Problem 51580. Construct a continued fraction for a square root

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Numbers can be expressed as continued fractions of the form</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"x = a_0 + 1/(a_1 + 1/(a_2 + 1/(a_3 + 1/...)))\"/></w:customXmlPr><w:r><w:t>x =a_0+\\frac{1}{a_1+\\frac{1}{a_2+\\frac{1}{a_3+\\frac{1}{\\ddots}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Some continued fractions—such as those for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"e\"/></w:customXmlPr><w:r><w:t>e</w:t></w:r></w:customXml><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:t>and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t>--continue forever without a discernable pattern in the coefficients, while the coefficients of continued fractions for square roots eventually repeat. For example, </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"sqrt(2) = 1+(1/(2+1/(2+1/(2+1/...))))\"/></w:customXmlPr><w:r><w:t>\\sqrt{2} =1+\\frac{1}{2+\\frac{1}{2+\\frac{1}{2+\\frac{1}{\\ddots}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>or</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"sqrt(14) = 3+1/(1+1/(2+1/(1+1/(6+1/...))))\"/></w:customXmlPr><w:r><w:t>\\sqrt{14} =3+\\frac{1}{1+\\frac{1}{2+\\frac{1}{1+\\frac{1}{6+\\frac{1}{\\ddots}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that takes a non-square integer and returns the values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a_i</w:t></w:r></w:customXml><w:r><w:t> until the values repeat. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>This problem celebrates my finally cracking </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/groups/25/problems/1215\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 1215</w:t></w:r></w:hyperlink><w:r><w:t> by James. If you struggle with Test 8, as I did, remember that MATLAB cannot represent decimals with infinite precision.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 483.917px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 241.958px; transform-origin: 407px 241.958px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 190.983px 7.91667px; transform-origin: 190.983px 7.91667px; unicode-bidi: normal; "><span style="">Numbers can be expressed as continued fractions of the form</span></span></div><div style="block-size: 85.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.6667px; text-align: left; transform-origin: 384px 42.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-69px"><img src="data:image/png;base64,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" alt="x = a_0 + 1/(a_1 + 1/(a_2 + 1/(a_3 + 1/...)))" style="width: 160.5px; height: 85.5px;" width="160.5" height="85.5"></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.192px 7.91667px; transform-origin: 141.192px 7.91667px; unicode-bidi: normal; "><span style="">Some continued fractions—such as those for </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">e</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style="font-style: italic; "> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; "><span style="">and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">π</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 206.033px 7.91667px; transform-origin: 206.033px 7.91667px; unicode-bidi: normal; "><span style="">--continue forever without a discernable pattern in the coefficients, while the coefficients of continued fractions for square roots eventually repeat. For example, </span></span></div><div style="block-size: 85.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.6667px; text-align: left; transform-origin: 384px 42.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-69px"><img src="data:image/png;base64,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" alt="sqrt(2) = 1+(1/(2+1/(2+1/(2+1/...))))" style="width: 153px; height: 85.5px;" width="153" height="85.5"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 7.91667px; transform-origin: 6.225px 7.91667px; unicode-bidi: normal; "><span style="">or</span></span></div><div style="block-size: 103px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 51.5px; text-align: left; transform-origin: 384px 51.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-87px"><img src="data:image/png;base64,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" alt="sqrt(14) = 3+1/(1+1/(2+1/(1+1/(6+1/...))))" style="width: 185px; height: 103px;" width="185" height="103"></span></div><div style="block-size: 21.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.625px; text-align: left; transform-origin: 384px 10.625px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 228.192px 7.91667px; transform-origin: 228.192px 7.91667px; unicode-bidi: normal; "><span style="">Write a function that takes a non-square integer and returns the values of </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAAoCAYAAAAPOoFWAAABC0lEQVRYhe2VTRGDMBBGnwcc1AAGogAFOMABDrCABiTEQy2gAQv0wO6Q0oSkkB46kzeTA1nYb3/YBAqFQqHwN9RAB/SAkT0jz3VOEQvMImaAEXgCq6xHDqFWnM1AdbCtju2nQq7YcFeocZwZj/0RsSdTAYs4mgLvaNYL/qyT6dmjDv1lVuw24qOLiWlWIUfGCaa/I5biyBLPPAm3hI3H3rFnnrVfR7GWbQwm3ss8sldBT5RJ9k+pHTErkVfiQOdNMxtlHf9YDThp/jRyd03sJVsO+6HvfW3w0kiEHZ/nXuzwXdiqkOW8PEPbEBqbrCQPcw50Bmu+6NlV9JZoyXAbxNALdeDmwBcKhWu8ANZGYx8ObIHOAAAAAElFTkSuQmCC" style="width: 13.5px; height: 20px;" width="13.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2917px 7.91667px; transform-origin: 74.2917px 7.91667px; unicode-bidi: normal; "><span style=""> until the values repeat. </span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 135.758px 7.91667px; transform-origin: 135.758px 7.91667px; unicode-bidi: normal; "><span style="">This problem celebrates my finally cracking </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/groups/25/problems/1215"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Cody Problem 1215</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.892px 7.91667px; transform-origin: 178.892px 7.91667px; unicode-bidi: normal; "><span style=""> by James. If you struggle with Test 8, as I did, remember that MATLAB cannot represent decimals with infinite precision.</span></span></div></div></div>

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Problem 51580. Construct a continued fraction for a square root

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