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<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<!DOCTYPE html>
<html lang="en">
<head>
<title>Avoin taulukkokirja</title>
<meta charset="utf-8"/>
<script type="text/javascript">
WebFontConfig = {
google: { families: [ 'Open+Sans:300:latin' ] }
};
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var wf = document.createElement('script');
wf.src = ('https:' == document.location.protocol ? 'https' : 'http') +
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var s = document.getElementsByTagName('script')[0];
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</script>
<link rel="stylesheet" type="text/css" href="tyyli.css">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
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inlineMath: [['$','$'], ['\\(','\\)']],
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MathJax.Hub.Config({
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displayAlign: "center"
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</script>
</head>
<body>
<!--
\(
\newcommand{\dd}{\operatorname{d\!}{}}
\)
-->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<!DOCTYPE html>
<html lang="en">
<head>
<title>Avoin taulukkokirja | Matematiikka</title>
<meta charset="utf-8"/>
<script type="text/javascript">
WebFontConfig = {
google: { families: [ 'Open+Sans:300:latin' ] }
};
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var wf = document.createElement('script');
wf.src = ('https:' == document.location.protocol ? 'https' : 'http') +
'://ajax.googleapis.com/ajax/libs/webfont/1/webfont.js';
wf.type = 'text/javascript';
wf.async = 'true';
var s = document.getElementsByTagName('script')[0];
s.parentNode.insertBefore(wf, s);
})();
</script>
<link rel="stylesheet" type="text/css" href="tyyli.css">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [['$','$'], ['\\(','\\)']],
processEscapes: true
}
});
</script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
MathJax.Hub.Config({
jax: ["input/TeX","output/HTML-CSS"],
displayAlign: "center"
});
</script>
</head>
<body>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h2>Merkintöjä ja symboleja</h2>
<section>
<h3>Kreikkalaiset aakkoset </h3>
<div class="viisPalstaa">
<table class="tg" id="kreikkalaiset">
<tr>
<td class="tg-031e">$$ \alpha, A $$</td>
<td class="tg-031e">Alfa</td>
</tr>
<tr>
<td class="tg-031e">$$\beta, B$$</td>
<td class="tg-031e">Beeta</td>
</tr>
<tr>
<td class="tg-031e">$$\gamma, \Gamma$$</td>
<td class="tg-031e">Gamma</td>
</tr>
<tr>
<td class="tg-031e">$$\delta, \Delta$$</td>
<td class="tg-031e">Delta</td>
</tr>
<tr>
<td class="tg-031e">$$ \epsilon,\varepsilon, E$$</td>
<td class="tg-031e">Epsilon</td>
</tr>
<tr>
<td class="tg-031e">$$\zeta, Z$$</td>
<td class="tg-031e">Zeeta</td>
</tr>
<tr>
<td class="tg-031e">$$\eta, H$$</td>
<td class="tg-031e">Eeta</td>
</tr>
<tr>
<td class="tg-031e">$$\theta, \vartheta, \Theta$$</td>
<td class="tg-031e">Theeta</td>
</tr>
<tr>
<td class="tg-031e">$$\iota, I$$</td>
<td class="tg-031e">Ioota</td>
</tr>
<tr>
<td class="tg-031e">$$\kappa, K$$</td>
<td class="tg-031e">Kappa</td>
</tr>
<tr>
<td class="tg-031e">$$\lambda, \Lambda$$</td>
<td class="tg-031e">Lambda</td>
</tr>
<tr>
<td class="tg-031e">$$\mu, M$$</td>
<td class="tg-031e">Myy</td>
</tr>
<tr>
<td class="tg-031e">$$\nu, N$$</td>
<td class="tg-031e">Nyy</td>
</tr>
<tr>
<td class="tg-031e">$$\xi, \Xi$$</td>
<td class="tg-031e">Ksii</td>
</tr>
<tr>
<td class="tg-031e">$$o, O$$</td>
<td class="tg-031e">Omikron</td>
</tr>
<tr>
<td class="tg-031e">$$\pi, \Pi$$</td>
<td class="tg-031e">Pii</td>
</tr>
<tr>
<td class="tg-031e">$$\rho, \varrho, P$$</td>
<td class="tg-031e">Roo</td>
</tr>
<tr>
<td class="tg-031e">$$\sigma, \Sigma$$</td>
<td class="tg-031e">Sigma</td>
</tr>
<tr>
<td class="tg-031e">$$\tau, T$$</td>
<td class="tg-031e">Tau</td>
</tr>
<tr>
<td class="tg-031e">$$\upsilon, \Upsilon$$</td>
<td class="tg-031e">Ypsilon</td>
</tr>
<tr>
<td class="tg-031e">$$\phi, \varphi, \Phi$$</td>
<td class="tg-031e">Fii</td>
</tr>
<tr>
<td class="tg-031e">$$\chi, X$$</td>
<td class="tg-031e">Khii</td>
</tr>
<tr>
<td class="tg-031e">$$\psi, \Psi$$</td>
<td class="tg-031e">Psii</td>
</tr>
<tr>
<td class="tg-031e">$$\omega, \Omega$$</td>
<td class="tg-031e">Omega</td>
</tr>
</table>
<!--
<table class="tg" id="kreikkalaiset">
<tr>
<td class="tg-031e">$$ \alpha, A $$</td>
<td class="tg-031e">Alfa</td>
</tr>
<tr>
<td class="tg-031e">$$\beta, B$$</td>
<td class="tg-031e">Beeta</td>
</tr>
<tr>
<td class="tg-031e">$$\gamma, \Gamma$$</td>
<td class="tg-031e">Gamma</td>
</tr>
<tr>
<td class="tg-031e">$$\delta, \Delta$$</td>
<td class="tg-031e">Delta</td>
</tr>
<tr>
<td class="tg-031e">$$ \epsilon,\varepsilon, E$$</td>
<td class="tg-031e">Epsilon</td>
</tr>
<tr>
<td class="tg-031e">$$\zeta, Z$$</td>
<td class="tg-031e">Zeeta</td>
</tr>
<tr>
<td class="tg-031e">$$\eta, H$$</td>
<td class="tg-031e">Eeta</td>
</tr>
<tr>
<td class="tg-031e">$$\theta, \vartheta, \Theta$$</td>
<td class="tg-031e">Theeta</td>
</tr>
<tr>
<td class="tg-031e">$$\iota, I$$</td>
<td class="tg-031e">Ioota</td>
</tr>
<tr>
<td class="tg-031e">$$\kappa, K$$</td>
<td class="tg-031e">Kappa</td>
</tr>
<tr>
<td class="tg-031e">$$\lambda, \Lambda$$</td>
<td class="tg-031e">Lambda</td>
</tr>
<tr>
<td class="tg-031e">$$\mu, M$$</td>
<td class="tg-031e">Myy</td>
</tr>
<tr>
<td class="tg-031e">$$\nu, N$$</td>
<td class="tg-031e">Nyy</td>
</tr>
<tr>
<td class="tg-031e">$$\xi, \Xi$$</td>
<td class="tg-031e">Ksii</td>
</tr>
<tr>
<td class="tg-031e">$$o, O$$</td>
<td class="tg-031e">Omikron</td>
</tr>
<tr>
<td class="tg-031e">$$\pi, \Pi$$</td>
<td class="tg-031e">Pii</td>
</tr>
<tr>
<td class="tg-031e">$$\rho, \varrho, P$$</td>
<td class="tg-031e">Roo</td>
</tr>
<tr>
<td class="tg-031e">$$\sigma, \Sigma$$</td>
<td class="tg-031e">Sigma</td>
</tr>
<tr>
<td class="tg-031e">$$\tau, T$$</td>
<td class="tg-031e">Tau</td>
</tr>
<tr>
<td class="tg-031e">$$\upsilon, \Upsilon$$</td>
<td class="tg-031e">Ypsilon</td>
</tr>
<tr>
<td class="tg-031e">$$\phi, \varphi, \Phi$$</td>
<td class="tg-031e">Fii</td>
</tr>
<tr>
<td class="tg-031e">$$\chi, X$$</td>
<td class="tg-031e">Khii</td>
</tr>
<tr>
<td class="tg-031e">$$\psi, \Psi$$</td>
<td class="tg-031e">Psii</td>
</tr>
<tr>
<td class="tg-031e">$$\omega, \Omega$$</td>
<td class="tg-031e">Omega</td>
</tr>
</table>
-->
</div>
<h3>Symboleja</h3>
<div class="kaksPalstaa">
<table class="tg">
<tr>
<td class="tg-031e">$$=$$</td>
<td class="tg-031e">on yhtä suuri kuin</td>
</tr>
<tr>
<td class="tg-031e">$$\neq$$</td>
<td class="tg-031e">ei ole yhtä suuri kuin</td>
</tr>
<tr>
<td class="tg-031e">$$<,>$$</td>
<td class="tg-031e">on suurempi kuin,<br>on pienempi kuin</td>
</tr>
<tr>
<td class="tg-031e">$$\leq,\neq$$</td>
<td class="tg-031e">on suurempi tai yhtä suuri kuin,<br>on pienempi tai yhtä suuri kuin</td>
</tr>
<tr>
<td class="tg-031e">$$\varpropto$$</td>
<td class="tg-031e">on verrannollinen</td>
</tr>
<tr>
<td class="tg-031e">$$+$$</td>
<td class="tg-031e">yhteenlasku</td>
</tr>
<tr>
<td class="tg-031e">$$-$$</td>
<td class="tg-031e">vähennyslasku</td>
</tr>
<tr>
<td class="tg-031e">$$\cdot$$</td>
<td class="tg-031e">kertolasku, pistetulo</td>
</tr>
<tr>
<td class="tg-031e">$$\times$$</td>
<td class="tg-031e">kertolasku, ristitulo</td>
</tr>
<tr>
<td class="tg-031e">$$/, \frac{""}{""}, :, \div$$ (laskimissa)</td>
<td class="tg-031e">jakolasku</td>
</tr>
<tr>
<td class="tg-031e">$$\pm$$</td>
<td class="tg-031e">plus-miinus</td>
</tr>
<tr>
<td class="tg-031e">$$\mp$$</td>
<td class="tg-031e">miinus-plus</td>
</tr>
<tr>
<td class="tg-031e">$$\sqrt{}$$</td>
<td class="tg-031e">neliöjuuri</td>
</tr>
<tr>
<td class="tg-031e">$$|...|$$</td>
<td class="tg-031e">itseisarvo, etäisyys</td>
</tr>
<tr>
<td class="tg-031e">$$!$$</td>
<td class="tg-031e">kertoma</td>
</tr>
<tr>
<td class="tg-031e">$${,}$$</td>
<td class="tg-031e">joukko</td>
</tr>
</table>
</div>
<h3>Numeromerkit</h3>
<table class="tg">
<tr>
<th class="tg-031e">Arabialaiset</th>
<th class="tg-031e">Armenialaiset</th>
<th class="tg-031e">Babylonialaiset</th>
<th class="tg-031e">Heprealaiset</th>
<th class="tg-031e">Kiinalaiset</th>
<th class="tg-031e">Kreikkalaiset</th>
<th class="tg-031e">Roomalaiset</th>
</tr>
<tr>
<td class="tg-031e">0</td>
<td class="tg-031e"></td>
<td class="tg-031e"></td>
<td class="tg-031e"></td>
<td class="tg-031e">零</td>
<td class="tg-031e"></td>
<td class="tg-031e"></td>
</tr>
<tr>
<td class="tg-031e">1</td>
<td class="tg-031e">Ա</td>
<td class="tg-031e"></td>
<td class="tg-031e">א</td>
<td class="tg-031e">壹</td>
<td class="tg-031e">α´</td>
<td class="tg-031e">I</td>
</tr>
<tr>
<td class="tg-031e">2</td>
<td class="tg-031e">Բ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ב</td>
<td class="tg-031e">贰</td>
<td class="tg-031e">β´</td>
<td class="tg-031e">II</td>
</tr>
<tr>
<td class="tg-031e">3</td>
<td class="tg-031e">Գ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ג</td>
<td class="tg-031e">叁</td>
<td class="tg-031e">γ´</td>
<td class="tg-031e">III</td>
</tr>
<tr>
<td class="tg-031e">4</td>
<td class="tg-031e">Դ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ד</td>
<td class="tg-031e">肆</td>
<td class="tg-031e">δ´</td>
<td class="tg-031e">IV</td>
</tr>
<tr>
<td class="tg-031e">5</td>
<td class="tg-031e">Ե</td>
<td class="tg-031e"></td>
<td class="tg-031e">ה</td>
<td class="tg-031e">伍</td>
<td class="tg-031e">ε´</td>
<td class="tg-031e">V</td>
</tr>
<tr>
<td class="tg-031e">6</td>
<td class="tg-031e">Զ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ו</td>
<td class="tg-031e">陆</td>
<td class="tg-031e">στ´</td>
<td class="tg-031e">VI</td>
</tr>
<tr>
<td class="tg-031e">7</td>
<td class="tg-031e">Է</td>
<td class="tg-031e"></td>
<td class="tg-031e">ז</td>
<td class="tg-031e">柒</td>
<td class="tg-031e">ζ´</td>
<td class="tg-031e">VII</td>
</tr>
<tr>
<td class="tg-031e">8</td>
<td class="tg-031e">Ը</td>
<td class="tg-031e"></td>
<td class="tg-031e">ח</td>
<td class="tg-031e">捌</td>
<td class="tg-031e">η´</td>
<td class="tg-031e">VIII</td>
</tr>
<tr>
<td class="tg-031e">9</td>
<td class="tg-031e">Թ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ט</td>
<td class="tg-031e">玖</td>
<td class="tg-031e">θ´</td>
<td class="tg-031e">IX</td>
</tr>
<tr>
<td class="tg-031e">10</td>
<td class="tg-031e">Ժ</td>
<td class="tg-031e"></td>
<td class="tg-031e">י</td>
<td class="tg-031e">拾</td>
<td class="tg-031e">ι´</td>
<td class="tg-031e">X</td>
</tr>
<tr>
<td class="tg-031e">50</td>
<td class="tg-031e">Ծ</td>
<td class="tg-031e"></td>
<td class="tg-031e">נ</td>
<td class="tg-031e">伍拾</td>
<td class="tg-031e">ν´</td>
<td class="tg-031e">L</td>
</tr>
<tr>
<td class="tg-031e">100</td>
<td class="tg-031e">Ճ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ק</td>
<td class="tg-031e">佰</td>
<td class="tg-031e">ρ´</td>
<td class="tg-031e">C</td>
</tr>
<tr>
<td class="tg-031e">500</td>
<td class="tg-031e">Շ</td>
<td class="tg-031e"></td>
<td class="tg-031e">ך</td>
<td class="tg-031e"></td>
<td class="tg-031e">φ´</td>
<td class="tg-031e"></td>
</tr>
</table>
<h3>Lukujoukot</h3>
<div class="kaksPalstaa">
<table class="tg">
<tr>
<td class="tg-031e">$$\mathbb{N}$$</td>
<td class="tg-031e">Luonnollisten lukujen joukko</td>
</tr>
<tr>
<td class="tg-031e">$$\mathbb{Z}$$</td>
<td class="tg-031e">Kokonaislukujen joukko</td>
</tr>
<tr>
<td class="tg-031e">$$\mathbb{Q}$$</td>
<td class="tg-031e">Rationaalilukujen joukko</td>
</tr>
<tr>
<td class="tg-031e">$$\bar{\mathbb{Q}}, \mathbb{R} - \mathbb{Q}, \mathbb{R} \setminus \mathbb{Q}$$</td>
<td class="tg-031e">Irrationaalilukujen joukko</td>
</tr>
<tr>
<td class="tg-031e">$$\mathbb{R}$$</td>
<td class="tg-031e">Reaalilukujen joukko</td>
</tr>
<tr>
<td class="tg-031e">$$\mathbb{C}$$</td>
<td class="tg-031e">Kompleksilukujen joukko</td>
</tr>
</table>
</div>
<h3>Kantalukujärjestelmät</h3>
<table class="tg">
<tr>
<th>Desimaali</th>
<th>Binaari</th>
<th>Oktaali</th>
<th>Heksadesimaali</th>
</tr>
<tr>
<td>$$0$$</td>
<td>$$0$$</td>
<td>$$0$$</td>
<td>$$0$$</td>
</tr>
<tr>
<td>$$1$$</td>
<td>$$1$$</td>
<td>$$1$$</td>
<td>$$1$$</td>
</tr>
<tr>
<td>$$2$$</td>
<td>$$10$$</td>
<td>$$2$$</td>
<td>$$2$$</td>
</tr>
<tr>
<td>$$3$$</td>
<td>$$11$$</td>
<td>$$3$$</td>
<td>$$3$$</td>
</tr>
<tr>
<td>$$4$$</td>
<td>$$100$$</td>
<td>$$4$$</td>
<td>$$4$$</td>
</tr>
<tr>
<td>$$5$$</td>
<td>$$101$$</td>
<td>$$5$$</td>
<td>$$5$$</td>
</tr>
<tr>
<td>$$6$$</td>
<td>$$110$$</td>
<td>$$6$$</td>
<td>$$6$$</td>
</tr>
<tr>
<td>$$7$$</td>
<td>$$111$$</td>
<td>$$7$$</td>
<td>$$7$$</td>
</tr>
<tr>
<td>$$8$$</td>
<td>$$1000$$</td>
<td>$$10$$</td>
<td>$$8$$</td>
</tr>
<tr>
<td>$$9$$</td>
<td>$$1001$$</td>
<td>$$11$$</td>
<td>$$9$$</td>
</tr>
<tr>
<td>$$10$$</td>
<td>$$1010$$</td>
<td>$$12$$</td>
<td>$$A$$</td>
</tr>
<tr>
<td>$$50$$</td>
<td>$$110010$$</td>
<td>$$62$$</td>
<td>$$32$$</td>
</tr>
<tr>
<td>$$100$$</td>
<td>$$1100100$$</td>
<td>$$144$$</td>
<td>$$64$$</td>
</tr>
<tr>
<td>$$1000$$</td>
<td>$$1111101000$$</td>
<td>$$1750$$</td>
<td>$$3E8$$</td>
</tr>
<tr>
<td>$$10000$$</td>
<td>$$10011100010000$$</td>
<td>$$23420$$</td>
<td>$$2710$$</td>
</tr>
<!--- TODO lisää muita --->
</table>
</section>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h2>Lajittelematon</h2>
<section>
<h3>Vakioita</h3>
$$\pi = 3,14159265359$$
$$\tau = 2 \pi = 6,28318530718$$
$$e = 2,71828182846$$
<h3><!---ALAKOHTA --></h3>
<!---SISÄLTÖ -->
</section>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h2>Reaalilukujen määrittely aksiomaattisesti</h2>
<section>
<div>
Reaaliluvut voidaan määritellä täydellisenä järjestettynä kuntana $$(\mathbb{R}, +, \cdot, \leq)$$, eli algebrallisena rakenteena, joka toteuttaa seuraavat ominaisuudet:
</div>
<h3>Kunta-aksioomat</h3>
<table class="tg">
<tr>
<td class="tg-031e">$a+b=b+a$</td>
<td class="tg-031e">Yhteenlaskun vaihdannaisuus</td>
</tr>
<tr>
<td class="tg-031e">$(a+c)+c=a+(b+c)$</td>
<td class="tg-031e">Yhteenlaskun liitännäisyys</td>
</tr>
<tr>
<td class='tg-031e'>$a + 0 = a$</td>
<td class='tg-031e'>Nolla-alkio (yhteenlaskun neutraalilalkio)</td>
</tr>
<tr>
<td class='tg-031e'>$a + a' = 0,$ merkitään $a' = -a$</td>
<td class='tg-031e'>Vastaluku</td>
</tr>
<tr>
<td class='tg-031e'>$ a \cdot b = b \cdot a$</td>
<td class='tg-031e'>Kertolaskun vaihdannaisuus</td>
</tr>
<tr>
<td class='tg-031e'>$ (a \cdot b) \cdot c = a \cdot (b \cdot c)$</td>
<td class='tg-031e'>Kertolaskun liitännäisyys</td>
</tr>
<tr>
<td class='tg-031e'>$ a \cdot 1 = a$</td>
<td class='tg-031e'>Ykkösalkio (kertolaskun neutraalialkio)</td>
</tr>
<tr>
<td class='tg-031e'>$ a \cdot a' = 1,$ merkitään $a' = a^{-1} = \frac{1}{a}$</td>
<td class='tg-031e'>Käänteisluku</td>
</tr>
<tr>
<td class='tg-031e'>$ a \cdot (b + c) = a \cdot b + a \cdot c$</td>
<td class='tg-031e'>Osittelulaki</td>
</tr>
</table>
<h3>Järjestysrelaation ominaisuudet</h3>
<table class="tg">
<tr>
<td class="tg-031e">$a\leq a$</td>
<td class="tg-031e">Refleksiivisyys</td>
</tr>
<tr>
<td class="tg-031e">Jos $a\leq b$ ja $ b \leq a$, niin $a = b$.</td>
<td class="tg-031e">Antisymmetrisyys</td>
</tr>
<tr>
<td class="tg-031e">Jos $a\leq b$ ja $ b \leq c$, niin $ a \leq c$.</td>
<td class="tg-031e">Transitiivisuus.</td>
</tr>
<tr>
<td class="tg-031e">$a\leq b$ tai $ b \leq a$.</td>
<td class="tg-031e">Totaalisuus.</td>
</tr>
</table>
<h3>Järjestysrelaation ja kunnan yhteensopivuus</h3>
<table class="tg">
<tr>
<td class="tg-031e">Jos $a\leq b$, niin $ a + c \leq b + c$.</td>
<td class="tg-031e">Järjestyksen säilyminen yhteenlaskussa.</td>
</tr>
<tr>
<td class="tg-031e">Jos $0\leq a$ ja $0 \leq b$, niin $ 0 \leq a\cdot b$.</td>
<td class="tg-031e">Ei-negatiivisuuden säilyminen kertolaskussa.</td>
</tr>
</table>
<h3>Täydellisyysominaisuus</h3>
Jos joukon $\mathbb{R}$ epätyhjällä osajoukolla $A$ on yläraja, osajoukolla $A$ on pienin yläraja.
</section>
<h2>Yleistä</h2>
<section>
<h3>Rationaalilukujen laskutoimitukset</h3>
<div class="kaksPalstaa">
\[ \text{laventaminen}\]
\[ \longrightarrow\]
\[
\frac{a}{b} = \frac{ca}{cb}
\]
\[ \longleftarrow\]
\[ \text{supistaminen}\]
missä $b \neq 0$ ja $c \neq 0$.
$$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$
$$\frac{a}{b} : \frac{c}{d} = \frac{ad}{bc}$$
</div>
</section>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h2>Kompleksiluvut</h2>
<section>
<header>
<h3>Kompleksiluvun määritelmä</h3>
</header>
Kompleksilukujen joukko $\mathbb{C}$ eli <i>kompleksitaso</i> ovat kunta, jonka alkioille $$z = x + yi = (x, y) \in \mathbb{R}^2$$ on määritelty
<ul>
<li>yhteenlasku $ (a, b) + (c, d) = (a + c, b + d)$</li>
<li>kertolasku $(a, b)(c, d) = (ac - bd, bc + ad)$.</li>
</ul>
Kompleksilukua $z \in \mathbb{C}$ merkitään tavallisesti $z = x + yi$, jolloin
<ul>
<li>$(a + bi) + (c + di) = a + c + (b + d )i$</li>
<li>$(a + bi) (c + di) = (ac - bd) + (bc + ad)i $.</li>
</ul>
<h3>Muita määritelmiä</h3>
<table class="tg">
<tr>
<td class="tg-031e">Vaihtoehtoisia merkintöjä kompleksiluvulle $z$:</td>
<td class="tg-031e">$ x + y, \quad ix + iy, \quad x + y\textrm{i},\quad x + yj$</td>
</tr>
<tr>
<td class="tg-031e">Kompleksiluvun reaaliosa</td>
<td class="tg-031e">$\operatorname{Re}(z) = \Re(z) = x$</td>
</tr>
<tr>
<td class="tg-031e">Imaginääriosa</td>
<td class="tg-031e">$\operatorname{Im}(z) = \Im(z) = y$</td>
</tr>
<tr>
<td class="tg-031e">Iteisarvo eli moduli (myös magnitudi)</td>
<td class="tg-031e">$r = |z| = |x+ yi| = \sqrt{x^2 + y^2}$</td>
</tr>
<tr>
<td class="tg-031e">Vaihekulma eli argumentti</td>
<td class="tg-031e">$\phi = \operatorname{arg}(z); \quad \tan \phi = \frac{y}{x}, a \not = 0$</td>
</tr>
<tr>
<td class="tg-031e">Liittoluku eli kompleksikonjugaatti</td>
<td class="tg-031e">$z^\ast = \overline{z} = \overline{x + yi} = x - yi$</td>
</tr>
</table>
<h3>Kompleksiluvun napaesitys</h3>
$$z = x + yi = |z| e^{i\phi} = |z| (\cos \phi + i \sin \phi) $$
Tunnetaan muodossa $ e^{i\phi} = \cos \phi + i \sin \phi$ myös
<i>Eulerin kaavana</i> tai <i>Eulerin lauseena (kompleksiluvuille).</i>
<h3>Ominaisuuksia ja laskutoimituksia</h3>
<table class="tg">
<tr>
<td class="tg-031e">Ykkösen neliöjuuri</td>
<td class="tg-031e">$i^2 = -1; \quad i = \sqrt{-1}$</td>
</tr>
<tr>
<td class="tg-031e">Jakolasku</td>
<td class="tg-031e">$\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i$</td>
</tr>
<tr>
<td class="tg-031e">Itseisarvon neliö</td>
<td class="tg-031e">$|z|^2 = x^2 + y^2 = (x + yi)(x - yi) = z\overline{z}$</td>
</tr>
<tr>
<td class="tg-031e">Liittoluvun liittoluku</td>
<td class="tg-031e">$(z^\ast)^\ast = \overline{\overline{z}} = z$</td>
</tr>
<tr>
<td class="tg-031e">Kokonaislukupotenssi (De Moivren kaava)</td>
<td class="tg-031e">$(z^n) = (r(\cos \phi + i \sin \phi))^n = r^n(\cos n\phi + i \sin n\phi)$</td>
</tr>
</table>
</section>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h2>Euklideen geometria</h2>
<section>
<h3>Piste, suora, taso(?)</h3>
<!---SISÄLTÖ -->
<h3>Euklidisen geometrian aksioomat</h3>
<ul style="list-style-type:circle">
<li>Mitkä tahansa kaksi pistettä voidaan yhdistää janalla.</li>
<li>Mitä tahansa janaa voidaan jatkaa.</li>
<li>Kaikki suorat kulmat ovat yhtä suuria.</li>
</ul>
<h3>Ympyrät</h3>
<!---SISÄLTÖ -->
<h3>Kolmiot, lauseet</h3>
<!---SISÄLTÖ -->
<h3>Nelikulmiot, säännölliset monikulmiot</h3>
<!---SISÄLTÖ -->
<h3>Avaruuskappaleet</h3>
<!---SISÄLTÖ -->
</section>
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA LOPPUU -->
<!---PÄÄDOKUMENTTIIN KOPIOITAVA OSA ALKAA -->
<h1>Geometria</h1>
<h2>Neliö</h2>
<img width="50%" src="kuvat/nelio.svg"/>
$$A=a^{2}$$
$$d=a \sqrt{2}
<h2>Puolisuunnikas</h2>
<img width="50%" src="kuvat/puolisuunnikas.svg"/>
$$kaavaa$$ <!-- TODO: lisää kaavat -->
<h1>Trigometria</h1>
<section>
<h2>Suorakulmaisen kolmion trigonometriaa</h2>
<div>
<img width="100%" src="kuvat/maFig1.svg"/>
<ol>
<li>$$ \sin ( \alpha ) = \frac{a}{c} $$</li>
<li>$$ \cos ( \alpha ) = \frac{b}{c} $$</li>
<li>$$ \tan ( \alpha ) = \frac{a}{b} $$</li>
<li>$$ \cot ( \alpha ) = \frac{b}{a} $$</li>
</ol>
</div>
</section>
<section>
<header>
<h2>Trigonometristen funktioiden tarkkoja arvoja</h2>
</header>
<!--Taulukko arvoista TODO: puuttuvat 45:n monikerrat-->
<table class="tg">
<tr>
<th class="tg-031e">asteet</th>
<th class="tg-031e">radiaanit</th>
<th class="tg-031e">sin</th>
<th class="tg-031e">cos</th>
<th class="tg-031e">tan</th>
<!-- <th class="tg-031e">cot</th> -->
</tr>
<tr>
<td class="tg-031e">$$0$$</td>
<td class="tg-031e">$$0$$</td>
<td class="tg-031e">$$0$$</td>
<td class="tg-031e">$$1$$</td>
<td class="tg-031e">$$0$$</td>
<!-- <td class="tg-031e">$$-$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$30$$</td>
<td class="tg-031e">$$\frac{\pi}{6}$$</td>
<td class="tg-031e">$$\frac{1}{2}$$</td>
<td class="tg-031e">$$\frac{\sqrt{3}}{2}$$</td>
<td class="tg-031e">$$\frac{1}{\sqrt{3}}$$</td>
<!-- <td class="tg-031e">$$\sqrt{3}$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$45$$</td>
<td class="tg-031e">$$\frac{\pi}{4}$$</td>
<td class="tg-031e">$$\frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$\frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$1$$</td>
<!-- <td class="tg-031e">$$1$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$60$$</td>
<td class="tg-031e">$$\frac{\pi}{3}$$</td>
<td class="tg-031e">$$\frac{\sqrt{3}}{2}$$</td>
<td class="tg-031e">$$\frac{1}{2}$$</td>
<td class="tg-031e">$$\sqrt{3}$$</td>
<!-- <td class="tg-031e">$$\frac{1}{\sqrt{3}}$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$90$$</td>
<td class="tg-031e">$$\frac{\pi}{2}$$</td>
<td class="tg-031e">$$1$$</td>
<td class="tg-031e">$$0$$</td>
<td class="tg-031e">$$-$$</td>
<!-- <td class="tg-031e">$$0$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$120$$</td>
<td class="tg-031e">$$\frac{2\pi}{2}$$</td>
<td class="tg-031e">$$\frac{\sqrt{3}}{2}$$</td>
<td class="tg-031e">$$-\frac{1}{2}$$</td>
<td class="tg-031e">$$-\sqrt{3}$$</td>
<!-- <td class="tg-031e">$$-\frac{1}{\sqrt{3}}$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$135$$</td>
<td class="tg-031e">$$\frac{3\pi}{4}$$</td>
<td class="tg-031e">$$\frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$- \frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$-1$$</td>
<!-- <td class="tg-031e">$$-1$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$150$$</td>
<td class="tg-031e">$$\frac{5 \pi}{6}$$</td>
<td class="tg-031e">$$\frac{1}{2}$$</td>
<td class="tg-031e">$$-\frac{\sqrt{3}}{2}$$</td>
<td class="tg-031e">$$-\frac{1}{\sqrt{3}}$$</td>
<!-- <td class="tg-031e">$$-\sqrt{3}$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$180$$</td>
<td class="tg-031e">$$\pi$$</td>
<td class="tg-031e">$$0$$</td>
<td class="tg-031e">$$-1$$</td>
<td class="tg-031e">$$0$$</td>
<!-- <td class="tg-031e">$$-$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$210$$</td>
<td class="tg-031e">$$\frac{7 \pi}{6}$$</td>
<td class="tg-031e">$$-\frac{1}{2}$$</td>
<td class="tg-031e">$$-\frac{\sqrt{3}}{2}$$</td>
<td class="tg-031e">$$\frac{1}{\sqrt{3}}$$</td>
<!-- <td class="tg-031e">$$\sqrt{3}$$</td> -->
</tr>
<tr>
<td class="tg-031e">$$225$$</td>
<td class="tg-031e">$$\frac{5 \pi}{4}$$</td>
<td class="tg-031e">$$- \frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$- \frac{1}{\sqrt{2}}$$</td>
<td class="tg-031e">$$1$$</td>