[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657323-105":3,"detail-sidebar-cat-0-en-105":74,"doc-detail-657323-en":124},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":67,"head_meta":69,"extra_data":71,"updated_unix":73},105,"en","complex-numbers-and-the-argand-plane-graphical-representation-real-and-imaginary-parts-modulus-argument-and-polarcartesian-conversion","Complex Numbers and the Argand plane - Graphical representation - Real and imaginary parts, modulus, argument, and polar/Cartesian conversion","","Complex Numbers and the Argand plane explains how to represent complex numbers graphically using the Argand diagram, where the real part and imaginary part determine coordinates in a plane. It defines the real and imaginary part functions Re(z) and Im(z), discusses when two complex numbers must be equal using equality of their projections, and introduces polar representation in terms of modulus r and argument arg(z). It further provides conversion rules between Cartesian and polar forms, illustrated with worked examples and plotted results.",{"@graph":14,"@context":66},[15,34,49],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/complex-numbers-and-the-argand-plane-graphical-representation-real-and-imaginary-parts-modulus-argument-and-polarcartesian-conversion/657323/",4,{"url":32,"name":10,"@type":35,"author":36,"headline":10,"publisher":39,"fileFormat":42,"inLanguage":8,"description":12,"dateModified":43,"datePublished":43,"encodingFormat":42,"isAccessibleForFree":44,"interactionStatistic":45},"DigitalDocument",{"name":37,"@type":38},"The UK Girl","Person",{"url":19,"name":40,"@type":41},"DocShare","Organization","application/pdf","2026-10-26",true,{"@type":46,"interactionType":47,"userInteractionCount":4},"InteractionCounter",{"@type":48},"ViewAction",{"@type":50,"mainEntity":51},"FAQPage",[52,58,62],{"name":53,"@type":54,"acceptedAnswer":55},"How does the Argand diagram represent a complex number?","Question",{"text":56,"@type":57},"A complex number is shown as a point in the Argand plane, with one coordinate for the real part and the other for the imaginary part.","Answer",{"name":59,"@type":54,"acceptedAnswer":60},"What do Re(z) and Im(z) mean?",{"text":61,"@type":57},"Re(z) gives the real part x of z, and Im(z) gives the imaginary part y of z. They act like projection functions from the plane to one dimension.",{"name":63,"@type":54,"acceptedAnswer":64},"How are polar coordinates and Cartesian coordinates related for complex numbers?",{"text":65,"@type":57},"For z = x + iy, the polar form uses r (the modulus) and arg z (the argument). Conversion formulas include arg z = tan^-1(y/x) and x = r cos(arg z), y = r sin(arg z).","https://schema.org",{"og:url":32,"og:type":68,"og:title":10,"og:site_name":40,"og:description":12},"article",{"robots":70,"canonical":32},"index,follow",{"doc_id":72,"site_id":7},657323,1792977390,{"code":4,"msg":75,"data":76},"success",[77,81,85,89,94,99,104,108,113,116,120],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":78,"show_sort_weight":79,"slug":80},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":82,"show_sort_weight":83,"slug":84},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},"Exam",70,"exam",{"id":90,"doc_module":4,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},5,"Comic",60,"comic",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},6,"Technology",50,"technology",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},7,"Healthcare",40,"healthcare",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":106,"slug":107},8,30,"research-report",{"id":109,"doc_module":4,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},9,"Religion & Spirituality",20,"religion-spirituality",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":114,"show_sort_weight":111,"slug":115},"World Cup","world-cup",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":118,"show_sort_weight":117,"slug":119},10,"Lifestyle","lifestyle",{"id":121,"doc_module":4,"doc_module_name":25,"category_name":122,"show_sort_weight":90,"slug":123},19,"General","general",{"code":4,"msg":75,"data":125},{"doc_id":72,"user_id":126,"nickname":37,"user_avatar":127,"doc_module":4,"category_id":105,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":128,"file_id":129,"file_url":130,"file_type":131,"file_size":132,"view_count":4,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":100,"language":133,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":12,"update_tm":73,"read_time":137},2199028711991,"https://us-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Complex Numbers and the Argand plane  \nThe Argand Diagram  \nThe Argand diagram is a graphical representation of a complex number that enables us to go some way towards visualising what a complex number is. A complex number is an abstract entity with no physical counterpart, but we can represent a complex number as a point in a plane where one coordinate represents the real part of the number and the other coordinate represents the imaginary part. The plane is called the Argand plane and the diagram is called the Argand diagram.  \nimaginary  \nz  \ny  \nx real  \nExample (1)  \nPlot the complex numbers z1 􀀠 1 􀀎 2i  \nz2 􀀠 􀀐2 􀀎 iz3 􀀠 z1 􀀎 z2  \nz4 􀀠 z2 􀀐 z1  \non the Argand plane  \nSolution  \nz1  \nz2  \nz3  \nz4  \n􀀠 1 􀀎 2i  \n􀀠 􀀐2 􀀎 i  \n􀀠 z1 􀀎 z2 􀀠 􀀋 1 􀀎 2i 􀀌 􀀎 􀀋􀀐2 􀀎 i 􀀌 􀀠 􀀐1 􀀎 3i  \n􀀠 z2 􀀐 z1 􀀠 􀀐􀀋2 􀀎 i 􀀌 􀀐 􀀋 1 􀀎 2i 􀀌 􀀠 􀀐3 􀀐 i  \n© [blacksacademy.net](blacksacademy.net)[ ](blacksacademy.net)1  \nOn the Argand diagram these points are  \n\n|  |  | z 3 |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  | 3 |  | z 1 |  |  |\n|  |  |  | 2 |  |  |  |  |\n|  | z 2 |  | 1 |  |  |  | |\n|  | -3 | -2 | 1 0 |  | 1 | 2 | 3  |\n| z4 |  |  | -1 |  |  |  |  |\n|  |  |  | -2 |  |  |  |  |\n|  |  |  | -3 |  |  |  |  |\n\ny  \nx  \n-  \nReal and imaginary parts  \nThe complex number z 􀀠 x 􀀎 iy has a real part x and an imaginary part y. This shows that it is a two dimensional object. Since it has two dimensions it is, in fact, a vector. The set of all complex numbers is denoted by the symbol 􀁞 and is a two-dimensional space of vectors. It is useful to have functions that pick out the real and imaginary parts of a complex number. These functions called the real and imaginary parts of the complex number, z.  \nIf z 􀀠 x 􀀎 iy  \nRe 􀀋 z 􀀌 􀀠 x  \nIm 􀀋 z 􀀌 􀀠 y  \nRe 􀀋 z 􀀌  \nis read, \"the real part of z\" .  \nIm 􀀋 z 􀀌  \nis read, \"the imaginary part of z\" .  \nExample (2)  \nFind Re 􀀋 z 􀀌 and Im 􀀋 z 􀀌 where z 􀀠 􀀐2 􀀐 4i  \nSolution  \nRe 􀀋 z 􀀌 􀀠 􀀐2 Im 􀀋 z 􀀌 􀀠 􀀐4  \nThe functions Re 􀀋 z 􀀌 and Im 􀀋 z 􀀌 are also called projection functions because they create a onedimensional image of a two-dimensional object. The Argand diagram makes it clear that if two  \n© [blacksacademy.net](blacksacademy.net)[ ](blacksacademy.net)2  \ncomplex numbers, z and w, have the same real and imaginary parts, then they must be the same number. This is written formally  \nIf  \nthen  \nRe 􀀋 z 􀀌 􀀠 Re 􀀋 w 􀀌 z 􀀠 w  \nand  \nIm 􀀋 z 􀀌 􀀠 Im 􀀋 w 􀀌  \nand conversely  \nThe phrase “and conversely” means that the result works the other way around; that is  \nIf  \nthen  \nz 􀀠 w  \nRe 􀀋 z 􀀌 􀀠 Re 􀀋 w 􀀌  \nand  \nIm 􀀋 z 􀀌 􀀠 Im 􀀋 w 􀀌  \nAnother way of writing this is  \nRe 􀀋 z 􀀌 􀀠 Re 􀀋 w 􀀌 and Im 􀀋 z 􀀌 􀀠 Im 􀀋 w 􀀌 iff z 􀀠 w  \nwhere iff is short for “if, and only if”. There is also a symbolic way of writing this  \nRe 􀀋 z 􀀌 􀀠 Re 􀀋 w 􀀌 and Im 􀀋 z 􀀌 􀀠 Im 􀀋 w 􀀌 􀂜 z 􀀠 w  \nThey all mean the same thing, namely, that the left hand side is interchangeable with the righthand side.  \nThe polar coordinates of a complex number  \nSince a complex number is a vector, it is possible to specify the locus (position) of a complex number in terms of an angle and a range. In other words, there is a polar representation of a complex number.  \nThe polar representation is denoted r, 􀁔 , where r = the distance of the complex number z from the origin in the Argand plane. This distance is also called the modulus of the complex number z,  \n© [blacksacademy.net](blacksacademy.net)[ ](blacksacademy.net)3  \nand is denoted z  . That is r 􀀠 z  . 􀁔 􀀠 the angle at which z lies measured relative to the real axis, and taken in a clockwise sense. This angle is called the argument of the complex number zand is also denoted arg z.  \nThe argument of z is often measured in radians. As the diagram indicates, the polar representation of a complex number is not unique since the complex number with argument 􀁔 is the same as the complex number with argument 􀁔 􀀎 􀀕 􀁓 radians. This usually does not cause a difficulty, and can be a definite advantage, but if necessary we can c","cbCaiaXEvOVnRtgC","https://ap.wps.com/l/cbCaiaXEvOVnRtgC","pdf",483213,"English","# The Argand diagram\n## Real and imaginary parts\n## Projection functions and equality of complex numbers\n# The polar coordinates of a complex number\n## Modulus and argument\n## Principal value and negative arguments\n## Converting between Cartesian and polar coordinates","[{\"question\":\"How does the Argand diagram represent a complex number?\",\"answer\":\"A complex number is shown as a point in the Argand plane, with one coordinate for the real part and the other for the imaginary part.\"},{\"question\":\"What do Re(z) and Im(z) mean?\",\"answer\":\"Re(z) gives the real part x of z, and Im(z) gives the imaginary part y of z. They act like projection functions from the plane to one dimension.\"},{\"question\":\"How are polar coordinates and Cartesian coordinates related for complex numbers?\",\"answer\":\"For z = x + iy, the polar form uses r (the modulus) and arg z (the argument). Conversion formulas include arg z = tan^-1(y/x) and x = r cos(arg z), y = r sin(arg z).\"}]","Complex Numbers and the Argand plane - Graphical representation - Real and imaginary parts, modulus, argument, and polar/Cartesian conversion | PDF",18]