{"id":468,"date":"2018-08-28T14:23:35","date_gmt":"2018-08-28T12:23:35","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=468"},"modified":"2021-10-26T17:22:06","modified_gmt":"2021-10-26T15:22:06","slug":"lexistence-de-pi","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/28\/lexistence-de-pi\/","title":{"rendered":"L’existence de Pi"},"content":{"rendered":"\n

\u00c0 l’\u00e9cole primaire, les \u00e9l\u00e8ves prennent connaissance de l’existence d’un nombre myst\u00e9rieux nomm\u00e9 \u00ab\u00a0Pi\u00a0\u00bb et not\u00e9 par la lettre grecque \u00ab\u00a0\\(\\pi\\)\u00a0\u00bb par Archim\u00e8de, en rapport avec l’initiale du mot  \u00ab\u00a0\\(\\pi\\varepsilon\\rho\\iota\\mu\\varepsilon\\tau\\rho o\\zeta\\)\u00a0\u00bb (\u00ab\u00a0p\u00e9rim\u00e8tre\u00a0\u00bb en fran\u00e7ais).<\/p>\n\n\n\n

\u00c0 ce stade de l’apprentissage, les professeurs des \u00e9coles disent que la valeur de Pi est 3,14 et j’esp\u00e8re qu’ils ajoutent que ce n’est qu’une valeur approch\u00e9e de ce nombre qui admet une partie d\u00e9cimale infinie. D’ailleurs, tout nombre qui ne peut pas s’\u00e9crire enti\u00e8rement est d\u00e9sign\u00e9 par une lettre ou autre chose de \u00ab\u00a0rapide \u00e0 \u00e9crire\u00a0\u00bb et c’est la raison pour laquelle Pi est d\u00e9sign\u00e9 par une lettre : on ne peut pas l’\u00e9crire en entier.<\/p>\n\n\n\n

On d\u00e9finit le nombre \\(\\pi\\) comme \u00e9tant le rapport constant entre le p\u00e9rim\u00e8tre d’un cercle et son diam\u00e8tre (il faut entendre ici : dans le plan euclidien<\/em>). Mais pourquoi ce rapport est-il constant ? Comment a-t-on pu d\u00e9montrer que \\(\\pi\\) existait ? Nous allons le voir ici …<\/p>\n\n\n\n\n\n\n\n

D\u00e9monstration de l’existence de Pi<\/h1>\n\n\n\n

Consid\u00e9rons deux cercles concentriques \\(\\mathcal{C}_1\\), de rayon \\(r_1\\), et \\(\\mathcal{C}_2\\), de rayon \\(r_2\\), avec \\(r_1<r_2\\). On appelle O le centre de ces cercles.<\/p>\n\n\n\n

\"existence<\/a><\/figure><\/div>\n\n\n\n

On trace deux rayons [OC] et [OD] de \\(\\mathcal{C}_2\\) et on note A et B les points d’intersection respectifs de [OC] et [OD] avec \\(\\mathcal{C}_1\\).<\/p>\n\n\n\n

Par construction, \\[ \\frac{\\text{OA}}{\\text{OC}}=\\frac{\\text{OB}}{\\text{OD}}=\\frac{r_1}{r_2} \\]<\/p>\n\n\n\n

et donc, d’apr\u00e8s la r\u00e9ciproque du th\u00e9or\u00e8me de Thal\u00e8s, \\[ (\\text{AB})~\/\/~(\\text{CD}).\\]<\/p>\n\n\n\n

On peut ainsi conclure, d’apr\u00e8s le th\u00e9or\u00e8me de Thal\u00e8s,  que :  \\[ \\frac{r_1}{r_2}=\\frac{\\text{AB}}{\\text{CD}}.\\]<\/p>\n\n\n\n

Par cons\u00e9quent, les produits en croix sont \u00e9gaux : \\[ r_1\\times\\text{CD}=r_2\\times\\text{AB}\\]<\/p>\n\n\n\n

et donc, en divisant par \\(\\text{AB}\\times\\text{CD}\\) : \\[ \\frac{r_1}{\\text{AB}}=\\frac{r_2}{\\text{CD}}.\\qquad(1) \\]<\/p>\n\n\n\n

Construisons maintenant n<\/em> triangles identiques comme l’illustre le sch\u00e9ma suivant :<\/p>\n\n\n\n

\"existence<\/a><\/figure><\/div>\n\n\n\n

Tous les triangles sont identiques, inscrits dans les cercles et isoc\u00e8les en O.<\/p>\n\n\n\n

Les triangles inscrits dans \\(\\mathcal{C}_1\\) ont pour base a<\/em> et ceux inscrits dans \\(\\mathcal{C}_2\\) ont pour base b<\/em>.<\/p>\n\n\n\n

L’\u00e9galit\u00e9 (1) nous dit que : \\[  \\frac{r_1}{n\\times a}=\\frac{r_2}{n\\times b}.\\qquad (2) \\]<\/p>\n\n\n\n

Si on augmente le nombre de triangles, c’est-\u00e0-dire si l’on fait tendre n<\/em> vers l’infini, alors \\(n\\times a\\) se rapproche du p\u00e9rim\u00e8tre de \\(\\mathcal{C}_1\\) (que l’on va noter \\(\\mathcal{P}_1\\)) et \\(n\\times b\\) vers le p\u00e9rim\u00e8tre de \\(\\mathcal{C}_2\\) (que l’on va noter \\(\\mathcal{P}_2\\)).<\/p>\n\n\n\n

Ainsi, l’\u00e9galit\u00e9 (2) devient, par passage \u00e0 la limite : \\[ \\frac{r_1}{\\mathcal{P}_1}=\\frac{r_2}{\\mathcal{P}_2}, \\]<\/p>\n\n\n\n

ou, si l’on pr\u00e9f\u00e8re : \\[ \\frac{\\mathcal{P}_1}{r_1}=\\frac{\\mathcal{P}_2}{r_2}.\\]<\/p>\n\n\n\n

En multipliant cette derni\u00e8re \u00e9galit\u00e9 par \\(\\displaystyle\\frac{1}{2}\\), on obtient : \\[ \\frac{\\mathcal{P}_1}{2r_1}=\\frac{\\mathcal{P}_2}{2r_2} ,\\]<\/p>\n\n\n\n

soit : \\[ \\frac{\\mathcal{P}_1}{D_1}=\\frac{\\mathcal{P}_2}{D_2},\\]<\/p>\n\n\n\n

c’est-\u00e0-dire que, quel que soit le rayon du cercle que l’on prend, le rapport entre son p\u00e9rim\u00e8tre et son diam\u00e8tre est constant.<\/p>\n\n\n\n

C’est ce rapport que l’on a d\u00e9cid\u00e9 de noter \\(\\pi\\).<\/p>\n\n\n\n

Existence de Pi<\/a>T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n

Les sources \\(\\LaTeX\\) du document PDF :<\/p>\n\n\n\n

Existence de Pi<\/a>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

\u00c0 l’\u00e9cole primaire, les \u00e9l\u00e8ves prennent connaissance de l’existence d’un nombre myst\u00e9rieux nomm\u00e9 \u00ab\u00a0Pi\u00a0\u00bb et not\u00e9 par la lettre grecque \u00ab\u00a0\\(\\pi\\)\u00a0\u00bb par Archim\u00e8de, en rapport avec l’initiale du mot  \u00ab\u00a0\\(\\pi\\varepsilon\\rho\\iota\\mu\\varepsilon\\tau\\rho o\\zeta\\)\u00a0\u00bb (\u00ab\u00a0p\u00e9rim\u00e8tre\u00a0\u00bb en fran\u00e7ais). \u00c0 ce stade de l’apprentissage, les professeurs des \u00e9coles disent que la valeur de Pi est […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[25,29,35,36],"class_list":["post-468","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-demonstration","tag-limite","tag-pi","tag-thales"],"yoast_head":"\nL'existence de Pi - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/28\/lexistence-de-pi\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"L'existence de Pi - 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