{"id":599,"date":"2017-03-06T15:55:06","date_gmt":"2017-03-06T14:55:06","guid":{"rendered":"http:\/\/www2.mathnique.com\/site\/?page_id=599"},"modified":"2025-12-22T12:50:11","modified_gmt":"2025-12-22T11:50:11","slug":"formulaires","status":"publish","type":"page","link":"https:\/\/www.mathnique.com\/site\/formulaires\/","title":{"rendered":"Calculs num\u00e9riques et alg\u00e9briques"},"content":{"rendered":"<ul>\n<li><span style=\"color: #ff0000;\"><strong>12 R\u00e8gles de calcul sur les quotients de nombres r\u00e9els<br \/>\n<\/strong><\/span><span style=\"color: #ff0000;\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-824 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant-300x220.png\" alt=\"\" width=\"119\" height=\"87\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant-300x220.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant.png 542w\" sizes=\"auto, (max-width: 119px) 85vw, 119px\" \/><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 1 : existence d'un quotient<br \/>\nUn quotient de nombres r\u00e9els existe si et seulement si son d\u00e9nominateur est non nul<br \/>\n<\/strong><span style=\"color: #000000;\">Le quotient $ \\dfrac{N}{D} = N \\times \\dfrac{1}{D}$ existe $\\iff N$ existe et $D$ existe et $D \\neq 0$<\/span><strong><br \/>\n<\/strong><\/span><\/li>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 2 :<\/strong><\/span><br \/>\n$ N = \\dfrac{N}{1}$<\/li>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 3 :<\/strong><\/span><br \/>\nsi $D \\neq 0$ alors\u00a0$\\dfrac{D}{D} = 1$<\/li>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 4 :<\/strong><\/span><br \/>\n$\\dfrac{N}{D} = 1 \\iff N = D $ lorsque $D \\neq 0$<br \/>\n$\\dfrac{N}{D} &lt; 1 \\iff N &lt; D $ lorsque $D&gt;0$<br \/>\n$\\dfrac{N}{D} &gt; 1 \\iff N &gt; D $ lorsque $D &gt;0$<\/li>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 5 :<\/strong><\/span><br \/>\nsi $D \\neq 0$ alors<br \/>\n$\\dfrac{-N}{D} =\u00a0\\dfrac{N}{-D} = -\\dfrac{N}{D}$<\/li>\n<li><strong><span style=\"color: #008000;\">R\u00e8gle 7 :\u00a0<span style=\"caret-color: #008000;\">Simplification<\/span>\u00a0d'une fraction<\/span><\/strong><br \/>\nsi $D \\neq 0$ et $k \\neq 0$ alors<br \/>\n$\\dfrac{kN}{kD} =\u00a0\\dfrac{N}{D}$<\/li>\n<li><strong><span style=\"color: #008000;\">R\u00e8gle 8 : Somme de fractions ayant le m\u00eame d\u00e9nominateur<\/span><\/strong><br \/>\nsi $D \u00a0\\neq 0$ \u00a0alors\u00a0<span style=\"font-size: inherit;\">$\\dfrac{N_1}{D} +\u00a0\\dfrac{N_2}{D} = \u00a0\\dfrac{N_1 + N_2}{D}$<\/span><\/li>\n<li><strong><span style=\"color: #008000;\">R\u00e8gle 9 :<\/span><\/strong><br \/>\nsi $D_1 \\neq 0$ et \u00a0$D_2 \\neq 0$ alors<br \/>\n$\\dfrac{N_1}{D_1} +\u00a0\\dfrac{N_2}{D_2} =\u00a0\\dfrac{N_1 D_2}{D_1D_2} + \\dfrac{N_2D_1}{D_1D_2} = \\dfrac{N_1D_2 + N_2D_1}{D_1D_2}$<\/li>\n<li><strong><span style=\"color: #008000;\">R\u00e8gle 10 :<\/span> <\/strong><br \/>\nsi $N_2 \\neq 0$ et $D_1 \\neq 0$ et \u00a0$D_2 \\neq 0$ alors<br \/>\n$\\dfrac{N_1}{D_1} \\times \\dfrac{N_2}{D_2} = \\dfrac{N_1N_2}{D_1D_2}$<\/li>\n<li><strong><span style=\"color: #008000;\">R\u00e8gle 11 :<\/span><\/strong><br \/>\nsi $N_2 \\neq 0$ et $D_1 \\neq 0$ et \u00a0$D_2 \\neq 0$ alors<br \/>\n$\\dfrac{\\dfrac{N_1}{D_1}}{\\dfrac{N_2}{D_2}} = \\dfrac{N_1}{D_1} \\times \\dfrac{1}{\\dfrac{N_2}{D_2}} = \\dfrac{N_1}{D_1}\\dfrac{D_2}{N_2} = \\dfrac{N_1D_2}{N_2D_1}$<\/li>\n<li><span style=\"color: #008000;\"><strong>R\u00e8gle 12 : Le produit des<\/strong><\/span> <span style=\"color: #ff0000;\">termes extr\u00eames<\/span> <strong><span style=\"color: #008000;\">est \u00e9gale au \u00a0produit<\/span><\/strong> des <span style=\"color: #0000ff;\">termes moyens<\/span> :<br \/>\nsi $ B \\neq 0$ et $D \\neq 0$ \u00a0alors<br \/>\n$\\dfrac{A}{B} = \\dfrac{C}{D} \\iff A \\ D = B \\ C$<\/li>\n<\/ul>\n<\/li>\n<li>Puissances enti\u00e8res de $10$<\/li>\n<li>Puissances enti\u00e8res d'un r\u00e9el<\/li>\n<li>Puissances r\u00e9elles d'un nombre r\u00e9el positif<\/li>\n<li>D\u00e9veloppements, factorisation d'expressions alg\u00e9briques<\/li>\n<li><strong>3 Identit\u00e9s remarquables dans le corps des r\u00e9els $(R,+,\\times)$<br \/>\nCes 3 identit\u00e9s sont valables\u00a0 dans tout anneau commutatif $(A,+,\\times)$<br \/>\n<\/strong><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-824 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant-300x220.png\" alt=\"\" width=\"147\" height=\"108\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant-300x220.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/malvoyant.png 542w\" sizes=\"auto, (max-width: 147px) 85vw, 147px\" \/><\/p>\n<ul>\n<li>IR1 : $\\boxed{(a + b)^2 = a^2 + 2ab + b^2}$<br \/>\n<em><strong>Le carr\u00e9 d'une somme est la somme du double produit et de la somme des carr\u00e9s des deux nombres.\u00a0<\/strong><\/em><br \/>\n<em><strong>Le carr\u00e9 d'une somme n'est donc pas la somme des carr\u00e9s.<br \/>\nLa fonction carr\u00e9 : $x \\mapsto x^2$ n'est pas une fonction lin\u00e9aire !!!<\/strong><\/em><br \/>\n<em><strong>Attention, il ne faut jamais oublier le double produit : vous pouvez le v\u00e9rifier sur le sch\u00e9ma suivant :<br \/>\n<\/strong><\/em><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2713 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem1-300x133.png\" alt=\"\" width=\"300\" height=\"133\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem1-300x133.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem1-768x339.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem1-1024x453.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem1.png 1190w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/>D\u00e9monstration :<br \/>\n$(a +b)^2 = (a + b)(a + b) = aa +ab + ba + bb = a^2 + 2ab + b^2$ car $ab = ba$ \u00e0 cause de la commutativit\u00e9 de la loi $\\times$<\/li>\n<li>IR2 : $\\boxed{(a - b)^2 = a^2 - 2ab + b^2}$<br \/>\nD\u00e9monstration :<br \/>\nCette IR2 obtient facilement \u00e0 partir de la IR1 en rempla\u00e7ant $b$ par $-b$.<br \/>\n<em><strong>Attention, ici aussi il ne faut jamais oublier le double produit : vous pouvez le v\u00e9rifier sur le sch\u00e9ma suivant :<br \/>\n<\/strong><\/em><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2714 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem2-300x135.png\" alt=\"\" width=\"300\" height=\"135\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem2-300x135.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem2-768x346.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem2-1024x462.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem2.png 1162w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<li>IR3\u00a0 : $\\boxed{(a + b)(a \\ - \\ b) = a^2 \\ - \\ b^2}$<br \/>\nD\u00e9monstration :<br \/>\n$(a + b)(a \\ - \\ b) = a^2 \\ - \\ ab\u00a0 \\ + \\ ab\u00a0 \\ - \\ b^2 = a^2 \\ - \\ b^2$<br \/>\n<strong>On peut v\u00e9rifier cette identit\u00e9 remarquable sur le graphique suivant :<\/strong><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2715 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem3-300x185.png\" alt=\"\" width=\"300\" height=\"185\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem3-300x185.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem3-768x475.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2019\/05\/idrem3.png 874w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><br \/>\n<strong>Cette identit\u00e9 remarquable IR3 , la plus importante des trois est tr\u00e8s utile Par exemple, en calcul mental.<\/strong><br \/>\n<em>Exemple :<\/em><br \/>\n$101 \\times 99 = (100 + 1)(100 - \\ 1) = 100^2 - \\ 1^2 = 10000 - \\ 1 = 9999$<\/li>\n<li><strong>Il y a bien s\u00fbr d'autres identit\u00e9s remarquables :<\/strong><br \/>\nIR4 : $\\boxed{(a + b + c)^2 =a^2 + b^2 + c^2 + 2ab + 2ac + 2bc}$<br \/>\nIR5 : $\\boxed{a^3 + b^3 = (a + b)(a^2 \\ - \\ ab + b^2)}$<br \/>\nIR6 : $\\boxed{a^3 \\ - \\ b^3 = (a \\ - \\ b)(a^2 + ab +b^2)}$<br \/>\nIR^ : $\\boxed{\\forall n \\geq 2 \\qquad a^n - b^n = (a \\ - \\ b)(a^{n \\ - \\ 1} + a^{n \\ - \\ 2}b + a^{n \\ - \\ 3}b^2 + \\cdots + ab^{n - \\ 2} + b^{n \\ - \\ 1}}$<\/li>\n<li>On peut retrouver <strong>la c\u00e9l\u00e8bre formule du bin\u00f4me de Newton<\/strong> dans un anneau commutatif ou avec des \u00e9l\u00e9ments $a$ et $b$ permutables dans un anneau c'est-\u00e0-dire tels que $ab = ba$<br \/>\nIR8 : $\\boxed{\\displaystyle{(a + b)^n = \\sum_{k=1}^n \\begin{pmatrix} n \\\\ k \\end{pmatrix} \\ \u00a0a^k \\ b^{n \\ - \\ k}}}$<br \/>\ngr\u00e2ce au Triangle dit du fran\u00e7ais Blaise PASCAL \u00a0mais d\u00e9couvert dans les manuscrits ant\u00e9rieurs du\u00a0 chinois SHI SHU YIE et de l'indien BASKARA<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-609 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-300x186.png\" alt=\"\" width=\"376\" height=\"233\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-300x186.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-768x477.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-1024x636.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal.png 1178w\" sizes=\"auto, (max-width: 376px) 85vw, 376px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2994 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2020\/01\/trianglechinois.png\" alt=\"\" width=\"216\" height=\"254\" \/><\/li>\n<li><strong>Attention !<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1178 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/tetemort.png\" alt=\"\" width=\"103\" height=\"92\" \/><\/strong><br \/>\n<strong>Dans un anneau o\u00f9 la multiplication n'est pas commutative<\/strong> ,par exemple l'anneau des matrices carr\u00e9es d'ordre $n$, il existe des matrices $A$ et $B$ telles que $AB \\neq BA$ alors<br \/>\n$\\boxed{(A + B)^2 = (A + B)(A + B) = AA + AB + BA + BB = A^2 + AB + BA + B^2}$<\/li>\n<li><strong>Exercices<\/strong>\n<ul>\n<li>Calculer mentalement 101<sup>2<\/sup>; 204<sup>2<\/sup>; 101 x 99\u00a0; 394 x\u00a0406 ; 47<sup>2<\/sup>- 43<sup>2<\/sup><\/li>\n<li>Compl\u00e9ter les \u00e9galit\u00e9s suivantes \ud83d\ude21<sup>2<\/sup>+ ..................... + 25 = ( ...... + .......)<sup>2<\/sup><br \/>\nx<sup>2<\/sup>-..................... + 16 = ( ...... - .......)<sup>2<\/sup>x<sup>2<\/sup>-..................... + 16 a<sup>2<\/sup>= ( ...... + .......)<sup>2<\/sup>x<sup>2<\/sup>- ..................... + 16 a<sup>2<\/sup>= ( ...... -.......)<sup>2<\/sup>x<sup>2<\/sup>+ 8 x + ........ = ( ...... + .......)<sup>2<\/sup>x<sup>2<\/sup>- 10 x .....................= ( ...... -.......)<sup>2<\/sup>x<sup>2<\/sup>+ 6 x + ......... = ( ...... + .......)<sup>2<\/sup><\/p>\n<p>x<sup>2<\/sup>- 12 x ..................... + 16 = ( ...... -.......)<sup>2<\/sup><\/p>\n<p>x<sup>2<\/sup>+ ..................... + b<sup>4<\/sup>= ( ...... + .......)<sup>2<\/sup><\/p>\n<p>x<sup>2<\/sup>- ..................... + b<sup>4<\/sup>= ( ...... -.......)<sup>2<\/sup><\/p>\n<p>a<sup>2<\/sup>b<sup>8<\/sup>+ ..................... + 64 = ( ...... + .......)<sup>2<\/sup><\/p>\n<p>4x<sup>2<\/sup>y<sup>12<\/sup>- ..................... + 121 = ( ...... -.......)<sup>2<\/sup><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Racine carr\u00e9e d'un r\u00e9el positif<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">Partie enti\u00e8re d'un nombre r\u00e9el<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">Bin\u00f4mes et trin\u00f4mes<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">Manipulation des in\u00e9galit\u00e9s<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #008000;\"><span style=\"color: #ff0000;\">R\u00e9solution d'\u00e9quations et d'in\u00e9quations simples simples<\/span><\/span><\/strong><\/li>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1303 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/kabritbwa.png\" alt=\"\" width=\"254\" height=\"266\" \/><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--more--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>12 R\u00e8gles de calcul sur les quotients de nombres r\u00e9els R\u00e8gle 1 : existence d'un quotient Un quotient de nombres r\u00e9els existe si et seulement si son d\u00e9nominateur est non nul Le quotient $ \\dfrac{N}{D} = N \\times \\dfrac{1}{D}$ existe $\\iff N$ existe et $D$ existe et $D \\neq 0$ R\u00e8gle 2 : $ N &hellip; <a href=\"https:\/\/www.mathnique.com\/site\/formulaires\/\" class=\"more-link\">Continuer la lecture<span class=\"screen-reader-text\"> de &laquo;&nbsp;Calculs num\u00e9riques et alg\u00e9briques&nbsp;&raquo;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-599","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/599","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/comments?post=599"}],"version-history":[{"count":60,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/599\/revisions"}],"predecessor-version":[{"id":3640,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/599\/revisions\/3640"}],"wp:attachment":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/media?parent=599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}