[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657324-105":59,"doc-detail-657324-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","complex-numbers-mathematical-introductory-ideas","Complex Numbers - Mathematical Introductory Ideas","","Introductory discussion of complex numbers, focusing on their algebraic operations and geometric interpretation. Explains why quadratic equations with negative discriminant require extending the real number system, introducing i as the standard notation for √−1. Defines complex numbers a + bi and the concepts of real and imaginary parts, leading to key results such as the Fundamental Theorem of Algebra and exercises on using complex numbers to solve polynomial equations.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/complex-numbers-mathematical-introductory-ideas/657324/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/complex-numbers-mathematical-introductory-ideas/657324.png","ImageObject",300,407,{"name":92,"@type":93},"Jiven","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why do we need complex numbers when b2 − 4ac \u003C 0?","Question",{"text":112,"@type":113},"When b2 − 4ac \u003C 0, the quadratic has no real solutions because no real number squares to the negative discriminant. Complex numbers allow such roots to be represented and solved.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are complex numbers defined and how are real and imaginary parts identified?",{"text":117,"@type":113},"A complex number has the form a + bi with real a and b. If z = a + bi, then a is the real part (Re z) and b is the imaginary part (Im z).",{"name":119,"@type":110,"acceptedAnswer":120},"What does the Fundamental Theorem of Algebra guarantee?",{"text":121,"@type":113},"It states that any polynomial equation with real or complex coefficients has all its roots in C: a degree n polynomial has exactly n complex roots counting multiplicities.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657324,1792977397,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","Complex Numbers  \nRichard Earl∗  \nMathematical Institute, Oxford, OX1 2LB,  \nJuly 2004  \nAbstract  \nThis article discusses some introductory ideas associated with complex numbers, their algebra and geometry. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Finally we look at the nth roots of unity, that is, the solutions of the equations zn = 1 .  \n1 The Need For Complex Numbers  \nThe shortest path between two truths in the real domain passes through the complex domain  \nJacques Hadamard (1865-1963)  \nAll of you will know that the two roots of the equation ax2 + bx + c = 0 are  \n−b ±√b2 − 4ac  \nx = (1)  \n2a  \nand solving quadratic equations is something that mathematicians have been able to do since the time of the Babylonians. When b2 − 4ac > 0 then these two roots are real and distinct; graphically they are where the curve y = ax2 + bx + c cuts the x-axis. When b2 − 4ac = 0 then we have one real root and the curve just touches the x-axis here. But what happens when b2 − 4ac \u003C 0? In this case there are no real solutions to the equation, as no real number squares to give the negative b2 − 4ac. From the graphical point of view, the curve y = ax2 + bx + c lies entirely above or below the x-axis.  \nIt is only comparatively recently that mathematicians have been comfortable with these roots when b2 − 4ac \u003C 0. During the Renaissance the quadratic would have been considered unsolvable, or its roots would have been called imaginary 1 .  \nIf we imagine √−1 to exist, and that it behaves much like other numbers, then the two roots of the quadratic ax2 + bx + c = 0 can be written in the form  \nx = A ± B √−1 (2)  \n∗ These pages are produced by Richard Earl, who is the Schools Liaison and Access Officer for mathematics, statistics and computer science at Oxford University. Any comments, suggestions or requests for other material would be welcomed  \nat [earl@maths.ox.ac.uk](earl@maths.ox.ac.uk)  \n1 The term‘imaginary’was first used by the French Mathematician René Descartes (1596-1650) . Whilst he is known more as a philosopher, Descartes made many important contributions to mathematics and helped found co-ordinate geometry– hence the naming of Cartesian co-ordinates.  \nwhere A = −b/2a and B = √4ac − b2  /2a are real numbers. But what meaning can such roots have? It was this philosophical point which pre-occupied mathematicians until the start of the 19th century; afterwards these ‘imaginary’ numbers started proving so useful (especially in the work of Cauchy and Gauss) that these philosophical concerns were essentially forgotten.  \nNotation 1 We shall from now on write i for √−1 – this is standard notation amongst mathematicians2 , though many books, particularly those written for engineers and physicists, use j instead.  \nDefinition 2 A complex number 3 is a number of the form a + bi where a and b are real numbers. If z = a + bi then a is known as the real part of z and b as the imaginary part. We write a = Rez and b = Im z. Note that real numbers are complex – a real number is simply a complex number with zero imaginary part.  \nRemark 3 Note that two complex numbers are equal precisely when their real and imaginary parts are equal – that is a+bi = c+di if and only if a = c and b = d. This is called‘comparing real and imaginary parts’.  \nNotation 4 We write C for the set of all complex numbers.  \nOne of the first major results concerning complex numbers, and which conclusively demonstrated their usefulness, was proved by Gauss in 1799 . From the quadratic formula (1) we know that all quadratic equations can be solved using complex numbers, but what Gauss was the first to prove was the much more general result:  \nTheorem 5 (FUNDAMENTAL THEOREM OF ALGEBRA) The roots of any polynomial equation  \na0 + a1 x + a2x2 + ··· + anxn = 0 ,  \nwith real (or complex) coefficients ai , are complex. That is there are n (not necessarily distinct) complex numbers γ 1 , .","cbCaioMvc7ozcJrT","https://ap.wps.com/l/cbCaioMvc7ozcJrT","pdf",234336,13,"English","# Abstract\n# The Need For Complex Numbers\n## Notation and Definition of Complex Numbers\n## Fundamental Theorem of Algebra\n## Exercises","[{\"question\":\"Why do we need complex numbers when b2 − 4ac \\u003c 0?\",\"answer\":\"When b2 − 4ac \\u003c 0, the quadratic has no real solutions because no real number squares to the negative discriminant. Complex numbers allow such roots to be represented and solved.\"},{\"question\":\"How are complex numbers defined and how are real and imaginary parts identified?\",\"answer\":\"A complex number has the form a + bi with real a and b. If z = a + bi, then a is the real part (Re z) and b is the imaginary part (Im z).\"},{\"question\":\"What does the Fundamental Theorem of Algebra guarantee?\",\"answer\":\"It states that any polynomial equation with real or complex coefficients has all its roots in C: a degree n polynomial has exactly n complex roots counting multiplicities.\"}]","Complex Numbers - Mathematical Introductory Ideas | PDF",33]