http://wiki.math.se/wikis/forberedandematte1/index.php?action=history&feed=atom&title=L%C3%B6sning_1.1%3A7c 2026-04-06T04:31:47Z Versionshistorik för denna sida på wikin MediaWiki 1.11.1 http://wiki.math.se/wikis/forberedandematte1/index.php?title=L%C3%B6sning_1.1:7c&diff=1544&oldid=prev 2008-04-29T08:16:53Z

<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. 08.16</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:1_1_7c-1(</del>2<del style="color: red; font-weight: bold; text-decoration: none;">)</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;">Tittar vi närmare på talet så ser vi att sifferkombinationen 001 upprepas från och med den andra decimalen</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_7c</del>-2(<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>&lt;center&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">&lt;math&gt;0{,}</ins>2<ins style="color: red; font-weight: bold; text-decoration: none;">\ \underline{001}\ \underline{001}\ \underline{001}\,\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;">och det avslöjar att talet är 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><ins style="color: red; font-weight: bold; text-decoration: none;">Genom att sedan multiplicera talet med 10 ett antal gånger kan vi skifta decimalkommat stegvis 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;">::&lt;math&gt;\insteadof[right</ins>]<ins style="color: red; font-weight: bold; text-decoration: none;">{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\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}{10x}{}=2\,,\,\underline{001}\ \underline{001}\ \underline{001}\ 1\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}{100x}{}=20\,,\,01\ 001\ 001\ 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;">::&lt;math&gt;\insteadof</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">right]{10000x}{1000x}{}=200\,,\,1\ 001\ 001\ 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;">:&lt;math&gt;\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 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;">I denna lista ser vi att talen 10''x'' och 10000''x'' har samma decimalutveckling, och det betyder att</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-10x = 2001{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots </ins>- 2<ins style="color: red; font-weight: bold; text-decoration: none;">{,}\underline{001}\ \underline{001}\ \underline{001}\,\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-10x}{} = 1999\,\mbox{.}\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;">Eftersom &lt;math&gt;10000x-10x = 9990x&lt;/math&gt; så 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;9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\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</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_7c-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_7c-2(2).gif]] &lt;/center&gt;--&gt;</div></td></tr> </table>

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<p>Ny sida: {{NAVCONTENT_START}} &lt;center&gt; <a href="/wikis/forberedandematte1/index.php/Bild:1_1_7c-1%282%29.gif" title="Bild:1 1 7c-1(2).gif">Bild:1_1_7c-1(2).gif</a> &lt;/center&gt; &lt;center&gt; <a href="/wikis/forberedandematte1/index.php/Bild:1_1_7c-2%282%29.gif" title="Bild:1 1 7c-2(2).gif">Bild:1_1_7c-2(2).gif</a> &lt;/center&gt; {{NAVCONTENT_STOP}}</p> <p><b>Ny sida</b></p><div>{{NAVCONTENT_START}}<br /> &lt;center&gt; [[Bild:1_1_7c-1(2).gif]] &lt;/center&gt;<br /> &lt;center&gt; [[Bild:1_1_7c-2(2).gif]] &lt;/center&gt;<br /> {{NAVCONTENT_STOP}}</div>

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