{"id":1390,"date":"2026-01-27T20:15:41","date_gmt":"2026-01-27T10:15:41","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1390"},"modified":"2026-01-28T19:49:46","modified_gmt":"2026-01-28T09:49:46","slug":"year12-math-4-3-2-complex-numbers","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1390","title":{"rendered":"Year12 MATH 4-3-2 complex numbers"},"content":{"rendered":"\n

In QLD Year 12 Specialist Mathematics (Unit 3), the topic of Complex Numbers<\/strong> extends the foundational knowledge introduced in Units 1 & 2 (Cartesian form a<\/mi>+<\/mo>b<\/mi>i<\/mi><\/mrow>a plus b i<\/annotation><\/semantics><\/math>) into more advanced forms, including polar and exponential representations. This topic is a core component of the “Further Complex Numbers” module, which focuses on manipulating complex numbers to solve polynomial equations and geometric problems. <\/p>\n\n\n\n

Here is a breakdown of the complex numbers curriculum for Unit 3: <\/p>\n\n\n\n

Key Topics in Unit 3 Complex Numbers <\/strong><\/h4>\n\n\n\n
    \n
  • Forms of Complex Numbers:<\/strong> Beyond z<\/mi>=<\/mo>a<\/mi>+<\/mo>b<\/mi>i<\/mi><\/mrow>z equals a plus b i<\/annotation><\/semantics><\/math> (Cartesian), you will learn the Polar form (z<\/mi>=<\/mo>r<\/mi>(<\/mo>cos<\/mi>\u03b8<\/mi>+<\/mo>i<\/mi>sin<\/mi>\u03b8<\/mi>)<\/mo><\/mrow>z equals r open paren cosine theta plus i sine theta close paren<\/annotation><\/semantics><\/math> and Exponential form (z<\/mi>=<\/mo>r<\/mi>e<\/mi>i<\/mi>\u03b8<\/mi><\/mrow><\/msup><\/mrow>z equals r e raised to the i theta power<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
  • De Moivre’s Theorem:<\/strong> Applying z<\/mi>n<\/mi><\/msup>=<\/mo>[<\/mo>r<\/mi>(<\/mo>cos<\/mi>\u03b8<\/mi>+<\/mo>i<\/mi>sin<\/mi>\u03b8<\/mi>)<\/mo>]<\/mo>n<\/mi><\/msup>=<\/mo>r<\/mi>n<\/mi><\/msup>(<\/mo>cos<\/mi>n<\/mi>\u03b8<\/mi>+<\/mo>i<\/mi>sin<\/mi>n<\/mi>\u03b8<\/mi>)<\/mo><\/mrow>z to the n-th power equals open bracket r open paren cosine theta plus i sine theta close paren close bracket to the n-th power equals r to the n-th power open paren cosine n theta plus i sine n theta close paren<\/annotation><\/semantics><\/math> to raise complex numbers to powers.<\/li>\n\n\n\n
  • Roots of Complex Numbers:<\/strong> Using De Moivre\u2019s theorem to find the n<\/mi>n<\/annotation><\/semantics><\/math>-th roots of a complex number.<\/li>\n\n\n\n
  • Roots of Unity:<\/strong> Solving equations of the form z<\/mi>n<\/mi><\/msup>=<\/mo>1<\/mn><\/mrow>z to the n-th power equals 1<\/annotation><\/semantics><\/math>, which geometrically represent vertices of a regular polygon on the Argand plane.<\/li>\n\n\n\n
  • Factorising Polynomials over C:<\/strong> Factorising quadratic and cubic polynomials using complex roots, including the Conjugate Root Theorem (if P<\/mi>(<\/mo>z<\/mi>)<\/mo><\/mrow>cap P open paren z close paren<\/annotation><\/semantics><\/math> has real coefficients, complex roots occur in conjugate pairs).<\/li>\n\n\n\n
  • The Factor and Remainder Theorems:<\/strong> Extended for use in factorising polynomials over the complex field. <\/li>\n<\/ul>\n\n\n\n

    Key Skills and Applications <\/strong><\/h4>\n\n\n\n
      \n
    • Geometric Interpretation:<\/strong> Using Argand diagrams to represent complex numbers, conjugates, modulus (|<\/mo>z<\/mi>|<\/mo><\/mrow>the absolute value of z end-absolute-value<\/annotation><\/semantics><\/math>), and argument (arg<\/mtext>(<\/mo>z<\/mi>)<\/mo><\/mrow>arg open paren z close paren<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
    • Solving Equations:<\/strong> Finding complex solutions to quadratic and cubic equations.<\/li>\n\n\n\n
    • Transformations:<\/strong> Understanding multiplication by a fixed complex number as a rotation and dilation in the complex plane. <\/li>\n<\/ul>\n\n\n\n

      Context in QLD Curriculum <\/strong><\/h4>\n\n\n\n
        \n
      • Unit 3:<\/strong> Covers “Further Complex Numbers” as part of the three main topics (with Vectors in 3D and Functions).<\/li>\n\n\n\n
      • Assessment:<\/strong> Knowledge of this topic is assessed in the internal (IA2\/IA3) exams and the external assessment (EA).<\/li>\n\n\n\n
      • Prerequisites:<\/strong> Assumes knowledge of i<\/mi>=<\/mo>-1<\/mn><\/msqrt><\/mrow>i equals the square root of negative 1 end-root<\/annotation><\/semantics><\/math>, conjugate pairs, and Cartesian operations (a<\/mi>+<\/mo>b<\/mi>i<\/mi><\/mrow>a plus b i<\/annotation><\/semantics><\/math>) from Unit 2.\u00a0<\/li>\n<\/ul>\n\n\n\n

        ******************************************************************************************<\/p>\n\n\n\n

        QLD 12\u5e74\u751f\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c83\uff09\u306e\u8907\u7d20\u6570\u306f\u3001\u30e6\u30cb\u30c3\u30c81\u30682\u3067\u5c0e\u5165\u3055\u308c\u305f\u57fa\u790e\u77e5\u8b58\uff08\u76f4\u4ea4\u5ea7\u6a19\u5f62\u5f0f (a+bi)\uff09\u3092\u3001\u6975\u5ea7\u6a19\u3084\u6307\u6570\u95a2\u6570\u8868\u73fe\u3092\u542b\u3080\u3088\u308a\u9ad8\u5ea6\u306a\u5f62\u5f0f\u3078\u3068\u767a\u5c55\u3055\u305b\u307e\u3059\u3002\u3053\u306e\u30c8\u30d4\u30c3\u30af\u306f\u3001\u300c\u8907\u7d20\u6570\u5165\u9580\u300d\u30e2\u30b8\u30e5\u30fc\u30eb\u306e\u4e2d\u6838\u3092\u6210\u3059\u3082\u306e\u3067\u3001\u3053\u306e\u30e2\u30b8\u30e5\u30fc\u30eb\u3067\u306f\u8907\u7d20\u6570\u3092\u64cd\u4f5c\u3057\u3066\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f\u3084\u5e7e\u4f55\u5b66\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

        \u30e6\u30cb\u30c3\u30c83\u306e\u8907\u7d20\u6570\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n

        \u30e6\u30cb\u30c3\u30c83\u306e\u8907\u7d20\u6570\u4e3b\u8981\u30c8\u30d4\u30c3\u30af<\/p>\n\n\n\n

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        \u5358\u4f4d\u6839: (z<\/mi>n<\/mi><\/msup>=<\/mo>1<\/mn><\/mrow>z^{n}=1<\/annotation><\/semantics><\/math>) \u306e\u5f62\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u30a2\u30eb\u30ac\u30f3\u5e73\u9762\u4e0a\u306e\u6b63\u591a\u89d2\u5f62\u306e\u9802\u70b9\u3092\u5e7e\u4f55\u5b66\u7684\u306b\u8868\u3057\u307e\u3059\u3002
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        \u56e0\u6570\u5b9a\u7406\u3068\u5270\u4f59\u5b9a\u7406\uff1a\u8907\u7d20\u4f53\u4e0a\u306e\u591a\u9805\u5f0f\u306e\u56e0\u6570\u5206\u89e3\u306b\u5fdc\u7528\u3059\u308b\u305f\u3081\u306b\u62e1\u5f35\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

        \u4e3b\u8981\u30b9\u30ad\u30eb\u3068\u5fdc\u7528<\/strong><\/p>\n\n\n\n

        \u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\uff1a\u30a2\u30eb\u30ac\u30f3\u56f3\u3092\u7528\u3044\u3066\u8907\u7d20\u6570\u3001\u5171\u5f79\u3001\u6cd5 (|z|)\u3001\u504f\u89d2 (arg<\/mtext>(<\/mo>z<\/mi>)<\/mo><\/mrow>\\text{arg}(z)<\/annotation><\/semantics><\/math>) \u3092\u8868\u3057\u307e\u3059\u3002
        \u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\uff1a\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3068\u4e09\u6b21\u65b9\u7a0b\u5f0f\u306e\u8907\u7d20\u89e3\u3092\u6c42\u3081\u307e\u3059\u3002
        \u5909\u63db\uff1a\u56fa\u5b9a\u8907\u7d20\u6570\u306b\u3088\u308b\u4e57\u7b97\u3092\u3001\u8907\u7d20\u5e73\u9762\u306b\u304a\u3051\u308b\u56de\u8ee2\u3068\u62e1\u5927\u3068\u3057\u3066\u7406\u89e3\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

        QLD\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u306b\u304a\u3051\u308b\u6587\u8108<\/strong>
        \u30e6\u30cb\u30c3\u30c83\uff1a3\u3064\u306e\u4e3b\u8981\u30c8\u30d4\u30c3\u30af\uff083\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u3068\u95a2\u6570\u3092\u542b\u3080\uff09\u306e\u4e00\u90e8\u3068\u3057\u3066\u300c\u3055\u3089\u306a\u308b\u8907\u7d20\u6570\u300d\u3092\u6271\u3044\u307e\u3059\u3002
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        \u524d\u63d0\u6761\u4ef6: \u30e6\u30cb\u30c3\u30c8 2 \u306e (i<\/mi>=<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/msqrt><\/mrow>i=\\sqrt{-1}<\/annotation><\/semantics><\/math>)\u3001\u5171\u5f79\u5bfe\u3001\u304a\u3088\u3073\u76f4\u4ea4\u5ea7\u6a19\u6f14\u7b97 (a<\/mi>+<\/mo>b<\/mi>i<\/mi><\/mrow>a+bi<\/annotation><\/semantics><\/math>) \u306b\u95a2\u3059\u308b\u77e5\u8b58\u3092\u524d\u63d0\u3068\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

        <\/p>\n","protected":false},"excerpt":{"rendered":"

        In QLD Year 12 Specialist Mathematics (Unit 3), the topic of Complex Numbers extends the foundational knowledge introduced in Units 1 & 2 (Cartesian form a+bia plus b i) into more advanced forms, including polar and exponential representations. This topic is a core component of the “Further Complex Numbers” module, which focuses on manipulating complex […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1390","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1390","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1390"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1390\/revisions"}],"predecessor-version":[{"id":1456,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1390\/revisions\/1456"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1390"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1390"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}