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| <title>Ciało liczb zespolonych — Matematyka - dokumentacja</title> | ||
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| <div class="document"> | ||
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| <section id="cialo-liczb-zespolonych"> | ||
| <h1>Ciało liczb zespolonych<a class="headerlink" href="#cialo-liczb-zespolonych" title="Link to this heading">¶</a></h1> | ||
| <ul class="simple"> | ||
| <li><p>w przeciwieństwie do liczb rzeczywistych nie można ich przedstawić | ||
| w postaci linearnej</p></li> | ||
| <li><p>dlatego też nie można powiedzieć że dana liczba jest ujemna/dodatnia</p></li> | ||
| <li><p>liczba zespolona to para</p></li> | ||
| <li><p><span class="math notranslate nohighlight">\(i^2 = -1\)</span></p></li> | ||
| </ul> | ||
| <section id="dzielenie"> | ||
| <h2>Dzielenie<a class="headerlink" href="#dzielenie" title="Link to this heading">¶</a></h2> | ||
| <p>Przy dzieleniu liczb zespolonych należy pomnożyć przez czynnik sprzężony | ||
| (tak jak usuwanie niewymierności z mianownika)</p> | ||
| </section> | ||
| <section id="modul-liczby-zespolonej"> | ||
| <h2>moduł liczby zespolonej<a class="headerlink" href="#modul-liczby-zespolonej" title="Link to this heading">¶</a></h2> | ||
| <p>Utożsamiany z długością wektora będącego interpretacją liczby urojonej. | ||
| niech <span class="math notranslate nohighlight">\(z = (x, yi)\)</span></p> | ||
| <div class="math notranslate nohighlight"> | ||
| \[ | ||
| |z| = \sqrt{x^2 + y^2} | ||
| \]</div> | ||
| </section> | ||
| <section id="postac-trygonometryczna-liczyb-zespolonej"> | ||
| <h2>Postać trygonometryczna liczyb zespolonej<a class="headerlink" href="#postac-trygonometryczna-liczyb-zespolonej" title="Link to this heading">¶</a></h2> | ||
| <div class="math notranslate nohighlight"> | ||
| \[\begin{split} | ||
| niech~z = x + yi \neq 0 \\ | ||
| z = |z|(\frac{x}{|z|}+\frac{y}{|z|})\\ | ||
| \exists \phi ~ cos \phi = \frac{x}{|z|} \land sin \phi = \frac{y}{|z|} | ||
| z = |z|(cos \phi + i * sin \phi) | ||
| \end{split}\]</div> | ||
| <p><span class="math notranslate nohighlight">\(arg z\)</span> to tzw. argument główny gdzie <span class="math notranslate nohighlight">\(argz \in [0, 2 \pi]\)</span></p> | ||
| <section id="potegowanie-liczby-zespolonej-w-postaci-trygonometrycznej"> | ||
| <h3>potęgowanie liczby zespolonej w postaci trygonometrycznej<a class="headerlink" href="#potegowanie-liczby-zespolonej-w-postaci-trygonometrycznej" title="Link to this heading">¶</a></h3> | ||
| <p>Wzór de Moivre’a | ||
| $<span class="math notranslate nohighlight">\( | ||
| \frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} (cos(\alpha - \beta) + i sin(\alpha - \beta)) \\ | ||
| z^n = |z|^n (cos n \phi + i * sin n \phi) | ||
| \)</span>$</p> | ||
| </section> | ||
| </section> | ||
| <section id="postac-wykladnicza"> | ||
| <h2>postać wykładnicza<a class="headerlink" href="#postac-wykladnicza" title="Link to this heading">¶</a></h2> | ||
| <div class="math notranslate nohighlight"> | ||
| \[ | ||
| e^{i \phi} = cos \phi + i sin \phi | ||
| $\]</div> | ||
| <div class="math notranslate nohighlight"> | ||
| \[ | ||
| z_k = \nroot{n}{r}(cos \frac{\phi + 2k \pi}{n} + i sin{\phi + 2k \pi}{n}) ~~ k = 0 , 1 ... n | ||
| \]</div> | ||
| <div class="admonition important"> | ||
| <p class="admonition-title">Ważne</p> | ||
| <p>Pierwiastek <code class="docutils literal notranslate"><span class="pre">n</span></code> stopnia <strong>ma <code class="docutils literal notranslate"><span class="pre">n</span></code> rozwiązań</strong></p> | ||
| </div> | ||
| </section> | ||
| <section id="zasadnicze-twierdzenie-algebry"> | ||
| <h2>zasadnicze twierdzenie algebry``<a class="headerlink" href="#zasadnicze-twierdzenie-algebry" title="Link to this heading">¶</a></h2> | ||
| <p>Każdy wielomian dodatniego stopnia ma rozwiązania w <span class="math notranslate nohighlight">\(\mathbb{C}\)</span></p> | ||
| </section> | ||
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| <h3><a href="../../index.html">Spis treści:</a></h3> | ||
| <ul> | ||
| <li><a class="reference internal" href="#">Ciało liczb zespolonych</a><ul> | ||
| <li><a class="reference internal" href="#dzielenie">Dzielenie</a></li> | ||
| <li><a class="reference internal" href="#modul-liczby-zespolonej">moduł liczby zespolonej</a></li> | ||
| <li><a class="reference internal" href="#postac-trygonometryczna-liczyb-zespolonej">Postać trygonometryczna liczyb zespolonej</a><ul> | ||
| <li><a class="reference internal" href="#potegowanie-liczby-zespolonej-w-postaci-trygonometrycznej">potęgowanie liczby zespolonej w postaci trygonometrycznej</a></li> | ||
| </ul> | ||
| </li> | ||
| <li><a class="reference internal" href="#postac-wykladnicza">postać wykładnicza</a></li> | ||
| <li><a class="reference internal" href="#zasadnicze-twierdzenie-algebry">zasadnicze twierdzenie algebry``</a></li> | ||
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