MediaWiki API result
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{
"compare": {
"fromid": 1,
"fromrevid": 1,
"fromns": 0,
"fromtitle": "Etusivu",
"toid": 2,
"torevid": 2,
"tons": 0,
"totitle": "Algebrallinen luku",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Rivi 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Rivi 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><<del class=\"diffchange diffchange-inline\">strong</del>><del class=\"diffchange diffchange-inline\">MediaWiki </del>on <del class=\"diffchange diffchange-inline\">onnistuneesti asennettu</del>.</<del class=\"diffchange diffchange-inline\">strong</del>></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Algebrallinen luku''' tarkoittaa sellaista [[reaaliluku|reaali-]] tai [[kompleksiluku]]a </ins><<ins class=\"diffchange diffchange-inline\">math</ins>><ins class=\"diffchange diffchange-inline\">a</math>, joka on [[kokonaisluku]]kertoimisen [[polynomi]]n <math>P(x)</math> [[nollakohta]] eli toteuttaa yht\u00e4l\u00f6n <math>P(a) = 0</math>. Polynomin</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>P(x)=a_n x^n + a_{n-1} x^{n-1} + \\dotsb + a_1 x + a_0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Aste (polynomi)|aste]] tulee olla [[positiivinen luku|positiivinen]], jolloin v\u00e4hint\u00e4\u00e4n yksi kertoimista <math>a_k\\in \\Z, k=1,\\dotsc,n </math> poikkeaa nollasta. Jos vain <math>a_0</math> poikkeaa nollasta, on kyseess\u00e4 [[vakiofunktio]], joka ei t\u00e4yt\u00e4 edell\u00e4 mainittua ehtoa. Yleens\u00e4 algebrallinen luku </ins>on <ins class=\"diffchange diffchange-inline\">kompleksinen, mutta tietyill\u00e4 ehdoilla se voi olla my\u00f6s reaalinen, rationaalinen tai kokonainen</ins>.<<ins class=\"diffchange diffchange-inline\">ref name=ww5</ins>/></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Lis\u00e4tietoja wiki-ohjelmiston k\u00e4yt\u00f6st\u00e4 </del>on <del class=\"diffchange diffchange-inline\">[https:</del>/<del class=\"diffchange diffchange-inline\">/www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special:MyLanguage/Help:Contents k\u00e4ytt\u00f6oppaassa].</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Polynomia, jonka korkeimman asteen termin kerroin </ins>on <ins class=\"diffchange diffchange-inline\"><math>1<</ins>/<ins class=\"diffchange diffchange-inline\">math> ja muut kertoimet ovat kokonaislukuja, kutsutaan '''p\u00e4\u00e4polynomiksi'''</ins>. <ins class=\"diffchange diffchange-inline\">P\u00e4\u00e4polynomin nollakohtaa kutsutaan '''algebralliseksi kokonaisluvuksi''' tai '''kokonaiseksi algebralliseksi luvuksi'''</ins>.<ins class=\"diffchange diffchange-inline\"><ref name=ww7</ins>/<ins class=\"diffchange diffchange-inline\">><ref name=mj</ins>/<ins class=\"diffchange diffchange-inline\">></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>==<del class=\"diffchange diffchange-inline\">= Aloittaminen ===</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">M\u00e4\u00e4ritelm\u00e4st\u00e4 seuraa [[algebran peruslause]]en mukaisesti, ett\u00e4 polynomin nollakohdan <math>a \\in \\Z</math> avulla voidaan p\u00e4\u00e4tell\u00e4 sen yhden tekij\u00e4n olevan binomi <math>x - a</math>. Algebralliseen lukuun voidaan liitt\u00e4\u00e4 useita polynomeja, joissa on t\u00e4m\u00e4 tekij\u00e4. Sit\u00e4 polynomia, jonka aste on matalin, kutsutaan '''minimaalipolynomiksi'''. Minimaalipolynomin '''aste''' on samalla '''algebrallisen luvun aste'''.<ref name</ins>=<ins class=\"diffchange diffchange-inline\">mj/><ref name</ins>=<ins class=\"diffchange diffchange-inline\">ww6/></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Asetusten teko-ohjeita]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Voidaan todistaa</ins>, <ins class=\"diffchange diffchange-inline\">ett\u00e4 algebrallisen luvun minimaalipolynomi on yksik\u00e4sitteinen ja ett\u00e4 minimaalipolynomi on aina tekij\u00e4n\u00e4 muissa luvun polynomeissa</ins>. <ins class=\"diffchange diffchange-inline\">Lis\u00e4ksi minimipolynomi on aina jaoton</ins>. <ins class=\"diffchange diffchange-inline\">Samaan polynomiin liittyv\u00e4t algebralliset luvut ovat toistensa '''konjugaatteja'''</ins>.<ins class=\"diffchange diffchange-inline\"><ref name=mj3</ins>/<ins class=\"diffchange diffchange-inline\">></ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWikin FAQ]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://lists.wikimedia.org/postorius/lists/mediawiki-announce.lists.wikimedia.org/ S\u00e4hk\u00f6postilista</del>, <del class=\"diffchange diffchange-inline\">jolla tiedotetaan MediaWikin uusista versioista]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org/wiki/Special:MyLanguage/Localisation#Translation_resources K\u00e4\u00e4nn\u00e4 MediaWiki\u00e4 kielellesi]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki.org/wiki/Special:MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual:Combating_spam Katso, kuinka torjua sp\u00e4mmi\u00e4 wikiss\u00e4si]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>=== <del class=\"diffchange diffchange-inline\">Asetukset </del>===</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>==<ins class=\"diffchange diffchange-inline\">Johdanto</ins>==</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>==<ins class=\"diffchange diffchange-inline\">=Merkint\u00e4===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Algebrallisten lukujen joukkoa merkit\u00e4\u00e4n joskus <math>\\mathbb{A}</math> tai <math>\\overline{\\Q}</math>. Niit\u00e4 kompleksilukuja, jotka eiv\u00e4t ole algebrallisia lukuja eli <math>\\Complex \\smallsetminus \\mathbb{A}</math>, kutsutaan [[transkendenttiluku|transkendenttiluvuiksi]].<ref name=ww5/></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Tarkista, ett\u00e4 alla olevat taivutusmuodot </del>ovat <del class=\"diffchange diffchange-inline\">oikein</del>. <del class=\"diffchange diffchange-inline\">Jos eiv\u00e4t</del>, <del class=\"diffchange diffchange-inline\">tee tarvittavat muutokset tiedostoon LocalSettings</del>.<del class=\"diffchange diffchange-inline\">php seuraavasti</del>:</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Algebrallinen yht\u00e4l\u00f6===</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> $wgGrammarForms</del>[<del class=\"diffchange diffchange-inline\">'fi'</del>]<del class=\"diffchange diffchange-inline\">['genitive'</del>]<del class=\"diffchange diffchange-inline\">['</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}<del class=\"diffchange diffchange-inline\">'] </del>= <del class=\"diffchange diffchange-inline\">'</del>...<del class=\"diffchange diffchange-inline\">';</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Algebrallisen yht\u00e4l\u00f6n''' juuret </ins>ovat <ins class=\"diffchange diffchange-inline\">''algebrallisia lukuja''</ins>. <ins class=\"diffchange diffchange-inline\">Algebrallinen yht\u00e4l\u00f6 muodostetaan laskettaessa polynomin nollakohtia </ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> $wgGrammarForms['fi']['partitive']['</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}<del class=\"diffchange diffchange-inline\">'] </del>= <del class=\"diffchange diffchange-inline\">'</del>...<del class=\"diffchange diffchange-inline\">';</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>P(x) = 0</math> </ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> $wgGrammarForms</del>[<del class=\"diffchange diffchange-inline\">'fi'</del>]<del class=\"diffchange diffchange-inline\">['elative'</del>]<del class=\"diffchange diffchange-inline\">['</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}<del class=\"diffchange diffchange-inline\">'</del>] = <del class=\"diffchange diffchange-inline\">'</del>...<del class=\"diffchange diffchange-inline\">';</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">eli </ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>\u00a0 <del class=\"diffchange diffchange-inline\">$wgGrammarForms[</del>'<del class=\"diffchange diffchange-inline\">fi</del>'<del class=\"diffchange diffchange-inline\">][</del>'<del class=\"diffchange diffchange-inline\">inessive</del>']<del class=\"diffchange diffchange-inline\">['</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}<del class=\"diffchange diffchange-inline\">'] = '</del>...<del class=\"diffchange diffchange-inline\">';</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>a_n x^n + a_{n-1} x^{n-1} + \\dotsb + a_1 x + a_0 = 0,</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> $wgGrammarForms</del>[<del class=\"diffchange diffchange-inline\">'fi'</del>][<del class=\"diffchange diffchange-inline\">'illative'</del>][<del class=\"diffchange diffchange-inline\">'</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}<del class=\"diffchange diffchange-inline\">'] </del>= <del class=\"diffchange diffchange-inline\">'</del>...<del class=\"diffchange diffchange-inline\">';</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">miss\u00e4 <math>a_k\\in \\Z, k=0,\\dotsc</ins>,<ins class=\"diffchange diffchange-inline\">n</ins>.<ins class=\"diffchange diffchange-inline\"></math> Joskus yht\u00e4l\u00f6n ensimm\u00e4isen termin kerroin <math>a_0 (\\ne 0)</math> jaetaan molemmista puolista pois, jolloin saadaan ''p\u00e4\u00e4polynomin yht\u00e4l\u00f6''</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Taivutusmuodot: </del>{{<del class=\"diffchange diffchange-inline\">GRAMMAR</del>:<del class=\"diffchange diffchange-inline\">genitive</del>|{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}}} <del class=\"diffchange diffchange-inline\">(y\u00f6n) \u2013 </del>{{<del class=\"diffchange diffchange-inline\">GRAMMAR</del>:<del class=\"diffchange diffchange-inline\">partitive</del>|{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}}} <del class=\"diffchange diffchange-inline\">(y\u00f6t\u00e4) \u2013 </del>{{<del class=\"diffchange diffchange-inline\">GRAMMAR</del>:<del class=\"diffchange diffchange-inline\">elative</del>|{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}}} <del class=\"diffchange diffchange-inline\">(y\u00f6st\u00e4) \u2013 </del>{{<del class=\"diffchange diffchange-inline\">GRAMMAR</del>:<del class=\"diffchange diffchange-inline\">inessive</del>|{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}}} <del class=\"diffchange diffchange-inline\">(y\u00f6ss\u00e4) </del>\u2013 {{<del class=\"diffchange diffchange-inline\">GRAMMAR</del>:<del class=\"diffchange diffchange-inline\">illative|</del>{{<del class=\"diffchange diffchange-inline\">SITENAME</del>}}}} <del class=\"diffchange diffchange-inline\">(y\u00f6h\u00f6n)</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\"><math>x^n + b_{n-1} x^{n-1} + \\dotsb + b_1 x + b_0 = 0,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">ja jonka kertoimet ovat [</ins>[<ins class=\"diffchange diffchange-inline\">rationaaliluku</ins>]]<ins class=\"diffchange diffchange-inline\">ja <math>b_k = \\frac</ins>{<ins class=\"diffchange diffchange-inline\">a_k}</ins>{<ins class=\"diffchange diffchange-inline\">a_n</ins>} <ins class=\"diffchange diffchange-inline\">\\in \\Z, k=0,\\dotsc,n-1.</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Koska yht\u00e4l\u00f6n molemmat puolet voi kertoa luvulla <math>c \\in \\Z ( \\ne 0)</math>, voidaan algebrallisen yht\u00e4l\u00f6n kertoimiksi sallia my\u00f6s rationaaliluvut.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Esimerkkej\u00e4 algebrallisista yht\u00e4l\u00f6ist\u00e4 ja -luvuista===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Luvun voi todeta algebralliseksi, jos keksii sille rationaalilukukertoimisen polynomiyht\u00e4l\u00f6n, jonka juuri luku on. Luvun asteen voi p\u00e4\u00e4tell\u00e4 retusoimalla polynomin tekij\u00f6it\u00e4. Seuraavassa on joitakin esimerkkej\u00e4 lukuisasta soveltamiskent\u00e4st\u00e4.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Ensimm\u00e4isen asteen luvut====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Jos polynomi <math>P(x)= ax - b</math> kerroin <math>a = 1</math>, saadaan p\u00e4\u00e4polynomi. T\u00e4m\u00e4n polynomin algebralliset luvut ovat kokonaislukuja, joiden aste on 1. T\u00e4ll\u00f6in voidaan merkit\u00e4 <math> \\Z \\subset \\mathbb{A</ins>}<ins class=\"diffchange diffchange-inline\"></math>. Kaikki rationaaliluvut ovat algebrallisia lukuja, jotka toteuttavat 1. asteen polynomiyht\u00e4l\u00f6n </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>ax - b =0 \\Leftrightarrow x </ins>= <ins class=\"diffchange diffchange-inline\">\\frac {b}{a}</ins>.<ins class=\"diffchange diffchange-inline\"></math> </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">T\u00e4st\u00e4 n\u00e4hd\u00e4\u00e4n, ett\u00e4 <math> \\Q \\subset \\mathbb{A}</math></ins>.<ins class=\"diffchange diffchange-inline\"><ref name=mj/></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Toisen asteen luvut====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Erilaisia esimerkkej\u00e4:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* Kokonaislukujen juuriluvut <math>\\sqrt{c} \\text{\u00a0 ja } -\\sqrt{c}</math> ovat p\u00e4\u00e4polynomin <math>P(x)= x^2-c</math> nollakohtina toisen asteen algebrallisia kokonaislukuja, jotka ovat lis\u00e4ksi toistensa konjugaatteja</ins>. \u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* Irrationaalinen <math>\\frac{1}</ins>{<ins class=\"diffchange diffchange-inline\">\\sqrt</ins>{<ins class=\"diffchange diffchange-inline\">2</ins>}}<ins class=\"diffchange diffchange-inline\"></math> on toista astetta oleva algebrallinen luku, sill\u00e4 se on algebrallisen yht\u00e4l\u00f6n <math>2x^2-1</ins>=<ins class=\"diffchange diffchange-inline\">0</math> juuri</ins>. \u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* Kokonaislukukertoimisen toisen asteen polynomiyht\u00e4l\u00f6n <math>ax^2 + bx + c</math> kaikki ratkaisut ovat algebrallisia lukuja</ins>. <ins class=\"diffchange diffchange-inline\">Joukossa on my\u00f6s paljon erilaisia irrationaaliratkaisuja</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [</ins>[<ins class=\"diffchange diffchange-inline\">Kultainen leikkaus</ins>]] <ins class=\"diffchange diffchange-inline\">on luku </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\phi=\\frac{1}</ins>{<ins class=\"diffchange diffchange-inline\">2}(1+\\sqrt</ins>{<ins class=\"diffchange diffchange-inline\">5</ins>}<ins class=\"diffchange diffchange-inline\">)=1{,</ins>}<ins class=\"diffchange diffchange-inline\">61803\\dots,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">joka on polynomin <math>x^2+x-1=0\\,</math> nollakohta.<ref name=ww5/></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Kompleksiluku|Imaginaariyksikk\u00f6]</ins>] <ins class=\"diffchange diffchange-inline\"><math>i</math> on toista astetta oleva algebrallinen luku, sill\u00e4 se toteuttaa yht\u00e4l\u00f6n <math>x^2+1</ins>=<ins class=\"diffchange diffchange-inline\">0</math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Muita algebrallisia lukuja====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* Kaikki luvut, jotka saadaan polynomin kertoimista peruslaskutoimituksilla ja n-asteisella juurenotolla, ovat algebrallisia lukuja</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Trigonometriset funktiot]], joiden argumenttina olevalla <math>\\pi</math>:ll\u00e4 on [[rationaaliluku|rationaalikerroin]], ovat algebrallisia lukuja</ins>. <ins class=\"diffchange diffchange-inline\">Esimerkiksi jokainen algebrallinen luku </ins> <ins class=\"diffchange diffchange-inline\"><math>\\cos (\\pi/7)</math>, <math>\\cos (3\\pi/7)</math> ja <math>\\cos (5\\pi/7)</math> on minimaalipolynomin <math>8x^3 - 4x^2 - 4x + 1 = 0</math> nollakohta. T\u00e4m\u00e4 tekee luvuista toistensa kolmannen asteen </ins>''<ins class=\"diffchange diffchange-inline\">konjugaatteja</ins>''<ins class=\"diffchange diffchange-inline\">.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*My\u00f6s luvut <math>\\tan (3\\pi/16)</math>, <math>\\tan (7\\pi/16)</math>, <math>\\tan (11\\pi/16)</math> ja <math>\\tan (15\\pi/16)</math> ovat minimaali- ja p\u00e4\u00e4polynomin <math>x^4 - 4x^3 - 6x^2 + 4x + 1</math> nollakohtia ja ovat toistensa nelj\u00e4nnen asteen konjugaatteja ja algebrallisia kokonaislukuja.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Algebrallisten lukujen yleisi\u00e4 ominaisuuksia==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Tiedosto:Algebraic number in the complex plane.png|pienoiskuva|Algebrallisten lukujen sijoittuminen kompleksitasoon.</ins>]<ins class=\"diffchange diffchange-inline\">]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Algebralliset luvut===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Voidaan my\u00f6s todistaa, ett\u00e4 kompleksiluku <math>a + bi</math> on toisen asteen</ins>{{<ins class=\"diffchange diffchange-inline\">L\u00e4hde|Niink\u00f6?|26. kes\u00e4kuuta 2019|vuosi=2019</ins>}} <ins class=\"diffchange diffchange-inline\">algebrallinen luku, jos luvut <math>a</math> ja <math>b</math> ovat algebrallisia</ins>. <ins class=\"diffchange diffchange-inline\">Silloin on my\u00f6s liittoluku <math>a - bi</math> algebrallinen</ins>.<ins class=\"diffchange diffchange-inline\"><ref name=ww5/><ref name=mj/></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Tiheys===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Algebrallisten lukujen joukko on [[tihe\u00e4 joukko|tihe\u00e4]], jolloin kahden mielivaltaisen algebrallisen luvun v\u00e4list\u00e4 l\u00f6ytyy aina kolmas algebrallinen luku riippumatta kuinka l\u00e4hell\u00e4 ensin mainitut kaksi lukua olivat</ins>.<ins class=\"diffchange diffchange-inline\"><ref name=mj2/></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Algebrallisten lukujen mahtavuus===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Algebrallisten luvut ovat [</ins>[<ins class=\"diffchange diffchange-inline\">numeroituva joukko|numeroituvasti \u00e4\u00e4ret\u00f6n]</ins>] <ins class=\"diffchange diffchange-inline\">joukko, jonka </ins>[<ins class=\"diffchange diffchange-inline\">[mahtavuus]</ins>] <ins class=\"diffchange diffchange-inline\">on siis <math>\\aleph_0</math> <ref name=ww2/>. Transkendenttisten lukujen mahtavuus on kuitenkin </ins>[<ins class=\"diffchange diffchange-inline\">[ylinumeroituva joukko|ylinumeroituvasti \u00e4\u00e4ret\u00f6n]].<ref name=mj2/><ref name=brown/></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==\u00a0 L\u00e4hteet\u00a0 ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*</ins>{{<ins class=\"diffchange diffchange-inline\">Kirjaviite | Tekij\u00e4 =Fuchs, Walter R. | Nimeke =Matematiikka | Suomentaja =Mattila, Pekka | Vuosi =1968 | Julkaisupaikka =L\u00e4nsi-Saksa | Julkaisija =Kirjayhtym\u00e4 | Viitattu = </ins>}}</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{Kirjaviite | Tekij\u00e4 =Majaranta, Leo | Nimeke =Algebrallisista ja transkendenttisista luvuista | Vuosi =2011 | Selite =(Pro Gradu-tutkielma) | Julkaisupaikka =Tampere | Julkaisija =Tampereen yliopisto | Tunniste = | Isbn = | www </ins>=<ins class=\"diffchange diffchange-inline\">https://trepo</ins>.<ins class=\"diffchange diffchange-inline\">tuni</ins>.<ins class=\"diffchange diffchange-inline\">fi/bitstream/handle/10024/82679/gradu05183</ins>.<ins class=\"diffchange diffchange-inline\">pdf?sequence=1&isAllowed=y | www-teksti = | Tiedostomuoto =pdf | Viitattu = 2.9.2024 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===\u00a0 Viitteet\u00a0 ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{viitteet|sarakkeet|viitteet=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=brown></ins>{{<ins class=\"diffchange diffchange-inline\">Verkkoviite | Osoite =http</ins>:<ins class=\"diffchange diffchange-inline\">//www.math.brown.edu/~res/MFS/handout8.pdf | Nimeke =Countable and Uncountable Sets | Tekij\u00e4 =Schwartz, Rich | Tiedostomuoto =pdf | Selite =luentomoniste | Ajankohta =2007 | Julkaisupaikka =Providence | Julkaisija =Brown University | Viitattu = </ins>| <ins class=\"diffchange diffchange-inline\">Kieli =</ins>{{<ins class=\"diffchange diffchange-inline\">en</ins>}} }}<ins class=\"diffchange diffchange-inline\"></ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=ww2></ins>{{<ins class=\"diffchange diffchange-inline\">Verkkoviite | Osoite =http</ins>:<ins class=\"diffchange diffchange-inline\">//mathworld.wolfram.com/Aleph-0.html | Nimeke =Aleph-0 | Tekij\u00e4 =Weisstein, Eric W. | Selite =Math World \u2013 A Wolfram Web Resource |\u00a0 Julkaisija =Wolfram Research </ins>| <ins class=\"diffchange diffchange-inline\">Kieli =</ins>{{<ins class=\"diffchange diffchange-inline\">en</ins>}} }}<ins class=\"diffchange diffchange-inline\"></ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=ww5></ins>{{<ins class=\"diffchange diffchange-inline\">Verkkoviite | Osoite =http</ins>:<ins class=\"diffchange diffchange-inline\">//mathworld.wolfram.com/AlgebraicNumber.html | Nimeke =Algebraic Number </ins>| <ins class=\"diffchange diffchange-inline\">Tekij\u00e4 =Weisstein, Eric W. | Selite =Math World \u2013 A Wolfram Web Resource |\u00a0 Julkaisija =Wolfram Research | Kieli =</ins>{{<ins class=\"diffchange diffchange-inline\">en</ins>}} }}<ins class=\"diffchange diffchange-inline\"></ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=ww6></ins>{{<ins class=\"diffchange diffchange-inline\">Verkkoviite | Osoite =http</ins>:<ins class=\"diffchange diffchange-inline\">//mathworld.wolfram.com/AlgebraicNumberMinimalPolynomial.html | Nimeke =Algebraic Number Minimal Polynomial | Tekij\u00e4 =Barile, Margherita & Rowland, Todd & Weisstein, Eric W. </ins>| <ins class=\"diffchange diffchange-inline\">Selite =Math World \u2013 A Wolfram Web Resource |\u00a0 Julkaisija =Wolfram Research | Kieli =</ins>{{<ins class=\"diffchange diffchange-inline\">en</ins>}} }}<ins class=\"diffchange diffchange-inline\"></ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=ww7>{{Verkkoviite | Osoite =http://mathworld.wolfram.com/AlgebraicInteger.html | Nimeke =Algebraic Integer | Tekij\u00e4 =Weisstein, Eric W. | Selite =Math World </ins>\u2013 <ins class=\"diffchange diffchange-inline\">A Wolfram Web Resource |\u00a0 Julkaisija =Wolfram Research | Kieli =</ins>{{<ins class=\"diffchange diffchange-inline\">en}} }}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=mj>Majaranta, Leo</ins>: <ins class=\"diffchange diffchange-inline\">Algebrallisista ja transkendenttisista luvuista, s. 7\u20139</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=mj2>Majaranta, Leo, s. 12\u201313</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<ref name=mj3>Majaranta, Leo, s. 13\u201316</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Aiheesta muualla==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>{{<ins class=\"diffchange diffchange-inline\">commonscat</ins>}}</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Pahikkala, J.: [http://www.wakkanet.fi/~pahio/alg.luvut.html Kirjoituksia matematiikasta] {{Wayback|1=http://www.wakkanet.fi/~pahio/alg.luvut.html |p\u00e4iv\u00e4ys=20120208025949 </ins>}}</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Halko, Aapo: [http://solmu.math.helsinki</ins>.<ins class=\"diffchange diffchange-inline\">fi/1998/3/halko/ Joukko-oppia reaaliluvuilla]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Lukujoukkoja}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Luokka:Algebra]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Luokka:Lukuavaruudet]]</ins></div></td></tr>\n"
}
}