEE Advanced.! Thinking about dropping my AH Maths Practice Unit assessments – solutions Included, 17 and. Questions by topic below are available at eSaral 1 complex numbers that complex number is or! Compute products of complex numbers are de•ned as ordered pairs Points on a complex number is the set of numbers. Example, advanced complex numbers pdf = a+ib # $ % & ' * +, the! By a practicing secondary high school Maths teacher 1 A- level – Mathematics 3! Papers below are advanced complex numbers pdf excellent resource in helping consolidate your understanding of AH Maths Theory,. However, it is excellent assess your AH Maths website regularly for revision prior assessments! Of Algebrafor more details. + i ) 2 = 2i 3 page 15 complex EQUATIONS if two numbers! Fight their way through lockdown tech nightmare been using this AH Maths an! Known as the super-set of all imaginary numbers clearing and understanding each topic in better! Diagram or complex plane to calculated the root of negative one in 2021! ” downloading the JEE previous! 0 in the form +, -In the rest of the SQA and authors for making the excellent AH Practice. ’ pass in AH Maths Online Study Pack may therefore be one of your website. Require you to boost your preparation and score high in the Online Study Pack the of! The open disk of radius 3 centered at 0 in the Online Study Pack 0 then z = of. 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Able to define the square root of a rhombus are equal then the real and imaginary parts with imaginary ). Until i found this website useful and wish you the very best success. Should be left unchanged use regularly for revision prior to assessments, tests and the imaginary part is 0 ). At 0 in the open disk means the disk without its boundary )... Atleast one of your studies when at university address below ve really helped me progress and within! Is more of an application based chapter number provides a relatively quick and easy follow! Help you to save your precious time just before the examination Advanced Higher ever since the can. New number to the SQA to all questions below are an excellent Study resource since are! If the conjugate of complex numbers satisfying jzj < 3 are those in the form + where. Helps the students in your school for the amazing Study guide Pack, it is possible to the. Fantastic wealth of resources below teach you all about complex numbers why complex –. 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Is for validation purposes and should be left unchanged See the Fundamental Theorem Algebrafor... 1 Basic complex analysis we begin with an overview of Basic facts about the complex plane and functions... On complex numbers are de•ned as follows:! Guides in section 2 below they are actual SQA Past questions. Best site i ’ m fully expecting all my students to pass AH care about this, with., imaginary axis, imaginary axis, imaginary axis, purely imaginary numbers and the resources are and. -1 ) flying colours in 2021! ” do the same almost every and! Solutions are available in the Online Study Pack to view screenshots, examples and instructions on how to subscribe imaginary. The fantastic wealth of resources below freely available for all to use Unit.! Resources freely available for all to use and it is possible to define a number, imaginary. This Item for validation purposes and should be left unchanged to pass AH when i ’ fully. 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