[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657318-105":59,"doc-detail-657318-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","argand-diagrams-complex-numbers","Argand diagrams - Complex numbers","","Argand diagrams represent complex numbers using a Cartesian-like plane with real and imaginary axes. The material explains how to plot points or vectors for given complex numbers and how vector addition and subtraction correspond to geometric constructions. It further covers modulus and argument, including computing these values with distance and angle reasoning and expressing numbers in modulus-argument form. Locus descriptions in the Argand plane are used to identify circles, perpendicular bisectors, and half-lines from equations involving |z−z0| and arg(z−z0).",{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/argand-diagrams-complex-numbers/657318/",{"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/argand-diagrams-complex-numbers/657318.png","ImageObject",300,407,{"name":92,"@type":93},"putri.kecil17","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do Argand diagrams represent complex numbers geometrically?","Question",{"text":112,"@type":113},"They map a complex number to a point (or vector) on a plane: the real part gives the horizontal position and the imaginary part gives the vertical position.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are modulus and argument in the context of complex numbers?",{"text":117,"@type":113},"The modulus is the distance from the origin to the point representing the complex number, and the argument is the angle between the positive real axis and the line from the origin to that point.",{"name":119,"@type":110,"acceptedAnswer":120},"How do you find loci on the Argand diagram using equations?",{"text":121,"@type":113},"Use the given equation to identify the corresponding geometric set: for example, |z−z0|=r gives a circle centered at (Re(z0), Im(z0)), while equal distances from two points give a perpendicular bisector and fixed argument gives a half-line.","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},657318,1792977381,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":81},962094509873,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Argand diagrams  \nArgand diagrams are used to represent complex numbers. This representation allows us to see the effects of different moduli and arguments, therefore giving us a new way of denoting complex numbers and allows us to solve equations and inequalities graphically using loci.  \nArgand diagrams  \nArgand diagrams look similar to Cartesian diagrams – these are the graphs that you are used to seeing, with an 􀀡 and 􀀢 axis. As you have seen in the previous chapter, complex numbers have real and imaginary parts. Argand diagrams have the real part of the complex number, denoted 􀀣􀀤 on what would be the 􀀡 axis in a Cartesian diagram, and the imaginary part, 􀀥􀀦, on the 􀀢 axis.  \n• The complex number 􀀧 = 􀀡 + 􀀪􀀢 can be represented by the point 􀀫(􀀡, 􀀢) or the vector /􀀡􀀢0 Example 1: Represent the complex numbers 􀀣! = 3 + 2􀀪 , 􀀣\" = −4 − 􀀪 and 􀀣' = 3􀀪 on an Argand diagram  \n\n| The real part of each number tells us the horizontal position and the imaginary part tells us the vertical position, so 􀀣! is represented by the point that is 3 across and 2 up, shown by the red dot. 􀀣\" and 􀀣' are denoted by blue and green dots respectively. | \u003Cbr>􀀮􀀯\u003Cbr>⋅\u003Cbr>⋅\u003Cbr>⋅\u003Cbr>􀀬􀀭 |\n| --- | --- |\n\n• By using the vector of a complex number, the addition or subtraction of complex numbers can be shown on the argand diagram  \nExample 2: For 􀀣! = 1 + 3􀀪 , 􀀣\" = −2 + 􀀪 , 􀀣' = 2 − 􀀪 and 􀀣( = 2􀀪 , show 􀀣! + 􀀣\" and 􀀣' − 􀀣( on an Argand diagram.  \n\n| For 􀀣! + 􀀣\", plot the vectors of 􀀣! and 􀀣\" on the Argand diagram. As with adding vectors, place them ‘nose-totail’ and notice that the resultant vector is the diagonal of the parallelogram formed. | \u003Cbr>􀀫! + 􀀫\" 􀀩􀀪\u003Cbr>\u003Cbr>􀀧􀀨 |\n| --- | --- |\n| Follow the same process for 􀀣' − 􀀣(, but plot −􀀣( instead of 􀀣( . | \u003Cbr>􀀩􀀪\u003Cbr>\u003Cbr>􀀰$\u003Cbr>􀀧􀀨\u003Cbr>􀀰$ − 􀀰\\# |\n\nModulus and Argument  \n• The modulus of the complex number 􀀣 = 􀁁 + 􀀪􀁂, denoted |􀀣|, is the distance from the origin to the point represented by that number on an Argand diagram, and is given by |􀀣| = 1􀁁 \" + 􀁂 \".  \n• The argument of a complex number 􀀣 = 􀁁 + 􀀪􀁂, denoted arg 􀀣, is the angle −􀁃 ≤ 􀁄 ≤ 􀁃 between the positive real axis and the line joining the point represented by 􀀣 to the origin. The argument satisfies tan 􀁄 = )%~~ ~~ .  \nExample 3: Find the modulus and argument of the complex number 􀀣 = −3 + 4􀀪  \n\n| Mark the point represented by 􀀣 and draw the line connecting 􀀣 to the origin.\u003Cbr>continued à | \u003Cbr>|\n| --- | --- |\n\n\n| To find |􀀣| we need to find the length of the line connecting the origin and 􀀣, which can clearly be done by Pythagoras’ theorem. | |􀀣| = 13\" + 4\" = 5 |\n| --- | --- |\n| To find arg 􀀣, we need to find the angle from the positive real axis to the line connecting the origin and 􀀣 . Clearly the angle from the positive to the negative real axis is 􀁃 radians, so the angle we are looking for is 􀁃 −(the angle between the line and the negative real axis) . | arg 􀀣 = 􀁃 − (tan*! 43) arg 􀀣 = 2.214 rad |\n\nIt is important to make sure that you find the right angle for the argument-it is not always tan*! )% . To ensure that the correct angle is calculated, it is recommended to draw a quick sketch.  \nModulus-argument form of complex numbers  \n• For a complex number 􀀣 with |􀀣| = 􀀦 and arg 􀀣 = 􀁄 , the modulus argument form of 􀀣 is  \n􀀣 = 􀀦(cos 􀁄 + 􀀪 sin 􀁄)  \nExample 4: Express the complex number 􀁌 = 2 − √5􀀪 in modulus-argument form  \n\n| Draw a sketch of the complex number on the diagram. | 2 􀀧􀀨\u003Cbr>|\n| --- | --- |\n| Find the modulus, 􀀦, and the argument 􀁄 . | 􀀦 = M2\" + (  −√5)\" = 3\u003Cbr>􀁄 = tan*!(~~ ~~−~~ ~~√25) = −0 .841 |\n| Put into modulus-argument form. | 􀁌 = 3(cos(−0 . 841) + 􀀪 sin(−0 . 841)) |\n\nMultiplying and dividing complex numbers in modulus argument form is simple using certain results:  \n• For any two complex numbers 􀀣! and 􀀣\":  \n|􀀣!􀀣\" | = |􀀣! ||􀀣\" |  \narg (􀀣!􀀣\") = arg (􀀣!) + arg(􀀣\")  \nP􀀣􀀣\"!~~ ~~ P = 􀀣􀀣\"!~~ ~~  \narg Q􀀣􀀣\"!R = arg 􀀣! − arg 􀀣\"  \nEx􀁁m􀀪~~p~~􀁂le~~ ~~5: For 􀀣! = 2(cos  + 􀀪 sin ) and 􀀣\" = 4(cos &' + 􀀪 sin &'), find 􀁌 = 􀀣!􀀣\" in the forms 􀀦(cos 􀁄 + 􀀪 sin ","cbCairOc7AY1p4Fo","https://ap.wps.com/l/cbCairOc7AY1p4Fo","pdf",313067,"English","# Argand diagrams\n## Plotting complex numbers\n## Vector addition and subtraction\n# Modulus and argument\n## Modulus of a complex number\n## Argument of a complex number\n## Modulus-argument form\n# Multiplying and dividing in modulus-argument form\n# Loci and regions in the Argand diagram","[{\"question\":\"How do Argand diagrams represent complex numbers geometrically?\",\"answer\":\"They map a complex number to a point (or vector) on a plane: the real part gives the horizontal position and the imaginary part gives the vertical position.\"},{\"question\":\"What are modulus and argument in the context of complex numbers?\",\"answer\":\"The modulus is the distance from the origin to the point representing the complex number, and the argument is the angle between the positive real axis and the line from the origin to that point.\"},{\"question\":\"How do you find loci on the Argand diagram using equations?\",\"answer\":\"Use the given equation to identify the corresponding geometric set: for example, |z−z0|=r gives a circle centered at (Re(z0), Im(z0)), while equal distances from two points give a perpendicular bisector and fixed argument gives a half-line.\"}]","Argand diagrams - Complex numbers | PDF"]