{"id":"d6974acc-2bbd-46f1-b79f-5cc395c39fc9","type":"https://serlo.org/editor","variant":"serlo-org","domainOrigin":"serlo.org","version":2,"editorVersion":"0.22.0","dateModified":"2025-02-26T18:25:13.806Z","document":{"plugin":"course","state":{"content":{"plugin":"rows","state":[]},"pages":[{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Inhalt"}]},{"type":"p","children":[{"text":"Der folgende Kurs handelt von Annäherungsverfahren für Wurzelberechnungen. "}]},{"type":"p","children":[{"text":" "}]},{"type":"p","children":[{"text":"Am Ende kannst du zwei verschiedene Verfahren anwenden, die du für die Berechnung von Wurzeln nutzen kannst."}]}],"id":"3fe65046-3866-4799-b347-f79eadb2e75a"},{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Vorwissen"}]},{"type":"p","children":[{"text":"Du solltest wissen:"}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Was eine "},{"type":"a","href":"/83733#83740","children":[{"text":"Quadratwurzel"}]},{"text":" ist."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Wie man den "},{"type":"a","href":"/1887","children":[{"text":"Mittelwert"}]},{"text":" von zwei Zahlen berechnet."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Wie man den Flächeninhalt von "},{"type":"a","href":"/1863","children":[{"text":"Quadrat"}]},{"text":" und "},{"type":"a","href":"/1697","children":[{"text":"Rechteck"}]},{"text":" berechnet."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Was ein "},{"type":"a","href":"1947","children":[{"text":"Intervall"}]},{"text":" ist, und seine Bezeichnung."}]}]}]}],"id":"85546134-dd99-43d6-83c3-c2e11485d2c5"},{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Dauer"}]},{"type":"p","children":[{"text":"Der Kurs hat eine Dauer von ca. 2h."}]}],"id":"9d8a2502-aa34-4fed-a7a7-e28307c15497"}],"id":"cfe46fa4-22a4-4c86-b102-a6ace5514117"},"id":"84468-course-page","title":"Berechnung von Wurzeln"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Wie gut kannst du eigentlich Kopfrechnen?"}]},{"type":"p","children":[{"text":"Wenn du gerade die Befürchtung hattest, Kopfrechnen zu müssen: Keine Sorge - es geht um Taschenrechner. "}]},{"type":"p","children":[{"text":"Der praktische Helfer macht den Schulalltag oft sehr angenehm. Doch wie funktioniert er eigentlich?"}]}],"id":"66e04b87-40ef-4442-bb61-e81931ea6aa4"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Bei der Multiplikation ist das ganz klar. Er berechnet das Ergebnis, indem er "},{"type":"a","href":"/54829","children":[{"text":"schriftlich multipliziert"}]},{"text":"."}]}],"id":"98e9866f-4529-489d-adfd-037d97ec19e3"},{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"\\large{\\begin{array}{r}\\ &\\underline{38 \\cdot 42} \\\\&76\\\\&\\underline{+1520}\\\\&\\underline{\\underline{1596}}\\end{array}}","inline":true,"children":[{"text":"\\large{\\begin{array}{r}\\ &\\underline{38 \\cdot 42} \\\\&76\\\\&\\underline{+1520}\\\\&\\underline{\\underline{1596}}\\end{array}}"}]},{"text":""}]}],"id":"d66ca34a-d0ed-439c-a914-31d24d101e34"}],"id":"ef934d40-16d1-4f48-a24e-0332fdf1eda5"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/59c25bf826526_6a24763dbd64234ef26f1f185f8259c0c5721a15.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"991267a5-3678-43e1-aa1c-8f229f38ee25"}},"id":"9a58f4ed-3446-4543-98cc-f27bd1fc6240"},"illustrating":true,"width":50},"id":"3902dbba-5fa4-4deb-bd4f-707fc47b7032"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Auch die Addition von vielen Zahlen ist kein Problem. Per Hand würde man das durch "},{"type":"a","href":"/2153","children":[{"text":"schriftliches Addieren"}]},{"text":" machen."}]}],"id":"0da05286-4006-4e94-b23e-8d74d77d6b40"},{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"\\large{\\begin{array}{r}&736\\\\&1024\\\\&57 \\\\& \\underline{+ \\; 200} \\\\&\\underline{\\underline{2017}}\\end{array}}","inline":true,"children":[{"text":"\\large{\\begin{array}{r}&736\\\\&1024\\\\&57 \\\\& \\underline{+ \\; 200} \\\\&\\underline{\\underline{2017}}\\end{array}}"}]},{"text":""}]}],"id":"a2fe758b-c602-4981-8646-e09cc0d13e2a"}],"id":"48dde8be-6a57-46d9-a767-543f055ac9e6"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/59c25c29bc2a6_1f0c8dac1c913421055b91498ce63273889fb909.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"b721798e-e648-43a8-8819-0989d0a25c67"}},"id":"885360fd-8a2b-4e37-964a-2078d371da4f"},"illustrating":true,"width":50},"id":"1c69266b-dc2d-4588-bcf7-4dd9262374f9"}],"id":"89b6bd42-2ea9-4ea0-b21b-74c1dfb9c689"},"id":"84098-course-page","title":"Motivation"},{"content":{"plugin":"rows","state":[{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Doch wie berechnet der Taschenrechner Wurzeln? "}]},{"type":"p","children":[{"text":"Manche Werte, wie "},{"type":"math","src":"\\sqrt{4}","inline":true,"children":[{"text":"\\sqrt{4}"}]},{"text":", "},{"type":"math","src":"\\sqrt{9}","inline":true,"children":[{"text":"\\sqrt{9}"}]},{"text":" oder "},{"type":"math","src":"\\sqrt{1}","inline":true,"children":[{"text":"\\sqrt{1}"}]},{"text":" sind dir bestimmt schon bekannt.\nAber der Taschenrechner liefert uns auch einen Wert für Rechnungen wie "},{"type":"math","src":"\\sqrt{2}","inline":true,"children":[{"text":"\\sqrt{2}"}]},{"text":", "},{"type":"math","src":"\\sqrt{5}","inline":true,"children":[{"text":"\\sqrt{5}"}]},{"text":" oder "},{"type":"math","src":"\\sqrt{7}","inline":true,"children":[{"text":"\\sqrt{7}"}]},{"text":"."}]},{"type":"p","children":[{"text":"Wie berechnet er diese?"}]}],"id":"34669752-66f5-438a-8ca7-171789be32d8"}],"id":"942d5f1d-0e1b-449f-91a3-009b24cf807f"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/59b933e29abbf_6dfd44ded1e5e0a31fadd3b236c7e8d1a8fdb851.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"a8926a89-6790-480c-bd02-da6e557a7e96"}},"id":"224a1d3c-6887-477e-a3f2-f2e2c5a0879c"},"illustrating":true,"width":50},"id":"846ba9ac-a94f-48ea-a5f5-9d87c3d94be7"}],"id":"f77d8c6a-2a50-40f1-b1db-088ad9aee91d"},"id":"85293-course-page","title":"Motivation 2"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Es gibt zwei verschiedene Methoden, mit denen Werte von Wurzeln angenähert werden können. "}]},{"type":"p","children":[{"text":"Das Ergebnis ist dabei oft eine "},{"type":"a","href":"/56179","children":[{"text":"irrationale Zahl"}]},{"text":", die unendlich viele Stellen hinter dem Komma hat. Diese Zahlen kann auch der Taschenrechner nur annähern und anschließend runden."}]},{"type":"p","children":[{"text":"Die folgenden zwei Verfahren liefern Näherungswerte und werden auf den nächsten Kursseiten vorgestellt: "}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"a","href":"/84096#84127","children":[{"text":"Intervallschachtelung"}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"a","href":"/84096#84110","children":[{"text":"Heron-Verfahren"}]},{"text":""}]}]}]}],"id":"ec889f45-cb5a-418f-87f6-69223485f407"}],"id":"fa15f62f-df04-4212-bb61-b97391f3b244"},"id":"84105-course-page","title":"Zwei Vorgehensweisen"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Intervallschachtelung ist eine Methode, um die Werte von "},{"type":"a","href":"/83733#83740","children":[{"text":"Wurzeln"}]},{"text":" anzunähern, ohne die Wurzel direkt zu berechnen. "}]},{"type":"p","children":[{"text":"Dabei versuchst du, ein "},{"type":"a","href":"/1947","children":[{"text":"Intervall"}]},{"text":" zu finden, in dem der Wert der Wurzel liegen muss. Dieses Intervall kannst du bis zur gewünschten Genauigkeit schrittweise verkleinern."}]}],"id":"9aef352b-0ea7-4b6c-8f92-21ae62dcea4d"},{"plugin":"image","state":{"src":"https://assets.serlo.org/59b7c62d9d3e8_a4ca26ac362ba8dfdf457c260d515a63215e651b.png","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"047d8f8b-1a0f-42f6-96e7-0cc18b8850cd"}},"id":"faee0658-9c5f-40c4-b61c-24316edd3b8d"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Auf diesem Bild siehst du, wie sich solche Intervalle verkleinern."}]}],"id":"f0433320-b027-427f-9d63-68091676ecfe"}],"id":"c46ff47a-b9a0-42bc-99cc-90dd90d38929"},"id":"84127-course-page","title":"Intervallschachtelung Einführung"},{"content":{"plugin":"rows","state":[{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Erklärung"}]}],"id":"100e0acd-65b4-4b2a-afa1-7c6a001dea5e"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiel"}]}],"id":"d05d5638-e497-4be1-be2d-f9b480915464"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Betrachte die Wurzel, die du berechnen möchtest. Suche einen bekannten Wurzelwert, der kleiner als die gesuchte Wurzel ist. Suche einen zweiten, der größer als die gesuchte Wurzel ist."}]}],"id":"85b8a6a3-287c-4975-96e1-15c3808b65db"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"zu berechnen: "},{"type":"math","src":"\\sqrt{2}","inline":true,"children":[{"text":"\\sqrt{2}"}]},{"text":""}]},{"type":"p","children":[{"text":"Du kennst bereits: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\sqrt{1} =1","inline":true,"children":[{"text":"\\sqrt{1} =1"}]},{"text":", "},{"type":"math","src":"\\sqrt{4} = 2","inline":true,"children":[{"text":"\\sqrt{4} = 2"}]},{"text":""}]},{"type":"p","children":[{"text":"Außerdem weißt du: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"1^2 < 2 < 2^2","inline":true,"children":[{"text":"1^2 < 2 < 2^2"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"1 < 2 < 4","inline":true,"children":[{"text":"1 < 2 < 4"}]},{"text":""}]},{"type":"p","children":[{"text":"dann gilt, wenn du überall die Wurzel ziehst: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\sqrt{1} < \\sqrt{2} < \\sqrt{4}","inline":true,"children":[{"text":"\\sqrt{1} < \\sqrt{2} < \\sqrt{4}"}]},{"text":""}]}],"id":"d56c787b-3868-46f3-9cd1-82d14d16d04a"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Mit den zwei bekannten Wurzelwerten kannst du nun ein "},{"type":"a","href":"/1947","children":[{"text":"Intervall"}]},{"text":" "},{"type":"math","src":"I_0","inline":true,"children":[{"text":"I_0"}]},{"text":" definieren, in dem der gesuchte Wert liegen muss."}]}],"id":"30533157-b983-483e-8845-6633efe04456"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Intervall: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\sqrt{2}\\; \\; \\in \\; \\; I_0 = \\left] \\sqrt{1}; \\sqrt{4} \\right[ = \\left] 1; 2 \\right[","inline":true,"children":[{"text":"\\sqrt{2}\\; \\; \\in \\; \\; I_0 = \\left] \\sqrt{1}; \\sqrt{4} \\right[ = \\left] 1; 2 \\right["}]},{"text":""}]}],"id":"1e795fe6-318e-462e-8032-eba14a0e3ceb"}}]}],"tableType":"OnlyColumnHeader"},"id":"12475465-4363-4bc4-bc10-ce80a0c7fd27"},{"plugin":"image","state":{"src":"https://assets.serlo.org/59b7c8388c93b_e6343012781f00df077c0b8e0b77fb6f952276f4.JPG","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"78c0e28b-f6fa-4a13-aaa1-696cd80c1e57"}},"id":"967d16ee-5315-4f49-b7cf-5ef8a3efc3ee"}],"id":"dbdeb4e4-3bb2-4e7d-8c26-cc681b912b3f"},"id":"84196-course-page","title":"Intervallschachtelung (1/4)"},{"content":{"plugin":"rows","state":[{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Erklärung"}]}],"id":"8e42c689-f0ce-4936-b479-1ab1c51134e3"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiel"}]}],"id":"6e212602-7add-4dd0-afe0-6ce3633d220b"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Zunächst suchst du einen neuen Wert, dessen Quadrat in diesem Intervall liegt, aber näher am gesuchten "},{"type":"a","href":"/83740","children":[{"text":"Radikanden"}]},{"text":" ist."}]}],"id":"d9f85f42-8d8d-454c-8705-00d50967f2c9"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"altes Intervall: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"I_0 = \\left] 1\\; ; \\; 2 \\right[","inline":true,"children":[{"text":"I_0 = \\left] 1\\; ; \\; 2 \\right["}]},{"text":""}]}],"id":"5063d11c-0bfb-444e-a558-736db0d928da"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Aus diesem neuen Wert und einem der anderen Werte bildest du ein neues, kleineres Intervall."}]}],"id":"685e1e0e-9b32-46f1-ad2a-645f1cf398f5"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"7341b132-a03e-4f95-af98-da8693b36e27"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Den neuen Wert erhältst du, indem du den "},{"type":"a","href":"/1887","children":[{"text":"Mittelwert"}]},{"text":" der alten Werte bildest. Diesen Mittelwert verwendest du später als neue Unter- "},{"text":"oder","strong":true},{"text":" Obergrenze."}]}],"id":"b800289a-d3bf-4b0e-968b-575c82d504d5"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Neuer Wert:","strong":true}]},{"type":"p","children":[{"text":"Mittelwert von "},{"type":"math","src":"1","inline":true,"children":[{"text":"1"}]},{"text":" und "},{"type":"math","src":"2","inline":true,"children":[{"text":"2"}]},{"text":": "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\displaystyle{\\frac{1+2}{2}=1,5}","inline":true,"children":[{"text":"\\displaystyle{\\frac{1+2}{2}=1,5}"}]},{"text":""}]}],"id":"b2f76d6f-efd8-4ec5-aa41-93ef538a3171"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Um dies zu entscheiden, musst du noch testen, ob das Quadrat dieses Wertes größer als der Radikand oder kleiner ist."}]}],"id":"f124dc3f-0dbc-46ae-80cd-8de2aa89a3d6"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Teste, ob "},{"type":"math","src":"1,5^2","inline":true,"children":[{"text":"1,5^2"}]},{"text":" größer oder kleiner als "},{"type":"math","src":"2","inline":true,"children":[{"text":"2"}]},{"text":" ist."}]},{"type":"p","children":[{"text":""},{"type":"math","src":"1,5^2 = 2,25","inline":true,"children":[{"text":"1,5^2 = 2,25"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"1,5^2 = 2,25 > 2","inline":true,"children":[{"text":"1,5^2 = 2,25 > 2"}]},{"text":""}]}],"id":"b2788fe3-c0fd-4f0d-824b-2b4a32151698"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Wenn er kleiner ist, bildet er zusammen mit der Obergrenze von "},{"type":"math","src":"I_0","inline":true,"children":[{"text":"I_0"}]},{"text":" das neue Intervall "},{"type":"math","src":"I_1","inline":true,"children":[{"text":"I_1"}]},{"text":"."}]},{"type":"p","children":[{"text":"Wenn er größer ist, bildet er zusammen mit der Untergrenze von "},{"type":"math","src":"I_0","inline":true,"children":[{"text":"I_0"}]},{"text":" das neue Intervall "},{"type":"math","src":"I_1","inline":true,"children":[{"text":"I_1"}]},{"text":"."}]}],"id":"51eed3e4-ed42-4a22-a2c8-73ad2ee91dd6"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Neues Intervall:","strong":true}]},{"type":"p","children":[{"text":""},{"type":"math","src":"1 < \\sqrt{2} < 1,5","inline":true,"children":[{"text":"1 < \\sqrt{2} < 1,5"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\sqrt{2}\\; \\; \\in \\; \\;I_1 = \\left]1\\; ; \\; 1,5\\right[","inline":true,"children":[{"text":"\\sqrt{2}\\; \\; \\in \\; \\;I_1 = \\left]1\\; ; \\; 1,5\\right["}]},{"text":""}]}],"id":"c109275f-ff76-447e-8822-2bf903e28f93"}}]}],"tableType":"OnlyColumnHeader"},"id":"3a9afc06-5543-4b69-8440-6157255ee785"},{"plugin":"image","state":{"src":"https://assets.serlo.org/59b7c8515f936_567e65a06e7845894437194366eb2f4c85c5b7d3.JPG","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"e28eff97-0d1b-4874-a0f4-5a6e0c2652f6"}},"id":"d8f49e3b-a496-454a-948c-89b829e7544c"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Dieses Verfahren der Verkleinerung von Intervallen wird nun so lange wiederholt, bis du die gewünschte Genauigkeit erreicht hast."}]}],"id":"defd03df-fae6-4442-944a-86074f7235f0"}],"id":"b958bc91-ce1a-4fd8-81f5-dc40d7703120"},"id":"84698-course-page","title":"Intervallschachtelung (2/4)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"In dieser interaktiven Grafik siehst du die Veränderung der Intervallgröße nach jedem Schritt. 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"}]},{"type":"p","children":[{"text":"Wenn du hier abbrichst, erhältst du als Ergebnis, dass "},{"type":"math","src":"\\sqrt{2}","inline":true,"children":[{"text":"\\sqrt{2}"}]},{"text":" zwischen "},{"type":"math","src":"1,40625","inline":true,"children":[{"text":"1,40625"}]},{"text":" und "},{"type":"math","src":"1,4375","inline":true,"children":[{"text":"1,4375"}]},{"text":" liegt."}]},{"type":"p","children":[{"text":"Oft wird aber noch nach einem Schätzwert gefragt. Jeder Wert aus diesem Intervall wäre ein gültiger Schätzwert. Ein Kandidat wäre hier der Mittelwert. 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Wir suchen ein "},{"type":"a","href":"/1863","children":[{"text":"Quadrat"}]},{"text":", dessen Flächeninhalt genau dem Wert des "},{"type":"a","href":"/83733#83740","children":[{"text":"Radikanden"}]},{"text":" "},{"type":"math","src":"A","inline":true,"children":[{"text":"A"}]},{"text":" entspricht. Die Seitenlänge "},{"type":"math","src":"s","inline":true,"children":[{"text":"s"}]},{"text":" entspricht dabei genau dem Wert der Wurzel. 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Wie genau du dabei vorgehst, lernst du auf den nächsten Seiten."}]},{"type":"p","children":[{"text":"Dieses Vorgehen kannst du nach Belieben wiederholen, um immer näher an eine quadratische Form zu kommen."}]},{"type":"p","children":[{"text":"Je geringer die "},{"type":"a","href":"/2111","children":[{"text":"Differenz"}]},{"text":" zwischen "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" und "},{"type":"math","src":"b","inline":true,"children":[{"text":"b"}]},{"text":", desto genauer ist das Ergebnis, weil sich das Rechteck einem Quadrat annähert."}]}],"id":"ab76e8f0-8601-47a1-ae80-f6fc3b447f26"},{"plugin":"rows","state":[{"plugin":"geogebra","state":"SPdvzVx5","id":"209132c3-7d2a-4449-a7ea-5978db5afd42"}],"id":"5ea809c2-80c7-42df-abf6-a7d57317ac8b"}],"id":"8c7c70af-fcf9-4727-a13f-024596559de9"},"id":"84110-course-page","title":"Heron-Verfahren Einführung"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Vorgehen anhand des Beispiels "},{"type":"math","src":"\\sqrt{20}","inline":true,"children":[{"text":"\\sqrt{20}"}]},{"text":":"}]}],"id":"a11dc225-0b64-493c-8a01-45ade190e04e"},{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Vorgehen"}]}],"id":"cff434f6-0622-44b2-a5a3-7dfb9dc88722"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiel"}]}],"id":"39f5e398-4fde-45fb-ac9e-9ed71fe0408f"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Wähle ein Rechteck, das den Flächeninhalt des "},{"type":"a","href":"/83740","children":[{"text":"Radikanden"}]},{"text":" besitzt. "}]},{"type":"p","children":[{"text":"Dazu suchst du einen Teiler des Radikanden, den du als erste Seite des Rechtecks nutzt. "}]},{"type":"p","children":[{"text":"Die zweite Seite erhältst du durch: "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\frac{\\text{Radikand}}{\\text{erste Seite}}","inline":true,"children":[{"text":"\\dfrac{\\text{Radikand}}{\\text{erste Seite}}"}]},{"text":""}]}],"id":"8b404aa9-7a11-4134-a51e-dc9d91b9f004"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Suche ein Rechteck mit Flächeninhalt "},{"type":"math","src":"20","inline":true,"children":[{"text":"20"}]},{"text":"."}]},{"type":"p","children":[{"text":"Du kannst z. B. die Seitenlängen "},{"type":"math","src":"\\color{#cc0000}{a_0=10}","inline":true,"children":[{"text":"\\color{#cc0000}{a_0=10}"}]},{"text":" und "},{"type":"math","src":"\\color{#cc0000}{b_0=2}","inline":true,"children":[{"text":"\\color{#cc0000}{b_0=2}"}]},{"text":" wählen. "}]},{"type":"p","children":[{"text":"Ziel: Mache dieses Rechteck möglichst quadratisch. 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4,29"}]},{"text":""}]}],"id":"526b67fc-94b6-4e50-8d68-252ecf335b02"}}]}],"tableType":"OnlyColumnHeader"},"id":"d468ea43-841d-4007-ba06-66a8396938f6"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Wiederhole die letzten beiden Schritte."}]}],"id":"f90c6411-b40a-4eb2-8314-cb1c53c9a622"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Nun hast du ein Rechteck mit den Seitenlängen "},{"type":"math","src":"\\color{#ff6600}{a_2 = 4,67}","inline":true,"children":[{"text":"\\color{#ff6600}{a_2 = 4,67}"}]},{"text":" und "},{"type":"math","src":"\\color{#ff6600}{b_2 = 4,29}","inline":true,"children":[{"text":"\\color{#ff6600}{b_2 = 4,29}"}]},{"text":"."}]}],"id":"ab8d5b5e-41be-4c50-a365-341495b0f8f8"},{"plugin":"rows","state":[],"id":"2c836598-aa25-44db-b251-a46f80f28a6e"},{"plugin":"image","state":{"src":"https://assets.serlo.org/0d960c90-a7b5-11ed-95de-a79d3d5d7bc4/image.svg","alt":"Heronverfahren","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"cc5cf5c0-6b9b-406b-a8c1-275dc79ea8e2"}},"id":"96386d69-4d09-42dd-8cf7-226cff61a3e5"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Du kannst sehen, dass sich die beiden Werte annähern. "}]},{"type":"p","children":[{"text":"Das exakte Ergebnis liegt zwischen diesen beiden Zahlen.\nWiederholst du dieses Verfahren, so erhältst du "},{"type":"math","src":"\\sqrt{20}\\approx4,4721","inline":true,"children":[{"text":"\\sqrt{20}\\approx4,4721"}]},{"text":"."}]},{"type":"p","children":[{"text":"Zum Vergleich: Der Taschenrechner liefert den gerundeten Wert "},{"type":"math","src":"\\sqrt{20}\\approx 4,472135955","inline":true,"children":[{"text":"\\sqrt{20}\\approx 4,472135955"}]},{"text":"."}]}],"id":"e698d68c-bb68-4659-ae70-94fc0e1f177b"}],"id":"030d441e-1b3c-44c1-8b68-c52cf108d2ac"},"id":"84115-course-page","title":"Heron-Verfahren (3/3)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Im folgenden Applet kannst du eine beliebige Wurzel von 1 bis 50 eingeben und für diese das Heron-Verfahren ansehen."}]},{"type":"p","children":[{"text":" "}]},{"type":"p","children":[{"text":"Den nächsten Schritt des Heron-Verfahrens kannst du ansehen, indem du „nächster Schritt” anklickst."}]},{"type":"p","children":[{"text":" "}]},{"type":"p","children":[{"text":"Die Wurzel, die du berechnen möchtest, änderst du durch Auswählen des Feldes unter der Wurzel. Hier kannst du Zahlen mit einer Nachkommastelle im Bereich zwischen "},{"type":"math","src":"1","inline":true,"children":[{"text":"1"}]},{"text":" und "},{"type":"math","src":"50","inline":true,"children":[{"text":"50"}]},{"text":" eingeben."}]}],"id":"1139d8e9-496f-4bde-91db-9cb99a28b995"},{"plugin":"geogebra","state":"H2HDFmdx","id":"51ea6abb-a1a3-4fae-b205-a980720a37cc"}],"id":"33817ad2-60d4-4dff-aedf-571d17b7c5fd"},"id":"84119-course-page","title":"Veranschaulichung des Heron-Verfahrens"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Aufgabe 1"}]}],"id":"3ef9fa2a-0418-4ba7-b40a-90c5892282bb"},{"plugin":"injection","state":"/84391","id":"077d82f8-5ae8-4e52-bb3c-56a39980a22f"},{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Aufgabe 2"}]},{"type":"p","children":[{"text":"Vergleiche die "},{"type":"a","href":"/84096#84310","children":[{"text":"Aufgabe zur Intervallschachtelung"}]},{"text":" und die zum Heron-Verfahren. "}]},{"type":"p","children":[{"text":"Welche Methode der Abschätzung findest du besser und warum?"}]}],"id":"6a5dee87-d2c3-4735-b740-0e4c6012d1ad"},{"plugin":"spoiler","state":{"title":"Lösung","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Bei der Intervallschachtelung erhältst du nach fünf Schritten das Ergebnis:"}]},{"type":"p","children":[{"type":"math","src":"\\sqrt{7} \\approx 2,65625","inline":false,"children":[{"text":"\\sqrt{7} \\approx 2,65625"}]}]},{"type":"p","children":[{"text":"Beim Heron-Verfahren erhältst du nach vier Schritten das Ergebnis: "}]},{"type":"p","children":[{"type":"math","src":"\\sqrt{7} \\approx 2,64575","inline":false,"children":[{"text":"\\sqrt{7} \\approx 2,64575"}]}]},{"type":"p","children":[{"text":"Der Taschenrechner gibt das Ergebnis "},{"type":"math","src":"\\sqrt{7} = 2,64575","inline":true,"children":[{"text":"\\sqrt{7} = 2,64575"}]},{"text":" aus. "}]},{"type":"p","children":[{"text":"Dabei fällt auf, dass das Ergebnis der Intervallschachtelung nur bei der ersten Dezimalstelle übereinstimmt mit dem Ergebnis des Taschenrechners. Das Heron-Verfahren hat die ersten fünf Dezimalstellen richtig ausgegeben und das nach weniger Schritten, als bei der Intervallverschachtelung."}]},{"type":"p","children":[{"text":"Daran kannst du erkennen, dass das Heron-Verfahren prinzipiell eine bessere Näherung ergibt, die Intervallverschachtelung aber auch zu einer Annäherung genutzt werden kann."}]}],"id":"5ae3c391-7178-4a2e-8e81-2c1392b6aec1"}],"id":"5f82c7c1-df50-4754-96a2-9ed89b2df141"}},"id":"90f27f51-3282-4a75-b0b8-f958d9370e1b"}],"id":"d307bc05-756f-492e-8666-8058a73a7e85"},"id":"84329-course-page","title":"Heron-Verfahren Aufgaben"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"In den Aufgaben hast du zwei verschiedene Verfahren angewendet und dabei unterschiedliche Werte für "},{"type":"math","src":"\\sqrt{7}","inline":true,"children":[{"text":"\\sqrt{7}"}]},{"text":" erhalten: "}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"a","href":"\\84310","children":[{"text":"Intervallschachtelung:","strong":true}]},{"text":" "},{"type":"math","src":"\\sqrt{7} \\approx 2,65625","inline":true,"children":[{"text":"\\sqrt{7} \\approx 2,65625"}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"a","href":"/84096#84329","children":[{"text":"Heron-Verfahren","strong":true},{"text":":"}]},{"text":" "},{"type":"math","src":"\\sqrt{7} \\approx 2,64575","inline":true,"children":[{"text":"\\sqrt{7} \\approx 2,64575"}]},{"text":" "}]}]}]},{"type":"p","children":[{"text":"Wie kannst du jetzt entscheiden, welches Ergebnis genauer ist?"}]},{"type":"p","children":[{"text":"Du weißt: "},{"type":"math","src":"(\\sqrt{7})^2=7","inline":true,"children":[{"text":"(\\sqrt{7})^2=7"}]},{"text":"\nQuadriere die beiden berechneten Werte und vergleiche, welcher näher an "},{"type":"math","src":"7","inline":true,"children":[{"text":"7"}]},{"text":" ist."}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Intervallschachtelung"},{"type":"math","src":"^2","inline":true,"children":[{"text":"^2"}]},{"text":": "},{"type":"math","src":"\\; \\; \\; \\; 2,65625^2 = 7,05566","inline":true,"children":[{"text":"\\; \\; \\; \\; 2,65625^2 = 7,05566"}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Heron-Verfahren"},{"type":"math","src":"^2","inline":true,"children":[{"text":"^2"}]},{"text":": "},{"type":"math","src":"\\; \\; \\; \\quad \\quad2,64575^2 = 6,99999","inline":true,"children":[{"text":"\\; \\; \\; \\quad \\quad2,64575^2 = 6,99999"}]},{"text":""}]}]}]},{"type":"p","children":[{"text":"Wie du siehst, ist der Wert des Heron-Verfahrens näher an "},{"type":"math","src":"\\sqrt{7}","inline":true,"children":[{"text":"\\sqrt{7}"}]},{"text":".\nAuch der Taschenrechner gibt einen Wert von "},{"type":"math","src":"\\sqrt{7} = 2,64575","inline":true,"children":[{"text":"\\sqrt{7} = 2,64575"}]},{"text":" aus."}]},{"type":"p","children":[{"text":"Außerdem hast du für das Heron-Verfahren weniger Schritte gebraucht als für die Intervallschachtelung."}]},{"type":"p","children":[{"text":"Die meisten Taschenrechner nutzen das Heronverfahren, um Wurzeln anzunähern. Dieses erreicht mit gleich vielen Schritten eine höhere Genauigkeit.\nDen Näherungswert rundet er anschließend und zeigt eine bestimmte Anzahl an Dezimalstellen an."}]}],"id":"e097e7a5-bf75-45c3-bc70-8a6fde5805c2"}],"id":"637b7eb1-ed4b-4a75-92d3-f0610855e881"},"id":"85023-course-page","title":"Vergleich Intervallschachtelung - Heron-Verfahren"}]},"id":"9c3390dc-6853-48e7-9318-9b94f7d80ff8"}}

/mathe/84096/berechnungen-von-wurzeln