{"id":1211,"date":"2026-01-21T14:56:34","date_gmt":"2026-01-21T04:56:34","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1211"},"modified":"2026-01-21T14:56:34","modified_gmt":"2026-01-21T04:56:34","slug":"year11-math-4-1-6-real-and-complex-numbers","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1211","title":{"rendered":"Year11-MATH-4-1-6 Real and Complex Numbers"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In Queensland Year 11 Maths, &#8220;Real and Complex Numbers&#8221; is <mark>a core topic in the <strong><a href=\"https:\/\/www.google.com\/search?q=Mathematical+Methods&amp;sca_esv=fb6ec2c420ab31bb&amp;sxsrf=ANbL-n5JIhW89QJma7JyIO0-E62tdpRiAw%3A1768902611499&amp;ei=009vaZKKHvSk1e8P7o3HyQ8&amp;ved=2ahUKEwijsrDx65mSAxXcdvUHHQ11PGEQgK4QegQIARAC&amp;oq=What+is+%22Real+and+Complex+Numbers%22%2C+a+subject+taught+in+Unit+2+of+Mathematics+in+Grade+11+in+Queensland%3F&amp;gs_lp=Egxnd3Mtd2l6LXNlcnAiaFdoYXQgaXMgIlJlYWwgYW5kIENvbXBsZXggTnVtYmVycyIsIGEgc3ViamVjdCB0YXVnaHQgaW4gVW5pdCAyIG9mIE1hdGhlbWF0aWNzIGluIEdyYWRlIDExIGluIFF1ZWVuc2xhbmQ_SABQAFgAcAB4AZABAJgBAKABAKoBALgBDMgBAPgBAvgBAZgCAKACAJgDAOIDBRIBMSBAkgcAoAcAsgcAuAcAwgcAyAcAgAgA&amp;sclient=gws-wiz-serp&amp;mstk=AUtExfCKBxI72UIyXd4pUoUYxDrPFvApxM636U6oy3txXNQBfd5cSM92646Axsyx6bwV_dbp9GatToR_wko_ymoJ9QxcEHunAw4mkebRR-0dRv0ql4GI7ltza2s8TGv8hZvm-a-48h9agGcTqwN6mVXr2M1SIHrII3MnSf14hTTTlrSg_oI8in9lWp_AavBHccQMwOLORzfBKH6rgo7Qbu5Tf7i6Zg&amp;csui=3\">Mathematical Methods<\/a><\/strong> syllabus (Units 1 &amp; 2) and a foundational domain in <strong><a href=\"https:\/\/www.google.com\/search?q=Specialist+Mathematics&amp;sca_esv=fb6ec2c420ab31bb&amp;sxsrf=ANbL-n5JIhW89QJma7JyIO0-E62tdpRiAw%3A1768902611499&amp;ei=009vaZKKHvSk1e8P7o3HyQ8&amp;ved=2ahUKEwijsrDx65mSAxXcdvUHHQ11PGEQgK4QegQIARAD&amp;oq=What+is+%22Real+and+Complex+Numbers%22%2C+a+subject+taught+in+Unit+2+of+Mathematics+in+Grade+11+in+Queensland%3F&amp;gs_lp=Egxnd3Mtd2l6LXNlcnAiaFdoYXQgaXMgIlJlYWwgYW5kIENvbXBsZXggTnVtYmVycyIsIGEgc3ViamVjdCB0YXVnaHQgaW4gVW5pdCAyIG9mIE1hdGhlbWF0aWNzIGluIEdyYWRlIDExIGluIFF1ZWVuc2xhbmQ_SABQAFgAcAB4AZABAJgBAKABAKoBALgBDMgBAPgBAvgBAZgCAKACAJgDAOIDBRIBMSBAkgcAoAcAsgcAuAcAwgcAyAcAgAgA&amp;sclient=gws-wiz-serp&amp;mstk=AUtExfCKBxI72UIyXd4pUoUYxDrPFvApxM636U6oy3txXNQBfd5cSM92646Axsyx6bwV_dbp9GatToR_wko_ymoJ9QxcEHunAw4mkebRR-0dRv0ql4GI7ltza2s8TGv8hZvm-a-48h9agGcTqwN6mVXr2M1SIHrII3MnSf14hTTTlrSg_oI8in9lWp_AavBHccQMwOLORzfBKH6rgo7Qbu5Tf7i6Zg&amp;csui=3\">Specialist Mathematics<\/a><\/strong> (Units 1 &amp; 2)<\/mark>, focusing on extending real number concepts (like surds, indices) and introducing complex numbers (operations, forms, applications) for advanced problem-solving in algebra and beyond, bridging to calculus and vectors.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What it covers:<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Real Numbers:<\/strong> Deep dive into indices, surds, logarithms, and algebraic manipulation of these, building on earlier knowledge.<\/li>\n\n\n\n<li><strong>Complex Numbers:<\/strong> Introduction to the imaginary unit (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math>), operations (addition, subtraction, multiplication, division), complex conjugate, modulus, argument, and polar\/rectangular forms, often linked to quadratic equations with no real roots.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key components of this topic in Year 11 include:&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Imaginary Unit (\ud835\udc56):<\/strong> Defining <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math> as the square root of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mn>-1<\/mn><annotation encoding=\"text\/plain\">negative 1<\/annotation><\/semantics><\/math> (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>-1<\/mn><\/mrow><annotation encoding=\"text\/plain\">i squared equals negative 1<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Complex Numbers Structure:<\/strong> Expressing numbers in the Cartesian form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"text\/plain\">z equals a plus b i<\/annotation><\/semantics><\/math> where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>a<\/mi><annotation encoding=\"text\/plain\">a<\/annotation><\/semantics><\/math> (real part) and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>b<\/mi><annotation encoding=\"text\/plain\">b<\/annotation><\/semantics><\/math> (imaginary part) are real numbers.<\/li>\n\n\n\n<li><strong>Operations:<\/strong> Performing addition, subtraction, multiplication, and division of complex numbers.<\/li>\n\n\n\n<li><strong>Complex Conjugates:<\/strong> Finding and using the conjugate (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mover accent=\"true\"><mi>z<\/mi><mo>\u0304<\/mo><\/mover><annotation encoding=\"text\/plain\">z bar<\/annotation><\/semantics><\/math>) of a complex number.<\/li>\n\n\n\n<li><strong>Argand Diagrams:<\/strong> Representing complex numbers graphically on a complex plane (horizontal axis for real numbers, vertical axis for imaginary numbers).<\/li>\n\n\n\n<li><strong>Solving Equations:<\/strong> Solving quadratic and polynomial equations that have complex solutions.<\/li>\n\n\n\n<li><strong>Polar Form:<\/strong> Expressing complex numbers in polar form, including modulus and argument.<\/li>\n\n\n\n<li><strong>Introduction to De Moivre\u2019s Theorem:<\/strong> Using De Moivre\u2019s Theorem for powers and roots.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Context in Queensland Curriculum&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Subject:<\/strong> This topic is specific to <strong>Specialist Mathematics<\/strong> (Unit 1 &amp; 2), which is often studied alongside Mathematical Methods.<\/li>\n\n\n\n<li><strong>Purpose:<\/strong> It acts as a tool for explaining abstract, complex relationships in scientific and technological endeavors.<\/li>\n\n\n\n<li><strong>Structure:<\/strong> It introduces the concepts in Year 11, setting the stage for further, more advanced applications in Year 12 Specialist Mathematics.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Skills Developed&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Representation:<\/strong> Representing complex numbers in both Cartesian (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"text\/plain\">a plus b i<\/annotation><\/semantics><\/math>) and polar (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><msup><mi>e<\/mi><mrow><mi>i<\/mi><mi>\u03b8<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"text\/plain\">r e raised to the i theta power<\/annotation><\/semantics><\/math>) forms.<\/li>\n\n\n\n<li><strong>Visualization:<\/strong> Graphing complex numbers and understanding their geometric interpretation (e.g., rotation, dilation).<\/li>\n\n\n\n<li><strong>Algebraic Manipulation:<\/strong> Solving equations and manipulating complex expressions.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it&#8217;s important:<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It provides essential tools for solving problems that pure real numbers can&#8217;t handle, especially in areas like electrical engineering, physics, and advanced pure mathematics.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In short, it&#8217;s about mastering advanced number systems to tackle complex equations and models, a key stepping stone in senior mathematics pathways.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Problems and Solutions<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Queensland Year 11 Specialist Mathematics (Unit 2), the &#8220;Complex Numbers&#8221; topic covers Cartesian (rectangular) and polar forms, basic operations, Argand diagrams, and solving equations.\u00a0Here are sample problems and solutions based on Queensland Specialist Maths resources.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Operations with Complex Numbers (Cartesian Form)\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: Let <math data-latex=\"(u=3-4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>=<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(u=3-4i)<\/annotation><\/semantics><\/math> and <math data-latex=\"(v=4+5i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>v<\/mi><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(v=4+5i)<\/annotation><\/semantics><\/math>. Evaluate (<math data-latex=\"2u-3v\"><semantics><mrow><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2u-3v<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute the values: <math data-latex=\"2(3-4i)-3(4+5i)\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2(3-4i)-3(4+5i)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Expand brackets: <math data-latex=\"6-8i-12-15i\"><semantics><mrow><mn>6<\/mn><mo>\u2212<\/mo><mn>8<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>\u2212<\/mo><mn>15<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">6-8i-12-15i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Group real and imaginary parts: <math data-latex=\"(6-12)+(-8i-15i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mn>12<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>8<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(6-12)+(-8i-15i)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify: <math data-latex=\"-6-23i\"><semantics><mrow><mo>\u2212<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mn>23<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-6-23i<\/annotation><\/semantics><\/math>\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: Multiply (<math data-latex=\"z_{1}=2-i\"><semantics><mrow><msub><mi>z<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z_{1}=2-i<\/annotation><\/semantics><\/math>) and <math data-latex=\"(z_{2}=3+4i\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>z<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(z_{2}=3+4i<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Expand (FOIL): <math data-latex=\"(2)(3)+(2)(4i)+(-i)(3)+(-i)(4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(2)(3)+(2)(4i)+(-i)(3)+(-i)(4i)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(<math data-latex=\"6+8i-3i-4i^{2}\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>8<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">6+8i-3i-4i^{2}<\/annotation><\/semantics><\/math>)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Use (<math data-latex=\"i^{2}=-1\"><semantics><mrow><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">i^{2}=-1<\/annotation><\/semantics><\/math>): <math data-latex=\"6+5i-4(-1)\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">6+5i-4(-1)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"6+5i+4\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6+5i+4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Result: \u00a0<math data-latex=\"10+5i\"><semantics><mrow><mn>10<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">10+5i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Conjugates and Division\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: Divide (<math data-latex=\"z=\\frac{1+i}{2-i}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{1+i}{2-i}<\/annotation><\/semantics><\/math>) and write the answer in (<math data-latex=\"a+bi\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+bi<\/annotation><\/semantics><\/math>) form.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Multiply<\/strong> numerator and denominator by the conjugate of the denominator (<math data-latex=\"2+i\"><semantics><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2+i<\/annotation><\/semantics><\/math>):          <math data-latex=\"\\frac{1+i}{2-i}\\times \\frac{2+i}{2+i}\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><\/mfrac><mo>\u00d7<\/mo><mfrac><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1+i}{2-i}\\times \\frac{2+i}{2+i}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Expand numerator<\/strong>: <math data-latex=\"(1)(2)+1(i)+i(2)+i^{2}=2+i+2i-1=1+3i\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>i<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(1)(2)+1(i)+i(2)+i^{2}=2+i+2i-1=1+3i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Expand denominator<\/strong>: <math data-latex=\"(2-i)(2+i)=4+2i-2i-i^{2}=4-(-1)=5\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2-i)(2+i)=4+2i-2i-i^{2}=4-(-1)=5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Result<\/strong>: \u00a0<math data-latex=\"\\frac{1+3i}{5}=\\frac{1}{5}+\\frac{3}{5}i\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><mn>5<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>5<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1+3i}{5}=\\frac{1}{5}+\\frac{3}{5}i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. Modulus and Argument (Polar Form)\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: Find the modulus and argument of <math data-latex=\"z=1-\\sqrt{3}i\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=1-\\sqrt{3}i<\/annotation><\/semantics><\/math> and write in <math data-latex=\"r(\\cos \\theta +i\\sin \\theta ) or (r\\text{cis}\\theta\"><semantics><mrow><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>+<\/mo><mi>i<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>o<\/mi><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>r<\/mi><mtext>cis<\/mtext><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(\\cos \\theta +i\\sin \\theta ) or (r\\text{cis}\\theta<\/annotation><\/semantics><\/math>) form.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Find modulus (<math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>)<\/strong>: <math data-latex=\"|z|=\\sqrt{a^{2}+b^{2}}=\\sqrt{1^{2}+(-\\sqrt{3})^{2}}=\\sqrt{1+3}=\\sqrt{4}=2\"><semantics><mrow><mi>|<\/mi><mi>z<\/mi><mi>|<\/mi><mo>=<\/mo><msqrt><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><msup><mn>1<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>4<\/mn><\/msqrt><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">|z|=\\sqrt{a^{2}+b^{2}}=\\sqrt{1^{2}+(-\\sqrt{3})^{2}}=\\sqrt{1+3}=\\sqrt{4}=2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Find argument<\/strong> <math data-latex=\"(\\theta )\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(\\theta )<\/annotation><\/semantics><\/math>: <math data-latex=\"\\tan \\theta =\\frac{b}{a}=\\frac{-\\sqrt{3}}{1}\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>b<\/mi><mi>a<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><\/mrow><mn>1<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta =\\frac{b}{a}=\\frac{-\\sqrt{3}}{1}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since (<math data-latex=\"a&gt;0\"><semantics><mrow><mi>a<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a&gt;0<\/annotation><\/semantics><\/math>) and (<math data-latex=\"b<0\"><semantics><mrow><mi>b<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b&lt;0<\/annotation><\/semantics><\/math>), (<math data-latex=\"z\"><semantics><mi>z<\/mi><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math>) is in the 4th quadrant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"\\theta =\\arctan (-\\sqrt{3})=-\\frac{\\pi }{3}\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mrow><mi>arctan<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta =\\arctan (-\\sqrt{3})=-\\frac{\\pi }{3}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Polar form<\/strong>: <math data-latex=\"2\\left(\\cos \\left(-\\frac{\\pi }{3}\\right)+i\\sin \\left(-\\frac{\\pi }{3}\\right)\\right)) or (2\\text{cis}\\left(-\\frac{\\pi }{3}\\right)\"><semantics><mrow><mn>2<\/mn><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>+<\/mo><mi>i<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>o<\/mi><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mtext>cis<\/mtext><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">2\\left(\\cos \\left(-\\frac{\\pi }{3}\\right)+i\\sin \\left(-\\frac{\\pi }{3}\\right)\\right)) or (2\\text{cis}\\left(-\\frac{\\pi }{3}\\right)<\/annotation><\/semantics><\/math>.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. Solving Complex Equations\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: Solve (<math data-latex=\"z^{2}+4z+13=0\"><semantics><mrow><msup><mi>z<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>z<\/mi><mo>+<\/mo><mn>13<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">z^{2}+4z+13=0<\/annotation><\/semantics><\/math>) over the complex field, giving answers in      (<math data-latex=\"a+bi\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+bi<\/annotation><\/semantics><\/math>) form.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Use Quadratic Formula <math data-latex=\"z=\\frac{-b\\pm \\sqrt{b^{2}-4ac}}{2a}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-b\\pm \\sqrt{b^{2}-4ac}}{2a}<\/annotation><\/semantics><\/math>  :  <math data-latex=\"z=\\frac{-4\\pm \\sqrt{4^{2}-4(1)(13)}}{2(1)}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><msup><mn>4<\/mn><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>13<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm \\sqrt{4^{2}-4(1)(13)}}{2(1)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify discriminant: <math data-latex=\"z=\\frac{-4\\pm \\sqrt{16-52}}{2}=\\frac{-4\\pm \\sqrt{-36}}{2}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><mn>16<\/mn><mo>\u2212<\/mo><mn>52<\/mn><\/mrow><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>36<\/mn><\/mrow><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm \\sqrt{16-52}}{2}=\\frac{-4\\pm \\sqrt{-36}}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Use <math data-latex=\"i\"><semantics><mi>i<\/mi><annotation encoding=\"application\/x-tex\">i<\/annotation><\/semantics><\/math>: <math data-latex=\"z=\\frac{-4\\pm 6i}{2}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm 6i}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Result: <math data-latex=\"z=-2\\pm 3i\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>2<\/mn><mo>\u00b1<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=-2\\pm 3i<\/annotation><\/semantics><\/math> (roots are <math data-latex=\"-2+3i\"><semantics><mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-2+3i<\/annotation><\/semantics><\/math> and <math data-latex=\"-2-3i)\"><semantics><mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">-2-3i)<\/annotation><\/semantics><\/math>\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>5. Exam-Style Question (QCAA Style)\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question<\/strong>: The complex number (<math data-latex=\"w\"><semantics><mi>w<\/mi><annotation encoding=\"application\/x-tex\">w<\/annotation><\/semantics><\/math>) is defined as (<math data-latex=\"w=1-2i\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">w=1-2i<\/annotation><\/semantics><\/math>). Determine <math data-latex=\"\\text{Im}(w^{4})\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>4<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(w^{4})<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong>:\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate <math data-latex=\"w^{2}: (1-2i)^{2}=1-4i+4i^{2}=1-4i-4=-3-4i\"><semantics><mrow><msup><mi>w<\/mi><mn>2<\/mn><\/msup><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>4<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">w^{2}: (1-2i)^{2}=1-4i+4i^{2}=1-4i-4=-3-4i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate <math data-latex=\"w^{4}\"><semantics><msup><mi>w<\/mi><mn>4<\/mn><\/msup><annotation encoding=\"application\/x-tex\">w^{4}<\/annotation><\/semantics><\/math> <math data-latex=\"(w^{2}\\times w^{2}): (-3-4i)(-3-4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><msup><mi>w<\/mi><mn>2<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(w^{2}\\times w^{2}): (-3-4i)(-3-4i)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">=<math data-latex=\"9+12i+12i+16i^{2}\"><semantics><mrow><mn>9<\/mn><mo>+<\/mo><mn>12<\/mn><mi>i<\/mi><mo>+<\/mo><mn>12<\/mn><mi>i<\/mi><mo>+<\/mo><mn>16<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">9+12i+12i+16i^{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">=<math data-latex=\"9+24i-16\"><semantics><mrow><mn>9<\/mn><mo>+<\/mo><mn>24<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">9+24i-16<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"=-7+24i\"><semantics><mrow><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>7<\/mn><mo>+<\/mo><mn>24<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">=-7+24i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Identify<\/strong> <math data-latex=\"\\text{Im}(z)\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>z<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(z)<\/annotation><\/semantics><\/math>: The imaginary part is the coefficient of <math data-latex=\"i\"><semantics><mi>i<\/mi><annotation encoding=\"application\/x-tex\">i<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer<\/strong>: <math data-latex=\"\\text{Im}(w^{4})=24\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>4<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(w^{4})=24<\/annotation><\/semantics><\/math>.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Concepts Summary\u00a0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"i^{2}=-1\"><semantics><mrow><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">i^{2}=-1<\/annotation><\/semantics><\/math>: Fundamental to simplifying all calculations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjugate<\/strong> <math data-latex=\"z\"><semantics><mi>z<\/mi><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math>: If <math data-latex=\"z=a+bi\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=a+bi<\/annotation><\/semantics><\/math>, <math data-latex=\"\\={z}=a-bi\"><semantics><mrow><mover><mi>z<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0;\">\u02c9<\/mo><\/mover><mo>=<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\={z}=a-bi<\/annotation><\/semantics><\/math>. Used for division and finding real parts of products.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Argand Diagram<\/strong>: Plots <math data-latex=\"z=a+bi\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=a+bi<\/annotation><\/semantics><\/math> as the coordinate <math data-latex=\"(a,b)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(a,b)<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>De Moivre\u2019s Theorem<\/strong>: <math data-latex=\"(\\text{r\\ cis\\ }\\theta )^{n}=r^{n}\\text{cis\\ }(n\\theta )\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mtext>r&nbsp;cis&nbsp;<\/mtext><mi>\u03b8<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>=<\/mo><msup><mi>r<\/mi><mi>n<\/mi><\/msup><mtext>cis&nbsp;<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(\\text{r\\ cis\\ }\\theta )^{n}=r^{n}\\text{cis\\ }(n\\theta )<\/annotation><\/semantics><\/math> (used for powers\/roots).\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e11\u5e74\u751f\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u300c\u5b9f\u6570\u3068\u8907\u7d20\u6570\u300d\u306f\u6570\u5b66\u7684\u65b9\u6cd5\u8ad6\u30b7\u30e9\u30d0\u30b9\uff08\u30e6\u30cb\u30c3\u30c81\u304a\u3088\u30732\uff09\u306e\u4e3b\u8981\u30c8\u30d4\u30c3\u30af\u3067\u3042\u308a\u3001\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c81\u304a\u3088\u30732\uff09\u306e\u57fa\u790e\u9818\u57df\u3067\u3082\u3042\u308a\u307e\u3059\u3002\u5b9f\u6570\u306e\u6982\u5ff5\uff08\u7121\u7406\u6570\u3001\u6307\u6570\u306a\u3069\uff09\u306e\u62e1\u5f35\u3068\u8907\u7d20\u6570\uff08\u6f14\u7b97\u3001\u5f62\u5f0f\u3001\u5fdc\u7528\uff09\u306e\u5c0e\u5165\u306b\u7126\u70b9\u3092\u5f53\u3066\u3001\u4ee3\u6570\u3092\u306f\u3058\u3081\u3068\u3059\u308b\u9ad8\u5ea6\u306a\u554f\u984c\u89e3\u6c7a\u80fd\u529b\u3092\u990a\u3044\u3001\u5fae\u7a4d\u5206\u3084\u30d9\u30af\u30c8\u30eb\u3078\u3068\u6a4b\u6e21\u3057\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5b66\u7fd2\u5185\u5bb9<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5b9f\u6570\uff1a\u6307\u6570\u3001\u7121\u7406\u6570\u3001\u5bfe\u6570\u3001\u305d\u3057\u3066\u305d\u308c\u3089\u306e\u4ee3\u6570\u7684\u64cd\u4f5c\u306b\u3064\u3044\u3066\u3001\u65e2\u5b58\u306e\u77e5\u8b58\u3092\u57fa\u306b\u6df1\u304f\u6398\u308a\u4e0b\u3052\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8907\u7d20\u6570\uff1a\u865a\u6570\u5358\u4f4d\uff08\\\u200b\u200b(i)\uff09\u3001\u6f14\u7b97\uff08\u52a0\u7b97\u3001\u6e1b\u7b97\u3001\u4e57\u7b97\u3001\u9664\u7b97\uff09\u3001\u8907\u7d20\u5171\u5f79\u3001\u4fc2\u6570\u3001\u504f\u89d2\u3001\u6975\u5f62\u5f0f\uff0f\u76f4\u4ea4\u5ea7\u6a19\u5f62\u5f0f\u306e\u6982\u8981\u3002\u3053\u308c\u3089\u306f\u5b9f\u6839\u3092\u6301\u305f\u306a\u3044\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3068\u95a2\u9023\u4ed8\u3051\u3089\u308c\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">11\u5e74\u751f\u306b\u304a\u3051\u308b\u3053\u306e\u30c8\u30d4\u30c3\u30af\u306e\u4e3b\u8981\u306a\u69cb\u6210\u8981\u7d20\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u865a\u6570\u5358\u4f4d <\/strong>: <math data-latex=\"(i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(i)<\/annotation><\/semantics><\/math>\u3092 (-1) \u306e\u5e73\u65b9\u6839 \u3068\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<math data-latex=\"(i^{2}=-1)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(i^{2}=-1)<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>\u8907\u7d20\u6570\u69cb\u9020<\/strong>: \u6570\u3092\u76f4\u4ea4\u5ea7\u6a19\u5f62\u5f0f (<math data-latex=\"z=a+bi\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=a+bi<\/annotation><\/semantics><\/math>) \u3067\u8868\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001(a) (\u5b9f\u90e8) \u3068 (b) (\u865a\u90e8) \u306f\u5b9f\u6570\u3067\u3059\u3002<\/li>\n\n\n\n<li><strong>\u6f14\u7b97<\/strong>: \u8907\u7d20\u6570\u306e\u52a0\u7b97\u3001\u6e1b\u7b97\u3001\u4e57\u7b97\u3001\u9664\u7b97\u3092\u5b9f\u884c\u3057\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>\u8907\u7d20\u5171\u5f79<\/strong>: \u8907\u7d20\u6570\u306e\u5171\u5f79 (<math data-latex=\"z\"><semantics><mi>z<\/mi><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u3001\u4f7f\u7528\u3057\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>\u30a2\u30fc\u30ac\u30f3\u30c9\u56f3<\/strong>: \u8907\u7d20\u6570\u3092\u8907\u7d20\u5e73\u9762\u4e0a\u306b\u30b0\u30e9\u30d5\u3067\u8868\u3057\u307e\u3059 (\u6a2a\u8ef8\u306b\u5b9f\u6570\u3001\u7e26\u8ef8\u306b\u865a\u6570)\u3002<\/li>\n\n\n\n<li><strong>\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5<\/strong>: \u8907\u7d20\u89e3\u3092\u6301\u3064\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3068\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>\u6975\u5f62\u5f0f<\/strong>\uff1a\u8907\u7d20\u6570\u3092\u6975\u5f62\u5f0f\u3067\u8868\u73fe\u3057\u3001\u4fc2\u6570\u3068\u504f\u89d2\u3092\u542b\u3081\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u5165\u9580<\/strong>\uff1a\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u3079\u304d\u4e57\u3068\u6839\u53f7\u306b\u9069\u7528\u3057\u307e\u3059\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u306b\u304a\u3051\u308b\u6587\u8108<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u79d1\u76ee<\/strong>\uff1a\u3053\u306e\u30c8\u30d4\u30c3\u30af\u306f\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c81\u30682\uff09\u306b\u7279\u5316\u3057\u3066\u304a\u308a\u3001\u6570\u5b66\u7684\u65b9\u6cd5\u3068\u4f75\u305b\u3066\u5b66\u7fd2\u3055\u308c\u308b\u3053\u3068\u304c\u591a\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u76ee\u7684<\/strong>\uff1a\u79d1\u5b66\u7684\u304a\u3088\u3073\u6280\u8853\u7684\u306a\u53d6\u308a\u7d44\u307f\u306b\u304a\u3051\u308b\u62bd\u8c61\u7684\u3067\u8907\u96d1\u306a\u95a2\u4fc2\u3092\u8aac\u660e\u3059\u308b\u305f\u3081\u306e\u30c4\u30fc\u30eb\u3068\u3057\u3066\u6a5f\u80fd\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u69cb\u9020<\/strong>\uff1a11\u5e74\u751f\u306e\u6982\u5ff5\u3092\u7d39\u4ecb\u3057\u300112\u5e74\u751f\u306e\u5c02\u9580\u6570\u5b66\u3067\u306e\u3088\u308a\u9ad8\u5ea6\u306a\u5fdc\u7528\u306e\u571f\u53f0\u3092\u7bc9\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u306a\u7fd2\u5f97\u30b9\u30ad\u30eb&nbsp;\u8868\u73fe<\/strong>\uff1a\u8907\u7d20\u6570\u3092\u76f4\u4ea4\u5ea7\u6a19\uff08<math data-latex=\"a+bi\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+bi<\/annotation><\/semantics><\/math>\uff09\u3068\u6975\u5ea7\u6a19\uff08<math data-latex=\"re^{i\\theta }\"><semantics><mrow><mi>r<\/mi><msup><mi>e<\/mi><mrow><mi>i<\/mi><mi>\u03b8<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">re^{i\\theta }<\/annotation><\/semantics><\/math>\uff09\u306e\u4e21\u65b9\u306e\u5f62\u5f0f\u3067\u8868\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u8996\u899a\u5316<\/strong>\uff1a\u8907\u7d20\u6570\u3092\u30b0\u30e9\u30d5\u5316\u3057\u3001\u305d\u306e\u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\uff08\u56de\u8ee2\u3001\u62e1\u5927\u306a\u3069\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4ee3\u6570\u64cd\u4f5c<\/strong>\uff1a\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u3001\u8907\u96d1\u306a\u5f0f\u3092\u64cd\u4f5c\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u306a\u305c\u91cd\u8981\u306a\u306e\u304b<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7279\u306b\u96fb\u6c17\u5de5\u5b66\u3001\u7269\u7406\u5b66\u3001\u9ad8\u5ea6\u306a\u7d14\u7c8b\u6570\u5b66\u3068\u3044\u3063\u305f\u5206\u91ce\u306b\u304a\u3044\u3066\u3001\u7d14\u7c8b\u306a\u5b9f\u6570\u3067\u306f\u6271\u3048\u306a\u3044\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306e\u5fc5\u9808\u30c4\u30fc\u30eb\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u3064\u307e\u308a<\/strong>\u3001\u8907\u96d1\u306a\u65b9\u7a0b\u5f0f\u3084\u30e2\u30c7\u30eb\u306b\u53d6\u308a\u7d44\u3080\u305f\u3081\u306e\u9ad8\u5ea6\u306a\u6570\u4f53\u7cfb\u3092\u7fd2\u5f97\u3059\u308b\u3053\u3068\u304c\u76ee\u7684\u3067\u3042\u308a\u3001\u30b7\u30cb\u30a2\u6570\u5b66\u3078\u306e\u9053\u7b4b\u3078\u306e\u91cd\u8981\u306a\u8db3\u639b\u304b\u308a\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e 11 \u5e74\u751f\u5c02\u9580\u6570\u5b66 (\u30e6\u30cb\u30c3\u30c8 2) \u306e\u300c\u8907\u7d20\u6570\u300d\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u76f4\u4ea4\u5ea7\u6a19\u5f62\u5f0f\u3068\u6975\u5ea7\u6a19\u5f62\u5f0f\u3001\u57fa\u672c\u7684\u306a\u6f14\u7b97\u3001\u30a2\u30eb\u30ac\u30f3\u56f3\u3001\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306b\u3064\u3044\u3066\u6271\u3044\u307e\u3059\u3002\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u5c02\u9580\u6570\u5b66\u306e\u30ea\u30bd\u30fc\u30b9\u306b\u57fa\u3065\u3044\u305f\u30b5\u30f3\u30d7\u30eb\u554f\u984c\u3068\u89e3\u7b54\u3092\u4ee5\u4e0b\u306b\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. <strong>\u8907\u7d20\u6570\u306e\u6f14\u7b97 (\u76f4\u4ea4\u5ea7\u6a19\u5f62\u5f0f)<\/strong> \u554f\u984c:  <math data-latex=\"(u=3-4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>=<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(u=3-4i)<\/annotation><\/semantics><\/math> , <math data-latex=\"(v=4+5i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>v<\/mi><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(v=4+5i)<\/annotation><\/semantics><\/math>\u3068\u3057\u307e\u3059\u3002 \u3000\u3000\u3000 (<math data-latex=\"2u-3v\"><semantics><mrow><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2u-3v<\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u306a\u3055\u3044\u3002\u89e3\u7b54: \u5024\u3092\u4ee3\u5165\u3057\u307e\u3059: (2(3-4i)-3(4+5i)) \u62ec\u5f27\u3092\u5c55\u958b\u3057\u307e\u3059: (6-8i-12-15i) \u5b9f\u90e8\u3068\u865a\u90e8\u3092\u30b0\u30eb\u30fc\u30d7\u5316\u3057\u307e\u3059: ((6-12)+(-8i-15i)) \u7c21\u7d04\u3057\u307e\u3059: ( <math data-latex=\"-6-23i\"><semantics><mrow><mo>\u2212<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mn>23<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-6-23i<\/annotation><\/semantics><\/math>\u00a0)\u3002 \u3000\u3000\u554f\u984c: (<math data-latex=\" z_{1}=2-i\"><semantics><mrow><msub><mi>z<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\"> z_{1}=2-i<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\" z_{2}=3+4i\"><semantics><mrow><msub><mi>z<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\"> z_{2}=3+4i<\/annotation><\/semantics><\/math>) \u3092\u639b\u3051\u306a\u3055\u3044\u3002\u89e3\u7b54: \u5c55\u958b (FOIL): (<math data-latex=\"(2)(3)+(2)(4i)+(-i)(3)+(-i)(4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(2)(3)+(2)(4i)+(-i)(3)+(-i)(4i)<\/annotation><\/semantics><\/math>) <math data-latex=\"(6+8i-3i-4i^{2})\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>+<\/mo><mn>8<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(6+8i-3i-4i^{2})<\/annotation><\/semantics><\/math> &#8211;> <math data-latex=\" (i^{2}=-1)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\"> (i^{2}=-1)<\/annotation><\/semantics><\/math>\u3092\u4f7f\u7528\u3057\u307e\u3059: (<math data-latex=\"6+5i-4(-1)\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">6+5i-4(-1)<\/annotation><\/semantics><\/math>) &#8211;> (<math data-latex=\"6+5i+4\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6+5i+4<\/annotation><\/semantics><\/math>)        \u7d50\u679c: (<math data-latex=\"10+5i\"><semantics><mrow><mn>10<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">10+5i<\/annotation><\/semantics><\/math>)<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>\u5171\u5f79\u3068\u5272\u308a\u7b97<\/strong>\u3000\u554f\u984c: (<math data-latex=\"z=\\frac{1+i}{2-i}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{1+i}{2-i}<\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u3001\u7b54\u3048\u3092 (<math data-latex=\"a+bi\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+bi<\/annotation><\/semantics><\/math>) \u306e\u5f62\u5f0f\u3067\u66f8\u304d\u306a\u3055\u3044\u3002\u89e3\u7b54: \u5206\u5b50\u3068\u5206\u6bcd\u306b\u3001\u5206\u6bcd\u306e\u5171\u5f79 (<math data-latex=\"2+i\"><semantics><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2+i<\/annotation><\/semantics><\/math>) \u3092\u639b\u3051\u307e\u3059: (<math data-latex=\"\\frac{1+i}{2-i}\\times \\frac{2+i}{2+i}\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><\/mrow><\/mfrac><mo>\u00d7<\/mo><mfrac><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><mrow><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1+i}{2-i}\\times \\frac{2+i}{2+i}<\/annotation><\/semantics><\/math>)\u5206\u5b50\u3092\u5c55\u958b\u3057\u307e\u3059: (<math data-latex=\"(1)(2)+1(i)+i(2)+i^{2}=2+i+2i-1=1+3i\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>i<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(1)(2)+1(i)+i(2)+i^{2}=2+i+2i-1=1+3i<\/annotation><\/semantics><\/math>)\u5206\u6bcd\u3092\u5c55\u958b\u3057\u307e\u3059: (<math data-latex=\"(2-i)(2+i)=4+2i-2i-i^{2}=4-(-1)=5\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2-i)(2+i)=4+2i-2i-i^{2}=4-(-1)=5<\/annotation><\/semantics><\/math>)\u7d50\u679c: (<math data-latex=\"\\frac{1+3i}{5}=\\frac{1}{5}+\\frac{3}{5}i\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><mn>5<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>5<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1+3i}{5}=\\frac{1}{5}+\\frac{3}{5}i<\/annotation><\/semantics><\/math>)\u00a0<\/li>\n\n\n\n<li> <strong>\u7d76\u5bfe\u5024\u3068\u504f\u89d2<\/strong> (\u6975\u5f62\u5f0f)\u00a0\u554f\u984c: (<math data-latex=\"z=1-\\sqrt{3}i\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=1-\\sqrt{3}i<\/annotation><\/semantics><\/math>) \u306e\u7d76\u5bfe\u5024\u3068\u504f\u89d2\u3092\u6c42\u3081\u3001(<math data-latex=\"r(\\cos \\theta +i\\sin \\theta \"><semantics><mrow><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>+<\/mo><mi>i<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(\\cos \\theta +i\\sin \\theta <\/annotation><\/semantics><\/math>)) \u307e\u305f\u306f (<math data-latex=\"r\\text{cis}\\theta\"><semantics><mrow><mi>r<\/mi><mtext>cis<\/mtext><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r\\text{cis}\\theta<\/annotation><\/semantics><\/math> ) \u306e\u5f62\u5f0f\u3067\u66f8\u304d\u8868\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u89e3\u6c7a\u65b9\u6cd5: \u7d76\u5bfe\u5024 (<math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u307e\u3059\u3002 \u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000(<math data-latex=\"|z|=\\sqrt{a^{2}+b^{2}}=\\sqrt{1^{2}+(-\\sqrt{3})^{2}}=\\sqrt{1+3}=\\sqrt{4}=2\"><semantics><mrow><mi>|<\/mi><mi>z<\/mi><mi>|<\/mi><mo>=<\/mo><msqrt><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><msup><mn>1<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>4<\/mn><\/msqrt><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">|z|=\\sqrt{a^{2}+b^{2}}=\\sqrt{1^{2}+(-\\sqrt{3})^{2}}=\\sqrt{1+3}=\\sqrt{4}=2<\/annotation><\/semantics><\/math>)\u3002\u504f\u89d2 (<math data-latex=\"\\theta \"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta <\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u308b: (<math data-latex=\"\\tan \\theta =\\frac{b}{a}=\\frac{-\\sqrt{3}}{1}\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>b<\/mi><mi>a<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><\/mrow><mn>1<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta =\\frac{b}{a}=\\frac{-\\sqrt{3}}{1}<\/annotation><\/semantics><\/math>)\u3002(<math data-latex=\"a&gt;0\"><semantics><mrow><mi>a<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a&gt;0<\/annotation><\/semantics><\/math>) \u304b\u3064 (<math data-latex=\"b<0\"><semantics><mrow><mi>b<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b&lt;0<\/annotation><\/semantics><\/math>) \u3067\u3042\u308b\u305f\u3081\u3001(<math data-latex=\"z\"><semantics><mi>z<\/mi><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math>) \u306f\u7b2c 4 \u8c61\u9650\u306b\u3042\u308b\u3002(<math data-latex=\"\\theta =\\arctan (-\\sqrt{3})=-\\frac{\\pi }{3}\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mrow><mi>arctan<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta =\\arctan (-\\sqrt{3})=-\\frac{\\pi }{3}<\/annotation><\/semantics><\/math>)\u3002\u6975\u5f62\u5f0f: (<math data-latex=\"2\\left(\\cos \\left(-\\frac{\\pi }{3}\\right)+i\\sin \\left(-\\frac{\\pi }{3}\\right)\\right)\"><semantics><mrow><mn>2<\/mn><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>+<\/mo><mi>i<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">2\\left(\\cos \\left(-\\frac{\\pi }{3}\\right)+i\\sin \\left(-\\frac{\\pi }{3}\\right)\\right)<\/annotation><\/semantics><\/math>)\u307e\u305f\u306f(<math data-latex=\"2\\text{cis}\\left(-\\frac{\\pi }{3}\\right)\"><semantics><mrow><mn>2<\/mn><mtext>cis<\/mtext><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">2\\text{cis}\\left(-\\frac{\\pi }{3}\\right)<\/annotation><\/semantics><\/math>). <\/li>\n\n\n\n<li><strong>\u8907\u7d20\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5<\/strong> \u554f\u984c: \u8907\u7d20\u4f53\u4e0a\u3067 (<math data-latex=\"z^{2}+4z+13=0\"><semantics><mrow><msup><mi>z<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>z<\/mi><mo>+<\/mo><mn>13<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">z^{2}+4z+13=0<\/annotation><\/semantics><\/math>) \u3092\u89e3\u304d\u3001\u7b54\u3048\u3092 (<math data-latex=\"a+bi\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+bi<\/annotation><\/semantics><\/math>) \u306e\u5f62\u5f0f\u3067\u6c42\u3081\u306a\u3055\u3044\u3002\u89e3\u7b54: \u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u516c\u5f0f (<math data-latex=\"z=\\frac{-b\\pm \\sqrt{b^{2}-4ac}}{2a}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-b\\pm \\sqrt{b^{2}-4ac}}{2a}<\/annotation><\/semantics><\/math>) \u3092\u4f7f\u7528\u3057\u3001\u5224\u5225\u5f0f\u3092\u7c21\u7565\u5316\u3057\u307e\u3059: (<math data-latex=\"z=\\frac{-4\\pm \\sqrt{4^{2}-4(1)(13)}}{2(1)}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><msup><mn>4<\/mn><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>13<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm \\sqrt{4^{2}-4(1)(13)}}{2(1)}<\/annotation><\/semantics><\/math>)\u3002\u5224\u5225\u5f0f\u3092\u7c21\u7565\u5316\u3057\u307e\u3059: (<math data-latex=\"z=\\frac{-4\\pm \\sqrt{16-52}}{2}=\\frac{-4\\pm \\sqrt{-36}}{2}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><mn>16<\/mn><mo>\u2212<\/mo><mn>52<\/mn><\/mrow><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><msqrt><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>36<\/mn><\/mrow><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm \\sqrt{16-52}}{2}=\\frac{-4\\pm \\sqrt{-36}}{2}<\/annotation><\/semantics><\/math>)\u3002(<math data-latex=\"i\"><semantics><mi>i<\/mi><annotation encoding=\"application\/x-tex\">i<\/annotation><\/semantics><\/math>) \u3092\u4f7f\u7528\u3057\u307e\u3059: (<math data-latex=\"z=\\frac{-4\\pm 6i}{2}\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><mo>\u00b1<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">z=\\frac{-4\\pm 6i}{2}<\/annotation><\/semantics><\/math>)\u3002\u7d50\u679c: (<math data-latex=\"z=-2\\pm 3i\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>2<\/mn><mo>\u00b1<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=-2\\pm 3i<\/annotation><\/semantics><\/math>) (\u6839(<math data-latex=\"-2+3i\"><semantics><mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-2+3i<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\"-2-3i\"><semantics><mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-2-3i<\/annotation><\/semantics><\/math>) \u3067\u3059) <\/li>\n\n\n\n<li><strong>\u8a66\u9a13\u5f62\u5f0f\u306e\u554f\u984c<\/strong> (QCAA \u5f62\u5f0f) \u8cea\u554f: \u8907\u7d20\u6570 (<math data-latex=\"w\"><semantics><mi>w<\/mi><annotation encoding=\"application\/x-tex\">w<\/annotation><\/semantics><\/math>) \u306f (<math data-latex=\"w=1-2i\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">w=1-2i<\/annotation><\/semantics><\/math>) \u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002 (<math data-latex=\"\\text{Im}(w^{4})\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>4<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(w^{4})<\/annotation><\/semantics><\/math>) \u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002\u89e3\u7b54: (<math data-latex=\"w^{2}\"><semantics><msup><mi>w<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">w^{2}<\/annotation><\/semantics><\/math>) \u3092\u8a08\u7b97\u3057\u307e\u3059: <math data-latex=\"(1-2i)^{2}=1-4i+4i^{2}=1-4i-4=-3-4i)\u3002(w^{4}) (w^{2}\\times w^{2}\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>4<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>\u3002<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>4<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><msup><mi>w<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(1-2i)^{2}=1-4i+4i^{2}=1-4i-4=-3-4i)\u3002(w^{4}) (w^{2}\\times w^{2}<\/annotation><\/semantics><\/math>) \u3092\u8a08\u7b97\u3057\u307e\u3059: <math data-latex=\"(-3-4i)(-3-4i)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(-3-4i)(-3-4i)<\/annotation><\/semantics><\/math> = <math data-latex=\"9+12i+12i+16i^{2}\"><semantics><mrow><mn>9<\/mn><mo>+<\/mo><mn>12<\/mn><mi>i<\/mi><mo>+<\/mo><mn>12<\/mn><mi>i<\/mi><mo>+<\/mo><mn>16<\/mn><msup><mi>i<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">9+12i+12i+16i^{2}<\/annotation><\/semantics><\/math> = <math data-latex=\"9+24i-16\"><semantics><mrow><mn>9<\/mn><mo>+<\/mo><mn>24<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">9+24i-16<\/annotation><\/semantics><\/math><math data-latex=\"=-7+24i\"><semantics><mrow><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>7<\/mn><mo>+<\/mo><mn>24<\/mn><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">=-7+24i<\/annotation><\/semantics><\/math>\u3002\u3000<math data-latex=\"\\text{Im}(z)\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>z<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(z)<\/annotation><\/semantics><\/math> \u3092\u7279\u5b9a\u3057\u307e\u3059: \u865a\u6570\u90e8\u306f (<math data-latex=\"i\"><semantics><mi>i<\/mi><annotation encoding=\"application\/x-tex\">i<\/annotation><\/semantics><\/math>) \u306e\u4fc2\u6570\u3067\u3059\u3002\u89e3\u7b54: (<math data-latex=\"\\text{Im}(w^{4})=24\"><semantics><mrow><mtext>Im<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>w<\/mi><mn>4<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Im}(w^{4})=24<\/annotation><\/semantics><\/math>)\u3002\n<ul class=\"wp-block-list\">\n<li><\/li>\n\n\n\n<li><strong>\u4e3b\u8981\u6982\u5ff5\u306e\u307e\u3068\u3081<\/strong>\u00a0(<math data-latex=\"i^{2}=-1\"><semantics><mrow><msup><mi>i<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">i^{2}=-1<\/annotation><\/semantics><\/math>): \u3059\u3079\u3066\u306e\u8a08\u7b97\u3092\u7c21\u7565\u5316\u3059\u308b\u305f\u3081\u306e\u57fa\u672c\u3002\u5171\u5f79 <math data-latex=\"{z}\"><semantics><mi>z<\/mi><annotation encoding=\"application\/x-tex\">{z}<\/annotation><\/semantics><\/math>:    <math data-latex=\"z=a+bi, \\={z}=a-bi\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo separator=\"true\">,<\/mo><mover><mi>z<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0;\">\u02c9<\/mo><\/mover><mo>=<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=a+bi, \\={z}=a-bi<\/annotation><\/semantics><\/math> \u306e\u5834\u5408\u3001\u9664\u7b97\u3068\u7a4d\u306e\u5b9f\u90e8\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u4f7f\u7528\u3057\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u30a2\u30eb\u30ac\u30f3\u30c9\u56f3: (<math data-latex=\"z=a+bi\"><semantics><mrow><mi>z<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z=a+bi<\/annotation><\/semantics><\/math>) \u3092\u5ea7\u6a19 (<math data-latex=\"a,b\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b<\/annotation><\/semantics><\/math>) \u3068\u3057\u3066\u30d7\u30ed\u30c3\u30c8\u3057\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406:  <math data-latex=\"(\\text{r\\ cis\\ }\\theta )^{n}=r^{n}\\text{cis\\ }(n\\theta )\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mtext>r&nbsp;cis&nbsp;<\/mtext><mi>\u03b8<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>=<\/mo><msup><mi>r<\/mi><mi>n<\/mi><\/msup><mtext>cis&nbsp;<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(\\text{r\\ cis\\ }\\theta )^{n}=r^{n}\\text{cis\\ }(n\\theta )<\/annotation><\/semantics><\/math>(\u3079\u304d\u4e57\/\u6839\u53f7\u306b\u4f7f\u7528)\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Queensland Year 11 Maths, &#8220;Real and Complex Numbers&#8221; is a core topic in the Mathematical Methods syllabus (Units 1 &amp; 2) and a foundational domain in Specialist Mathematics (Units 1 &amp; 2), focusing on extending real number concepts (like surds, indices) and introducing complex numbers (operations, forms, applications) for advanced problem-solving in algebra and [&hellip;]<\/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-1211","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1211","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=1211"}],"version-history":[{"count":12,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1211\/revisions"}],"predecessor-version":[{"id":1242,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1211\/revisions\/1242"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}