http://wiki.math.se/wikis/forberedandematte1/index.php?action=history&feed=atom&title=L%C3%B6sning_1.1%3A7b 2026-04-06T04:55:58Z Versionshistorik för denna sida på wikin MediaWiki 1.11.1 http://wiki.math.se/wikis/forberedandematte1/index.php?title=L%C3%B6sning_1.1:7b&diff=1543&oldid=prev 2008-04-29T08:14:02Z

<p></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr> <td colspan='2' style="background-color: white; color:black;">← Äldre version</td> <td colspan='2' style="background-color: white; color:black;">Versionen från 29 april 2008 kl. 08.14</td> </tr> <tr><td colspan="2" class="diff-lineno">Rad 1:</td> <td colspan="2" class="diff-lineno">Rad 1:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{NAVCONTENT_START}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{NAVCONTENT_START}}</div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;center&gt; [[<del style="color: red; font-weight: bold; text-decoration: none;">Bild</del>:<del style="color: red; font-weight: bold; text-decoration: none;">1_1_7b-1(2).gif</del>]] &lt;/<del style="color: red; font-weight: bold; text-decoration: none;">center</del>&gt;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Ett rationellt tal har alltid en decimalutveckling som från och med en viss decimal upprepar sig periodiskt.</ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;<del style="color: red; font-weight: bold; text-decoration: none;">center</del>&gt; [<del style="color: red; font-weight: bold; text-decoration: none;">[Bild</del>:<del style="color: red; font-weight: bold; text-decoration: none;">1_1_7b</del>-<del style="color: red; font-weight: bold; text-decoration: none;">2</del>(<del style="color: red; font-weight: bold; text-decoration: none;">2</del>).<del style="color: red; font-weight: bold; text-decoration: none;">gif]] </del>&lt;/<del style="color: red; font-weight: bold; text-decoration: none;">center</del>&gt;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">I vårt fall så upprepas sekvensen 1416 i all oändlighet</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;center&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">&lt;math&gt;3{,}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots&lt;/math&gt;&lt;/center&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Med andra ord är talet rationellt.</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Nästa problem är att skriva om talet som ett bråktal och då utnyttjar vi att multiplikation med 10 flyttar decimalkommat ett steg åt höger.</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Om vi skriver</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;\insteadof</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">right]{10000x}{x}{} = 3\,\color{red}{&amp;#130;}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">så är därför</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;\insteadof</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">right]{10000x}{10x}{} = 31\,\color{red}{&amp;#130;}\,4161\ 4161\ 4161\,\ldots&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<ins style="color: red; font-weight: bold; text-decoration: none;">:&lt;math&gt;\insteadof[right</ins>]<ins style="color: red; font-weight: bold; text-decoration: none;">{10000x}{100x}{} = 314\,\color{red}{&amp;#130;}\,1614\ 1614\ 161\,\ldots&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;\insteadof[right</ins>]<ins style="color: red; font-weight: bold; text-decoration: none;">{10000x}{1000x}{} = 3141\,\color{red}{&amp;#130;}\,6141\ 6141\ 61\,\ldots</ins>&lt;/<ins style="color: red; font-weight: bold; text-decoration: none;">math</ins>&gt;</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::</ins>&lt;<ins style="color: red; font-weight: bold; text-decoration: none;">math</ins>&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">\insteadof</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">right]{10000x}{10000x}{} = 31416\,\color{red}{&amp;#130;}\,\underline{1416}\ \underline{1416}\ 1\,\ldots&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Notera att i 10000''x'' har vi flyttat decimalkommat tillräckligt många steg så att decimalutvecklingen av 10000''x'' har kommit i fas med decimalutvecklingen av ''x'', dvs. de har samma decimalutveckling.</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Därför är</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<ins style="color: red; font-weight: bold; text-decoration: none;">:&lt;math&gt;10000x</ins>-<ins style="color: red; font-weight: bold; text-decoration: none;">x = 31416\,{,}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,{,}\,\underline{1416}\ \underline{1416}\,\ldots&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;\phantom{10000x-x}{}= 31413\quad&lt;/math&gt;</ins>(<ins style="color: red; font-weight: bold; text-decoration: none;">decimalerna tar ut varandra</ins>)</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">och eftersom &lt;math&gt;10000x-x = 9999x&lt;/math&gt; så har vi alltså sambandet</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;9999x = 31413\,\mbox{</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;">}</ins>&lt;/<ins style="color: red; font-weight: bold; text-decoration: none;">math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STEP}}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Löser vi ut ''x'' ur detta samband får vi ''x'' som en kvot mellan två heltal</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">::&lt;math&gt;x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}&lt;/math</ins>&gt;</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{NAVCONTENT_STOP}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{NAVCONTENT_STOP}}</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;!--&lt;center&gt; [[Bild:1_1_7b-1(2).gif]] &lt;/center&gt;</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;center&gt; [[Bild:1_1_7b-2(2).gif]] &lt;/center&gt;--&gt;</div></td></tr> </table>

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