[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657322-105":59,"doc-detail-657322-en":134},{"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":127,"head_meta":129,"extra_data":131,"updated_unix":133},105,"en","the-argand-diagram-definitions-modulus-argument-and-polynomial-roots","The Argand diagram - Definitions, modulus, argument, and polynomial roots","","An explanation of complex numbers using the Argand diagram: defining z = x + iy, identifying real and imaginary parts, and interpreting real/imaginary axes. It derives addition and multiplication rules geometrically and algebraically, then develops polar form with modulus |z| and argument arg(z), including the principal value and triangle inequality. The complex conjugate is introduced to compute real and imaginary parts and the multiplicative inverse. Finally, complex polynomials are defined and linked to the fundamental theorem of algebra, yielding factorization and examples.",{"@graph":69,"@context":126},[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/the-argand-diagram-definitions-modulus-argument-and-polynomial-roots/657322/",{"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/the-argand-diagram-definitions-modulus-argument-and-polynomial-roots/657322.png","ImageObject",300,407,{"name":92,"@type":93},"Helloiii","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118,122],{"name":109,"@type":110,"acceptedAnswer":111},"How is a complex number represented on the Argand diagram?","Question",{"text":112,"@type":113},"A complex number z = x + iy is placed at the point (x, y). Real numbers lie on the x-axis, and purely imaginary numbers (with real part 0) lie on the y-axis.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are the formulas for adding and multiplying complex numbers?",{"text":117,"@type":113},"Addition adds real parts and imaginary parts: (z1 + z2) = (x1 + x2) + i(y1 + y2). Multiplication uses (x1x2 − y1y2) + i(x1y2 + x2y1), based on i2 = −1.",{"name":119,"@type":110,"acceptedAnswer":120},"How are modulus and argument defined, and what is the principal value?",{"text":121,"@type":113},"The modulus |z| is the distance from (x, y) to the origin: |z| = sqrt(x^2 + y^2). The argument arg(z) is the angle the vector (x, y) makes with the x-axis, defined up to multiples of 2π; the principal value Arg(z) lies in (−π, π].",{"name":123,"@type":110,"acceptedAnswer":124},"What does the fundamental theorem of algebra guarantee for complex polynomials?",{"text":125,"@type":113},"For a complex polynomial p(z) of degree n > 0, there exists at least one complex root: some λ such that p(λ) = 0. This leads to factorization into linear factors and enables examples like z^2 + 1 = (z − i)(z + i).","https://schema.org",{"og:url":83,"og:type":128,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":130,"canonical":83},"index,follow",{"doc_id":132,"site_id":62},657322,1792977390,{"code":4,"msg":5,"data":135},{"doc_id":132,"user_id":136,"nickname":92,"user_avatar":137,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":138,"file_id":139,"file_url":140,"file_type":141,"file_size":142,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":143,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":144,"faqs":145,"seo_title":146,"seo_description":67,"update_tm":133,"read_time":52},5909897196768,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","2. The Argand diagram  \nDe􀀌nition 2.1 . A complex number z is an expression of the form z = x + iy, where x and y are real numbers.  \nFor example  \n2 + 3i and 􀀙 + 13i  \nare complex numbers.  \nDe􀀌nition 2.2 . If z = x + iy is a complex number, so that x and yare real numbers, x, denoted Re z, is called the real part of z and y , denoted Im z, is the called the imaginary part.  \nThus 2 is the real part of 2 􀀀 3i and 􀀀13 is the imaginary part of 􀀙 􀀀 13i. It is customary, and certainly convenient, to visualise a complex number as being a point of the Argand diagram, or the complex plane. We put the complex number z = x + iy at the point (x; y) . We can think of a real number x as a complex number whose imaginary part is zero. Thus real numbers correspond to the x-axis of the Argand diagram. We say a complex number is purely imaginary if its real part is zero. Imaginary numbers have the form iy, where y is a real number and the purely imaginary numbers correspond to they-axis of the Argand diagram.  \nDe􀀌nition 2.3 . If z1 = x 1 + iy 1 and z2 = x2 + iy2 are two complex numbers, their sum, denoted z1 + z2 , is  \n(x1 + x2 ) + i(y1 + y2 ) :  \nIn words, add the real and imaginary parts. For example,  \n(2 􀀀 3i) + (􀀙 + 13i) = (2 + 􀀙) + (􀀀3 + 13)i  \n= (2 + 􀀙) + 10i:  \nIn terms of the Argand diagram we just add the corresponding vectors.  \nDe􀀌nition 2.4 . If z1 = x 1 + iy 1 and z2 = x2 + iy2 are two complex numbers, their product, denoted z1 z2 , is  \n(x1 x2 􀀀 y1 y2 ) + (x1 y2 + x2 y 1 )i:  \nIn practice the way to compute the product of two complex numbers is just to apply the usual rules of arithmetic and use the rule i2 = 􀀀1 every time we need to get rid of a power of i. For example,  \n(2 􀀀 3i)(􀀙 + 13i) = (2􀀙 􀀀 (􀀀3) 􀀁 13) + (2 􀀁 13 􀀀 3 􀀁 􀀙)i = (2􀀙 + 39) + (26 􀀀 3􀀙)i:  \nOne can justify all of the formal rules from the de􀀌nitions (this is a somewhat tedious and uninformative exercise):  \n(z1 + z2 ) + z3 = z1 + (z2 + z3 ); z1 + z2 = z2 + z1 ; z1 + 0 = z1 ; z1 + (􀀀z1 ) = 0 (z1 z2 )z3 = z1 (z2 z3 ); z1 z2 = z2 z1 ; z1 􀀁 1 = z1 ; z1 (z2 + z3 ) = z1 z2 + z1 z3 :  \nHere z1 , z2 and z2 are general complex numbers. The most interesting identity is the last one, which says you can distribute multiplication over addition, in the usual way.  \nInstead of using cartesian coordinates to think about complex numbers, instead one can think in terms of polar coordinates. In polar coordinates the two relevant quantities are the distance r to the origin and the angle 􀀒 to the x-axis, going counter-clockwise.  \nDe􀀌nition 2.5 . The modulus, or absolute value of a complex number z, denoted jz j, is the distance of the point (x; y) to the origin.  \nThe argument of z, denoted arg(z), is the angle the vector (x; y) makes with the x-axis.  \nThe argument is only well-de􀀌ned up to a multiple of 2􀀙 . The principal value of the argument, denoted Arg(z), is constrained to lie in the interval (􀀀􀀙; 􀀙] . Note that  \njzj =px2 + y 2  and Arg(z) = arctan yx: The modulus of 3 + 4i is 5 and the principal value of the argument of  \n1 􀀀 i is 􀀀􀀙=4.  \nOf course we can go backwards, from polar to cartesian:  \nx = r cos 􀀒 and x = r sin 􀀒: Putting this together, we get  \nz = r cos 􀀒 + ir sin 􀀒:  \nThe most basic property of the modulus is the triangle inequality:  \nj z + w j 􀀔 jz j + jw j ;  \nwith equality if and only if either z = 0 or w is a positive real scalar multiple of z.  \nLemma 2.6 . If z and w are complex numbers then  \njz j 􀀀 jw j 􀀔 j z 􀀀 w j :  \nProof. As z = (z 􀀀 w) + w we have  \njz j 􀀔 j z 􀀀 w j + jw j ;  \nby the triangle inequality. Rearranging, we get  \njz j 􀀀 jw j 􀀔 j z 􀀀 w j : 􀀃  \nNote that i is a square root of 􀀀1. 􀀀i is the other square root of 􀀀1. In some sense the choice of square root is arbitrary.  \nDe􀀌nition 2.7 . Let z = x + iy be a complex number. The complex conjugate, denoted , is the complex number  \nx 􀀀 iy:  \nOne can use the complex conjugate to write down a formula for the real and imaginary parts:  \nz +    z 􀀀   \nRez = and Im z = :  \n2 2i  \nThere are agai","cbCaigrj1WYIEN7N","https://ap.wps.com/l/cbCaigrj1WYIEN7N","pdf",143876,"English","# The Argand diagram\n## Complex numbers and parts\n## Addition and multiplication\n## Polar form: modulus and argument\n## Triangle inequality and conjugates\n## Complex polynomials and the fundamental theorem of algebra","[{\"question\":\"How is a complex number represented on the Argand diagram?\",\"answer\":\"A complex number z = x + iy is placed at the point (x, y). Real numbers lie on the x-axis, and purely imaginary numbers (with real part 0) lie on the y-axis.\"},{\"question\":\"What are the formulas for adding and multiplying complex numbers?\",\"answer\":\"Addition adds real parts and imaginary parts: (z1 + z2) = (x1 + x2) + i(y1 + y2). Multiplication uses (x1x2 − y1y2) + i(x1y2 + x2y1), based on i2 = −1.\"},{\"question\":\"How are modulus and argument defined, and what is the principal value?\",\"answer\":\"The modulus |z| is the distance from (x, y) to the origin: |z| = sqrt(x^2 + y^2). The argument arg(z) is the angle the vector (x, y) makes with the x-axis, defined up to multiples of 2π; the principal value Arg(z) lies in (−π, π].\"},{\"question\":\"What does the fundamental theorem of algebra guarantee for complex polynomials?\",\"answer\":\"For a complex polynomial p(z) of degree n \\u003e 0, there exists at least one complex root: some λ such that p(λ) = 0. This leads to factorization into linear factors and enables examples like z^2 + 1 = (z − i)(z + i).\"}]","The Argand diagram - Definitions, modulus, argument, and polynomial roots | PDF"]