{"id":40,"date":"2025-01-02T23:03:46","date_gmt":"2025-01-02T23:03:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/fractals-get-stronger-answer-key\/"},"modified":"2025-01-02T23:03:46","modified_gmt":"2025-01-02T23:03:46","slug":"fractals-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/fractals-get-stronger-answer-key\/","title":{"raw":"Fractals: Get Stronger Answer Key","rendered":"Fractals: Get Stronger Answer Key"},"content":{"raw":"\n
    \n\t
  1. \"Step<\/li>\n\t
  2.  <\/li>\n\t
  3. \"Step<\/li>\n\t
  4.  <\/li>\n\t
  5. \"Step<\/li>\n\t
  6.  <\/li>\n\t
  7.  <\/li>\n\t
  8.  <\/li>\n\t
  9. Four copies of the Koch curve are needed to create a curve scaled by [latex]3[\/latex]
    \n[latex]\ud835\udc37=log(4)\/log\u2061(3)\u22481.262[\/latex]
    \n\"Step<\/li>\n\t
  10.  <\/li>\n\t
  11. \n

    Eight copies of the shape are needed to make a copy scaled by [latex]3[\/latex]. [latex]D=\\frac{\\log (8)}{\\log (3)} \\approx 1.893[\/latex]<\/p>\n<\/li>\n\t

  12.  <\/li>\n\t
  13. \"A<\/li>\n\t
  14.  <\/li>\n\t
  15. \n
      \n\t
    1. [latex]5-i[\/latex]<\/li>\n\t
    2. [latex]5-4i[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
    3. \n
        \n\t
      1. [latex]3-5i[\/latex]<\/li>\n\t
      2.  [latex]-(6+i)[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
      3. \n
          \n\t
        1. [latex]6+12i[\/latex]<\/li>\n\t
        2. [latex]10-2i[\/latex]<\/li>\n\t
        3. [latex]14+2i[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
        4. \n
            \n\t
          1. [latex]-2+6i[\/latex]<\/li>\n\t
          2. [latex]18+6i[\/latex]<\/li>\n\t
          3. [latex]7+3i[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
          4. [latex](2+3i)(1-i) = 5+i.[\/latex]It appears that multiplying by [latex]1-i[\/latex] both scaled the number away from the origin, and rotated it clockwise about [latex]45\u00b0[\/latex].\"A<\/li>\n\t
          5.  <\/li>\n\t
          6. [latex]z_{1}=i z_{0}+1=i(2)+1=1+2 i[\/latex]
            \n[latex]z_{2}=i z_{1}+1=i(1+2 i)+1=i-2+1=-1+i[\/latex]
            \n[latex]z_{3}=i z_{2}+1=i(-1+i)+1=-i-1+1=-i[\/latex]<\/li>\n\t
          7.  <\/li>\n\t
          8. [latex]z_{0}=0[\/latex]
            \n[latex]z_{1}=z_{0}^{2}-0.25=0-0.25=-0.25[\/latex]
            \n[latex]z_{2}=z_{1}^{2}-0.25=(-0.25)^{2}-0.25=-0.1875[\/latex]
            \n[latex]z_{3}=z_{2}^{2}-0.25=(-0.1875)^{2}-0.25=-0.21484[\/latex]
            \n[latex]z_{4}=z_{3}^{2}-0.25=(-0.21484)^{2}-0.25=-0.20384[\/latex]<\/li>\n\t
          9.  <\/li>\n\t
          10. attracted, to approximately [latex]-0.3776+0.14242i[\/latex]<\/li>\n\t
          11.  <\/li>\n\t
          12. periodic [latex]2[\/latex]-cycle<\/li>\n\t
          13.  <\/li>\n\t
          14. Escaping<\/li>\n\t
          15.  <\/li>\n\t
          16. periodic [latex]3[\/latex]-cycle<\/li>\n<\/ol>\n","rendered":"
              \n
            1. \"Step<\/li>\n
            2.  <\/li>\n
            3. \"Step<\/li>\n
            4.  <\/li>\n
            5. \"Step<\/li>\n
            6.  <\/li>\n
            7.  <\/li>\n
            8.  <\/li>\n
            9. Four copies of the Koch curve are needed to create a curve scaled by [latex]3[\/latex]
              \n[latex]\ud835\udc37=log(4)\/log\u2061(3)\u22481.262[\/latex]
              \n\"Step<\/li>\n
            10.  <\/li>\n
            11. \n

              Eight copies of the shape are needed to make a copy scaled by [latex]3[\/latex]. [latex]D=\\frac{\\log (8)}{\\log (3)} \\approx 1.893[\/latex]<\/p>\n<\/li>\n

            12.  <\/li>\n
            13. \"A<\/li>\n
            14.  <\/li>\n
            15. \n
                \n
              1. [latex]5-i[\/latex]<\/li>\n
              2. [latex]5-4i[\/latex]<\/li>\n<\/ol>\n<\/li>\n
              3. \n
                  \n
                1. [latex]3-5i[\/latex]<\/li>\n
                2.  [latex]-(6+i)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                3. \n
                    \n
                  1. [latex]6+12i[\/latex]<\/li>\n
                  2. [latex]10-2i[\/latex]<\/li>\n
                  3. [latex]14+2i[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                  4. \n
                      \n
                    1. [latex]-2+6i[\/latex]<\/li>\n
                    2. [latex]18+6i[\/latex]<\/li>\n
                    3. [latex]7+3i[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                    4. [latex](2+3i)(1-i) = 5+i.[\/latex]It appears that multiplying by [latex]1-i[\/latex] both scaled the number away from the origin, and rotated it clockwise about [latex]45\u00b0[\/latex].\"A<\/li>\n
                    5.  <\/li>\n
                    6. [latex]z_{1}=i z_{0}+1=i(2)+1=1+2 i[\/latex]
                      \n[latex]z_{2}=i z_{1}+1=i(1+2 i)+1=i-2+1=-1+i[\/latex]
                      \n[latex]z_{3}=i z_{2}+1=i(-1+i)+1=-i-1+1=-i[\/latex]<\/li>\n
                    7.  <\/li>\n
                    8. [latex]z_{0}=0[\/latex]
                      \n[latex]z_{1}=z_{0}^{2}-0.25=0-0.25=-0.25[\/latex]
                      \n[latex]z_{2}=z_{1}^{2}-0.25=(-0.25)^{2}-0.25=-0.1875[\/latex]
                      \n[latex]z_{3}=z_{2}^{2}-0.25=(-0.1875)^{2}-0.25=-0.21484[\/latex]
                      \n[latex]z_{4}=z_{3}^{2}-0.25=(-0.21484)^{2}-0.25=-0.20384[\/latex]<\/li>\n
                    9.  <\/li>\n
                    10. attracted, to approximately [latex]-0.3776+0.14242i[\/latex]<\/li>\n
                    11.  <\/li>\n
                    12. periodic [latex]2[\/latex]-cycle<\/li>\n
                    13.  <\/li>\n
                    14. Escaping<\/li>\n
                    15.  <\/li>\n
                    16. periodic [latex]3[\/latex]-cycle<\/li>\n<\/ol>\n","protected":false},"author":6,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/40"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/40\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=40"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=40"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=40"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}