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{"name": "Rechnen mit komplexen Zahlen I", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Rechnen mit komplexen Zahlen I", "tags": [], "metadata": {"description": "<p>Elementary examples of multiplication and addition of complex numbers. Four parts.</p>\n<p>Original: https://numbas.mathcentre.ac.uk/question/11784/arithmetics-of-complex-numbers-i/ by Newcastle University Mathematics ans Statistics</p>\n<p>Translated to German.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Schreiben Sie die folgenden komplexenz Zahlen in der Form $a+bi\\;$ mit $a$ und $b$ in $\\mathbb R$.</p>\n<p>Schreiben Sie $a$ und $b$ als Bruchzahlen oder als ganze Zahlen.</p>", "advice": "<p><strong>a)</strong></p>\n<p>Die Multiplikationsformel f&uuml;r komplexe Zahlen ist <br/>\\[ (a+bi)(c+di) = (ac-bd) + (ad+bc)i \\]</p>\n<p>Also gilt<br/>\\[ \\simplify[timesdot]{{a}*{b}} = (\\simplify[timesdot]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)}}) + (\\simplify[timesdot]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)}})i \\]<br/>Die L&ouml;sung ist also</p>\n<p>\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]</p>\n<p><strong>b)</strong></p>\n<p>Wir schreiben die zweite Potenz aus als</p>\n<p>$(\\simplify[std]{{a1}})^2=&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})$ und berechnen das Produkt:</p>\n<p>\\[\\begin{eqnarray*}(\\simplify[std]{{a1}})^2&amp;=&amp;&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})\\\\ &amp;=&amp; \\simplify[timesdot]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\\\ &amp;=&amp; \\simplify[std,timesdot]{{a1^2}} \\end{eqnarray*} \\]</p>\n<p><strong>c)</strong></p>\n<p>Es gilt $i^2=-1$, also $i^3=i^2i=-i$.</p>\n<p>Somit:<br/>\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&amp;=&amp;\\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\\\ &amp;=&amp;\\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\\\ &amp;=&amp;\\simplify[std]{{a3 -c3} + {b3 -d3} * i} \\end{eqnarray*} \\]</p>\n<p><strong>d)</strong></p>\n<p>Wir multiplizieren die Zahlen in zwei Schritten und erhalten:<br/>\\[ \\begin{eqnarray*} (\\var{z1})(\\var{z2})(\\var{z3})&amp;=&amp;((\\var{z1})(\\var{z2}))(\\var{z3})\\\\ &amp;=&amp;(\\var{z1*z2})(\\var{z3})\\\\ &amp;=&amp;\\var{z1*z2*z3} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "extensions": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "d3": {"name": "d3", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s3*random(1..9)+s4*random(1..9)*i", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped 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