{"id":1466,"date":"2023-04-10T14:54:33","date_gmt":"2023-04-10T14:54:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1466"},"modified":"2024-10-18T20:53:51","modified_gmt":"2024-10-18T20:53:51","slug":"fractals-generated-by-complex-numbers-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-learn-it-2\/","title":{"raw":"Fractals Generated by Complex Numbers: Learn It 2","rendered":"Fractals Generated by Complex Numbers: Learn It 2"},"content":{"raw":"

Complex Plane<\/h2>\r\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>\r\n

\r\n
\r\n

complex plane<\/h3>\r\n

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

 <\/p>\r\n

\"The<\/center><\/div>\r\n<\/section>\r\n
\r\n

How To: Given a complex number, represent its components on the complex plane.<\/strong><\/p>\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>\r\n
    \r\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.<\/p>\r\n<\/section>\r\n

    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
    \"A<\/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

    \"The<\/div>\n<\/div>\n<\/section>\n
    \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
      \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.<\/p>\n<\/section>\n

      Plot the number [latex]3-4i[\/latex] on the complex plane.<\/p>\n