{"id":59758,"date":"2023-01-06T04:20:10","date_gmt":"2023-01-06T04:20:10","guid":{"rendered":"https:\/\/euler.euclid.int\/a-propos-de-lidentite-deuler\/"},"modified":"2023-01-06T04:20:10","modified_gmt":"2023-01-06T04:20:10","slug":"a-propos-de-lidentite-deuler","status":"publish","type":"page","link":"https:\/\/euler.euclid.int\/fr\/a-propos-de-lidentite-deuler\/","title":{"rendered":"\u00c0 propos de l&#8217;identit\u00e9 d&#8217;Euler"},"content":{"rendered":"<p>L&#8217;identit\u00e9 d&#8217;Euler est souvent consid\u00e9r\u00e9e comme la formule la plus belle en math\u00e9matiques. Des individus la portent sur des T-shirts et se la font tatouer. Pourquoi donc ?  <\/p>\n<p><iframe title=\"Euler&#039;s Identity (Complex Numbers)\" width=\"800\" height=\"450\" src=\"https:\/\/www.youtube.com\/embed\/sKtloBAuP74?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p>L&#8217;identit\u00e9 s&#8217;\u00e9nonce comme suit<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/17eab3c768113bf642698c4da7edf422\/images\/img-0001.png\" alt=\"[ e^{ipi }+1=0, ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"rightimage\"><img fetchpriority=\"high\" decoding=\"async\" src=\"https:\/\/plus.maths.org\/issue45\/features\/sangwin\/Euler_portraitcolour.jpg\" alt=\"Leonhard Euler, 1707-1783. Portrait par Johann Georg Brucker.\" width=\"250\" height=\"313\">Leonhard Euler, 1707-1783. Portrait par Johann Georg Brucker. <\/p>\n<\/div>\n<p>o\u00f9 <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/8ef555e22dd3dc0de3bd377f42bfe5d6\/images\/img-0001.png\" alt=\"$e= 2,7182818284... $\"> est la base du logarithme naturel, <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/8ef555e22dd3dc0de3bd377f42bfe5d6\/images\/img-0002.png\" alt=\"$pi =3,1415926535...$\"> est le rapport entre la circonf\u00e9rence d&#8217;un cercle et son diam\u00e8tre, et <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/8ef555e22dd3dc0de3bd377f42bfe5d6\/images\/img-0003.png\" alt=\"$i =sqrt&lt;wpml_curved wpml_value='-1'&gt;&lt;\/wpml_curved&gt;$\">. Ces trois constantes sont d&#8217;une importance capitale en math\u00e9matiques \u2014 et puisque l&#8217;identit\u00e9 implique \u00e9galement <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/8ef555e22dd3dc0de3bd377f42bfe5d6\/images\/img-0004.png\" alt=\"$0$\"> et <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/8ef555e22dd3dc0de3bd377f42bfe5d6\/images\/img-0005.png\" alt=\"$1$\">, nous obtenons une formule qui relie cinq des nombres les plus importants en math\u00e9matiques en utilisant quatre des op\u00e9rations et relations math\u00e9matiques les plus fondamentales \u2013 l&#8217;addition, la multiplication, l&#8217;exponentiation et l&#8217;\u00e9galit\u00e9. C&#8217;est pour cette raison que les math\u00e9maticiens affectionnent tant l&#8217;identit\u00e9 d&#8217;Euler.  <\/p>\n<p>Mais d&#8217;o\u00f9 provient-elle et que signifie-t-elle ? Comme mentionn\u00e9 pr\u00e9c\u00e9demment, <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0001.png\" alt=\"$i =sqrt&lt;wpml_curved wpml_value='-1'&gt;&lt;\/wpml_curved&gt;$\">. Cela peut sembler surprenant car les nombres n\u00e9gatifs ne sont pas cens\u00e9s avoir de racines carr\u00e9es. Cependant, si nous d\u00e9cr\u00e9tons simplement que <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0002.png\" alt=\"$-1$\"> poss\u00e8de effectivement une racine carr\u00e9e et que nous la nommons <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0003.png\" alt=\"$i$\">, nous pouvons alors construire une toute nouvelle classe de nombres, appel\u00e9e les <em>nombres complexes<\/em>. Les nombres complexes ont la forme      <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0004.png\" alt=\"$x+iy,$\">  o\u00f9  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0005.png\" alt=\"$x$\">  et  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0006.png\" alt=\"$y$\">  sont des nombres r\u00e9els ordinaires (pour le nombre complexe  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0003.png\" alt=\"$i$\">  nous avons  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0007.png\" alt=\"$x=0$\">  et  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3be7c11382a410ca39080c3524be3014\/images\/img-0008.png\" alt=\"$y=1$\">). Veuillez consulter <a href=\"https:\/\/plus.maths.org\/content\/maths-minute-complex-numbers\">ici<\/a> pour une introduction rapide aux nombres complexes et \u00e0 leur calcul. Notez qu&#8217;un nombre r\u00e9el peut \u00e9galement \u00eatre consid\u00e9r\u00e9 comme un nombre complexe. Le nombre <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/649a99b95aeefbb3691de877b79746b1\/images\/img-0001.png\" alt=\"$-1$\">, par exemple, est un nombre complexe avec <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/649a99b95aeefbb3691de877b79746b1\/images\/img-0002.png\" alt=\"$x=-1$\"> et <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/649a99b95aeefbb3691de877b79746b1\/images\/img-0003.png\" alt=\"$y=0$\">.  <\/p>\n<p>De la m\u00eame mani\u00e8re qu&#8217;un nombre r\u00e9el est repr\u00e9sent\u00e9 par un point sur une droite num\u00e9rique, un nombre complexe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/9254d09b03f608eae975565ddde98879\/images\/img-0001.png\" alt=\"$z$\"> est repr\u00e9sent\u00e9 par un point dans le plan. Au nombre complexe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/9254d09b03f608eae975565ddde98879\/images\/img-0002.png\" alt=\"$z=x+iy$\">, nous associons le point de coordonn\u00e9es <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/9254d09b03f608eae975565ddde98879\/images\/img-0003.png\" alt=\"$(x,y)$\">. <\/p>\n<div class=\"centreimage\"><img decoding=\"async\" src=\"https:\/\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/Euler\/cartesian.png\" alt=\"Coordonn\u00e9es cart\u00e9siennes\" width=\"400\" height=\"332\">&nbsp;<\/p>\n<\/div>\n<p>Dans cette description, nous avons utilis\u00e9 les coordonn\u00e9es cart\u00e9siennes : elles d\u00e9crivent la position d&#8217;un point en indiquant la distance \u00e0 parcourir dans la direction horizontale et la distance \u00e0 parcourir dans la direction verticale. Cependant, il est parfois plus commode de d\u00e9crire la position d&#8217;un point en termes de vecteur partant du point d&#8217;intersection des deux axes, comme illustr\u00e9 ci-dessous. <\/p>\n<div class=\"centreimage\"><img decoding=\"async\" src=\"https:\/\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/Euler\/polar.png\" alt=\"Coordonn\u00e9es polaires\" width=\"400\" height=\"360\">&nbsp;<\/p>\n<\/div>\n<p>Pour d\u00e9finir ce vecteur, il est n\u00e9cessaire de conna\u00eetre sa longueur <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0001.png\" alt=\"$r$\"> et l&#8217;angle <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0002.png\" alt=\"$\theta $\"> qu&#8217;il forme avec l&#8217;axe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0003.png\" alt=\"$x$\"> positif (mesur\u00e9 dans le sens antihoraire). Ce sont les <em>coordonn\u00e9es polaires<\/em> de notre point. La trigonom\u00e9trie \u00e9l\u00e9mentaire (voir le diagramme ci-dessous) nous indique que si un point a pour coordonn\u00e9es cart\u00e9siennes <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0004.png\" alt=\"$(x,y)$\"> et pour coordonn\u00e9es polaires <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0005.png\" alt=\"$(r,\theta )$\">, alors  <\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0006.png\" alt=\"[ x=r cos {(\theta )} ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>et<\/p>\n<table id=\"a0000000003\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/37f3a5d111741224a911781164a3c4b7\/images\/img-0007.png\" alt=\"[ y=r sin {(\theta )}. ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"centreimage\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/Euler\/trig.png\" alt=\"trigonom\u00e9trie\" width=\"400\" height=\"353\">&nbsp;<\/p>\n<\/div>\n<p>Par cons\u00e9quent, le nombre complexe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0001.png\" alt=\"$z$\"> repr\u00e9sent\u00e9 par notre point, <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0002.png\" alt=\"$x+iy$\">, peut \u00e9galement s&#8217;\u00e9crire<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0003.png\" alt=\"[ z= r (cos {(\theta )}+isin {(\theta )}). ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Voici le point crucial. Il se trouve que pour les nombres r\u00e9els <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0004.png\" alt=\"$r$\"> et   <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0005.png\" alt=\"$\theta $\"><\/p>\n<table id=\"a0000000003\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/c9570eacdeb0833cf294aca44d62bf7e\/images\/img-0006.png\" alt=\"[ r(cos {(\theta )} + i sin {(\theta )}) = re^{i\theta }. ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Vous pouvez d\u00e9montrer cela en utilisant les <em>s\u00e9ries enti\u00e8res<\/em>, consultez <a href=\"https:\/\/plus.maths.org\/content\/beauty-mathematics\">ici<\/a> pour en savoir plus. C&#8217;est un fait remarquable que la fonction exponentielle et les deux fonctions trigonom\u00e9triques sinus et cosinus soient li\u00e9es de cette mani\u00e8re. Cela signifie que tout nombre complexe    <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0001.png\" alt=\"$z$\">  peut s&#8217;\u00e9crire sous la forme  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0002.png\" alt=\"$re^{i\theta }$\">  o\u00f9  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0003.png\" alt=\"$r$\">  est la longueur de la ligne reliant le point du plan associ\u00e9 \u00e0  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0001.png\" alt=\"$z$\">  au point d&#8217;intersection des axes, et  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0004.png\" alt=\"$\theta $\">  est l&#8217;angle que forme cette ligne avec l&#8217;axe  <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/3c05d0df825cecff17f7016919c5850a\/images\/img-0005.png\" alt=\"$x$\"> positif (mesur\u00e9 dans le sens antihoraire).<\/p>\n<p>Ceci rend d\u00e9sormais l&#8217;identit\u00e9 d&#8217;Euler parfaitement limpide. Le nombre complexe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b3e85a885255cb250d2331bfdf746ac0\/images\/img-0001.png\" alt=\"$e^{ipi } = 1 \times e^{ipi }$\"> repr\u00e9sente le point sur le plan \u00e0 une distance <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b3e85a885255cb250d2331bfdf746ac0\/images\/img-0002.png\" alt=\"$1$\"> du point d&#8217;intersection des axes avec un angle associ\u00e9 de <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b3e85a885255cb250d2331bfdf746ac0\/images\/img-0003.png\" alt=\"$pi $\">. Il s&#8217;agit du point aux coordonn\u00e9es cart\u00e9siennes <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b3e85a885255cb250d2331bfdf746ac0\/images\/img-0004.png\" alt=\"$(-1,0)$\"> qui repr\u00e9sente le nombre complexe <img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b3e85a885255cb250d2331bfdf746ac0\/images\/img-0005.png\" alt=\"$-1$\">.  <\/p>\n<div class=\"centreimage\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/Euler\/Euler.png\" alt=\"L'identit\u00e9 d'Euler\" width=\"400\" height=\"328\">&nbsp;<\/p>\n<\/div>\n<p>En r\u00e9unissant tous ces \u00e9l\u00e9ments, nous constatons que<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b7b172cee37d9c06da34d769912d2d57\/images\/img-0001.png\" alt=\"[ e^{ipi } = -1, ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>ce qui signifie que<\/p>\n<table id=\"a0000000003\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/plus.maths.org\/MI\/b7b172cee37d9c06da34d769912d2d57\/images\/img-0002.png\" alt=\"[ e^{ipi }+1 = 0. ]\"><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Et voil\u00e0 l&#8217;identit\u00e9 d&#8217;Euler.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>L&#8217;identit\u00e9 d&#8217;Euler est souvent consid\u00e9r\u00e9e comme la formule la plus belle en math\u00e9matiques. Des individus la portent sur des T-shirts et se la font tatouer. Pourquoi donc ? L&#8217;identit\u00e9 s&#8217;\u00e9nonce comme suit Leonhard Euler, 1707-1783. Portrait par Johann Georg Brucker. o\u00f9 est la base du logarithme naturel, est le rapport entre la circonf\u00e9rence d&#8217;un cercle [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-59758","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u00c0 propos de l&#039;identit\u00e9 d&#039;Euler - EFMU: The Euler-Franeker Memorial University and Institute<\/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:\/\/euler.euclid.int\/a-propos-de-lidentite-deuler\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u00c0 propos de l&#039;identit\u00e9 d&#039;Euler - EFMU: The Euler-Franeker Memorial University and Institute\" \/>\n<meta property=\"og:description\" content=\"L&#8217;identit\u00e9 d&#8217;Euler est souvent consid\u00e9r\u00e9e comme la formule la plus belle en math\u00e9matiques. 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