{"id":1835,"date":"2024-06-10T22:29:07","date_gmt":"2024-06-10T22:29:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1835"},"modified":"2025-08-14T01:06:20","modified_gmt":"2025-08-14T01:06:20","slug":"complex-numbers-and-operations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/complex-numbers-and-operations-learn-it-2\/","title":{"raw":"Complex Numbers and Operations: Learn It 2","rendered":"Complex Numbers and Operations: Learn It 2"},"content":{"raw":"

Complex Plane<\/h2>\r\nWe cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To plot a complex number like [latex]3-4i[\/latex], we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane<\/strong>.\r\n\r\n
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complex plane<\/h3>\r\nIn the complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.\r\n\r\n \r\n\r\n
\r\n\r\n[caption id=\"attachment_8047\" align=\"aligncenter\" width=\"300\"]\"The Graph with imaginary and real axes[\/caption]\r\n\r\n<\/center><\/div>\r\n<\/section>
How To: Given a complex number, represent its components on the complex plane.<\/strong>\r\n
    \r\n \t
  1. Determine the real part and the imaginary part of the complex number.<\/li>\r\n \t
  2. Move along the horizontal axis to show the real part of the number.<\/li>\r\n \t
  3. Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\r\n \t
  4. Plot the point.<\/li>\r\n<\/ol>\r\n<\/section>
    Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number [latex]z=a+bi[\/latex] as if we were plotting the point [latex](a, b)[\/latex] in Cartesian coordinates. Sometimes people write complex numbers as [latex]z=x+yi[\/latex] to highlight this relation.<\/section>
    Plot the number [latex]3-4i[\/latex] on the complex plane.\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703380\"]\r\nThe real part of this number is [latex]3[\/latex], and the imaginary part is [latex]-4[\/latex]. To plot this, we draw a point [latex]3[\/latex] units to the right of the origin in the horizontal direction and [latex]4[\/latex] units down in the vertical direction.\r\n
    \r\n\r\n[caption id=\"attachment_1732\" align=\"aligncenter\" width=\"275\"]\"A Graph with imaginary and real axes with plotted point[\/caption]\r\n\r\n<\/center>\r\n[\/hidden-answer]<\/section>","rendered":"

    Complex Plane<\/h2>\n

    We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To plot a complex number like [latex]3-4i[\/latex], we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane<\/strong>.<\/p>\n

    \n
    \n

    complex plane<\/h3>\n

    In the complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.<\/p>\n

     <\/p>\n

    \n
    \"The
    Graph with imaginary and real axes<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n
    How To: Given a complex number, represent its components on the complex plane.<\/strong><\/p>\n
      \n
    1. Determine the real part and the imaginary part of the complex number.<\/li>\n
    2. Move along the horizontal axis to show the real part of the number.<\/li>\n
    3. Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\n
    4. Plot the point.<\/li>\n<\/ol>\n<\/section>\n
      Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number [latex]z=a+bi[\/latex] as if we were plotting the point [latex](a, b)[\/latex] in Cartesian coordinates. Sometimes people write complex numbers as [latex]z=x+yi[\/latex] to highlight this relation.<\/section>\n
      Plot the number [latex]3-4i[\/latex] on the complex plane.<\/p>\n