" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "M2B8Eo_hC_9m", "outputId": "634500b7-6d15-480b-b2da-5c8986a87dff" }, "outputs": [], "source": [ "a = int(input(\"Donnez-moi un nombre entier : \"))\n", "print(\"a =\",a)\n", "est_entier(a)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "1NqTmguETl7j", "outputId": "1bc3cf47-caf3-4c66-a91b-68138b6b3e08" }, "outputs": [], "source": [ "# Entrer le même nombre mais l'écrire avec une virgule, comme 22.0\n", "\n", "a = int(input(\"Donnez-moi un nombre entier : \"))\n", "print(\"a =\", a)\n", "est_entier(a)\n" ] }, { "cell_type": "markdown", "metadata": { "id": "wEayY-6kTxNh" }, "source": [ "Python peut convertir \"22\" en int mais il ne peut pas convertir \"22.0\" car cette écriture ne représente pas un int" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "hd3L8CXrC_9m", "outputId": "2d339520-d5d8-4a08-e3bd-404ee294ee6d" }, "outputs": [], "source": [ "# Essayer ceci :\n", "a = int(\"12\")\n", "print(\"On convertit -> \", a)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 201 }, "id": "fOGjfQhaC_9m", "outputId": "60cfb5e8-4303-4e6e-dabf-4aaa27c4d754" }, "outputs": [], "source": [ "# mais en revanche :\n", "a = int(\"1.3\")\n", "print(\"on convertit -> \",a)" ] }, { "cell_type": "markdown", "metadata": { "id": "ZAIybgtxEjCW" }, "source": [ "Python ne peut pas convetir \"1.3\" en int, car 1.3 n'est pas un entier...\n", "\n", "---\n", "\n", "---\n", "\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "ZnykkdZeC_9m", "outputId": "569b1dd3-8943-4166-ba79-08221c7a2e91" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On convertit -> 1.3\n" ] } ], "source": [ "# On aurait pu convertir en float :\n", "a = float(\"1.3\")\n", "print(\"On convertit -> \", a)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "Qn93ZiWBExCX", "outputId": "ce1775e1-6ee4-4ab3-9312-4b1a96faf133" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1\n" ] } ], "source": [ "# Et pour courronner l'étrangeté :\n", "print(int(1.3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", " Partie 2 : Quels nombres existent et quels nombres n'existent pas en informatique ?\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "id": "YnKt7_ITC_9n" }, "source": [ "Une infinité de nombres ~n'existent pas~ ne sont pas représentés, et beaucoup ~existent~ sont représentés. \n", "En vérité, il existe $2^{32}$ nombres flottants différents, si on utilise la norme d'encodage (qui est nommée IEE-754) en 32 bits, $2^{64}$ nombres si on code en 64 bits, ce qui est aujourd'hui la norme la plus courrante. \n", "Il est possible de coder de façon plus précise mais il restera toujours des petites approximations.\n", "\n", "Combien ça fait $2^{64}$ ?" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "ROerLM8BC_9n", "outputId": "2d872b22-37a7-4248-b7cc-a83ab3ff578a" }, "outputs": [ { "data": { "text/plain": [ "18446744073709551616" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "2**64" ] }, { "cell_type": "markdown", "metadata": { "id": "0T8R6QhXC_9n" }, "source": [ "Soit environ 18 milliards de nombres. Ca reste bien peu, comparé à l'infinité de ceux qui existent....\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "id": "Hc_9qnPDqgHj" }, "source": [ "Essayons de calculer avec des entiers $2021^{2021}$ :" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "-f-1CCKoquay", "outputId": "b5dda2d9-e7e9-49a3-b6dc-33b13c73aa1d" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1070707501339186485735906879443009775934515847505507436018532469684060484156622556457940618910046467830496460657338854012970702899287555566826004516420133794728047184071142548492410443301659715439743625096193236087851529177791060888939828172127987187919574923360023338432674647681970057099520986338877776338279883387973701425369925459020411156250094194342515102162425104062929276329283880169188856404079753194924930969635473553870897476682038143931031573253433941733127645758730483703653471990820412314751443278022905625251903298491023565615744988202131047273530186175195631645770932228743760502207651547255118332248293010066737283999639868654389771744361871703954090517198554231862135047152934458481546495173285112090254834412707379743939483939258258060272883866205097104140727360891591292182415184434433559084966642080908014301263634129860823894267892881214040812098733359209927853178049284614778910789273746693494392251653493094721200263124696661580850468990327590470600696358028221286071214020251780884869770706526823025050133114202849364033187158900109994074636628428766519719453765509320594880981969503546846688842646609603375646409577195259949752093337091420361397407133652382998587213371875124731214490123682625978584375839425333747073698037042640320131314959866714363942277088811919393864468736305498528790928148779645137000051903808218437789383541977765870707605561686589983906272073470848028268488954825846773073974630313184857643890852008280832285412381930975152143690812623679268943516157674718411216324659187300126800601491543982222548575437661233255121436703157941185687243508082983009413148354830577548193030828620730569991044139013691351818339679893820579466041388404939447163375273055509857664608938556800532125741862653738579880984473424843099135225470111952600953195926372885743409363689620361424039391783227369945713098884059062008989698213237524214584358374582327810161390561143210174749414401108123111017162894036674507621878737030333929187011290180824855540449661826330388134098478485456092847318723461055449397487959598747155456585106012633744090505619866423764745908836893794013470156672063464948555944589377531256319600873009499341628534003244350910130279641510804087704582493388354170927344674288694442269865212446867858765582189394389311615323725615269095314732053759398073242800819875199383135550513685069985649206405996657405952041383938783165220614224027954067177830894706306235472073283579609878029327947402459368372641445958403280578610171774342560519119652378374775701213736035909437339123991899314479697214815935623043559453555319536837991472857821025116272871538527584444714299741838155265908835308405698119144283161361723411257707584016979579347880738732768644686133120876120656148193181215753107770092826715475403936415376369676214460706683623622199450653273576203928498841508258800908135114203104560599292595660793496025096812151236236053291529277057613163972682124861765135404063518748354801732065920017763618942145640646074008395344835649643762989766166509952115909554924261425270949651192172712265193064004209961942521604181870292547818594673308061368412878644331381074881401979642531061724185064108979726388765300455693814154226451057518416077857338430806721279171795377990216678228860261474248312511490779414842553731172378444774366368358518202179642540909707974727513259866682068710142495929366537883629176635119794633847497708438529585544774722626142388900980828532634569237550326675004061776467566681848665455623909453616926911878139065444195584231381564975212936232015867472774170166281178945555039337953532178897979477874314455471046454917790872768699843085175172386425831215422315485180786961310487769645403176237794790306683261463261407770501862321921702357409558279674727512420823345239141223049199350038730945879271248509140538823203699110476751430175288392919333015077819697241519409463925769863775910867860869854119887201959010098724870047289639756708693858780559074731146756076567895499601833532619303734555414312390035538572546510820300992744870207730206863106724244883713126511517019190510994529661355459251967977743906656420726491467416851182387002461750478810196412989005872052293894115725694877844131176435674937832231066294503743601630698525483477508661483426904187026128463512167448130786449990089632841130381770082903477921867321397453435404067590899381511540683168378614552468139538915584898084699295220133531410497480390960145014740363215683808859792676567052812827406635268103102448679778660069242317041851544503378031072649879877670949796192354409017039092826819346877850835968521800614791695706019894298644669794837154048852548503560210608844827442932580166765250508458427386959239713734981200185794904368043158518087380531922996107580861699536145965621094078276923402469501130467303508024204188054154178821938349166883550391130691142685746205398126458773834953707563237944546440491258933957889840020325461771172260217422069242718512437781313527538548209810133682599245049906800800411128206095235316751826436641628581460565487875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] } ], "source": [ "grand_nombre = 2023**2023\n", "print(grand_nombre)" ] }, { "cell_type": "markdown", "metadata": { "id": "fQ6v6sNMrH1c" }, "source": [ "Combien de chiffres Python a-t-il utilisé pour écrire le résultat de $2023^{2023}$ ?" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "E5BDReYSrWTZ", "outputId": "a2faa7da-5a58-43f1-907c-63ca484ea5b7" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "6689\n" ] } ], "source": [ "nbre_chiffres = len(str(grand_nombre))\n", "print(nbre_chiffres)" ] }, { "cell_type": "markdown", "metadata": { "id": "yZRjMSunqS3Y" }, "source": [ "🙂\n", "```\n", "Les int sont tous représentés (enfin, jusqu'à de très très grands entiers, on pourrait trouver une limite quand même) :\n", "\n", "... -2 -1 0 1 2 3 ...\n", " ---------°---------°---------°---------°---------°---------°---------°\n", "```\n" ] }, { "cell_type": "markdown", "metadata": { "id": "LhDd9Ar6C_9n" }, "source": [ "😟 \n", "```\n", "\n", "Mais pour les réels, on a quelques float, mais pas les nombres qui sont entre deux float :\n", "\n", "... -2.0 -1.0 0.0 1.0 2.0 3.0 ...\n", " °--°-°--|-°--°-°--|-°--°-°--|-°--°-°--|-°--°-°--|-°--°-°--|-°--°-°--|-\n", "```\n", "\n", "La différence entre deux float qui se suivent est de l'ordre de $10^{-45}$ pour les nombres entre 0 et 1, de $10^{-16}$ pour les nombres de l'ordre de quelques unités à quelques milliers, et elle augmente quand on s'éloigne de zéro.\n", "Quelle que soit cette différence, il y a toujours une infinité de nombre réels entre les deux.\n" ] }, { "cell_type": "markdown", "metadata": { "id": "D5CfDGh2WPse" }, "source": [ "🤔 Reprenons l'xemple du début où l'on voit qu' en Python 0.1 + 0.1 + 0.1 = 0.30000000000000004 \n", "En effet 0.1 est représenté par 0.1000000000000000055… et donc quand on fait 0.1 + 0.1 + 0.1 on obtient avec les arrondis la représentation 0.3000000000000000444… qui n’est pas celle de 0.3. \n", "En fait 0.3 est représenté en machine par 0.2999999999999999889…" ] }, { "cell_type": "markdown", "metadata": { "id": "zTwo_QgQWTKM" }, "source": [ "👉 Le plus grand float positif dans l'ordinateur :\n", "- avec cette norme IEEE-754 en 32 bits est $3,402 823 26 × 10^{38}$, c'est à dire en gros 300 mille miliards de miliards de miliards \n", "- avec cette norme IEEE-754 en 64 bits est $1,797 693 134 862 315 7 \\times 10^{308}$" ] }, { "cell_type": "markdown", "metadata": { "id": "xyQygvySrxGH" }, "source": [ "🤔 Recommençons notre calcul de tout à l'heure, mais avec des flottants : $2023.0^{2023.0}$" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 184 }, "id": "gskC5-8fsW4K", "outputId": "36c737d8-73c7-4051-9cb8-1f0f5fc2da2d" }, "outputs": [ { "ename": "OverflowError", "evalue": "(34, 'Result too large')", "output_type": "error", "traceback": [ "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[1;31mOverflowError\u001b[0m Traceback (most recent call last)", "\u001b[1;32m
\n", " Partie 3 : Comment tester l'égalité entre deux nombres ?\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "id": "HKeSgN2pC_9o" }, "source": [ "Quelques bizarreries dues à ces approximations :
" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "iiVQmBE-C_9o", "outputId": "4215fdfe-335a-4857-c177-6c2236b94a04" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(1/10) * 554 == 55.4 -> False\n", "(1/10) * 879 == 87.9 -> True\n", "(1/10) * 18 == 1.8 -> True\n", "(1/10) * 119 == 11.9 -> True\n", "(1/10) * 395 == 39.5 -> True\n", "(1/10) * 567 == 56.7 -> True\n", "(1/10) * 609 == 60.9 -> False\n", "(1/10) * 713 == 71.3 -> True\n", "(1/10) * 323 == 32.3 -> False\n", "(1/10) * 251 == 25.1 -> True\n" ] } ], "source": [ "from random import randint\n", "\n", "def fct_etrange(n) :\n", " return (1/10)*n == n/10 # True si 1/10 * n est égal à n/10 et False sinon\n", "\n", "\n", "for i in range(10) :\n", " n = randint(1, 1000) # on prend un nombre n aléatoire entre 1 et 1000\n", " print('(1/10) * ', n ,' == ', n/10 ,' ->', fct_etrange(n)) # et on affiche si l'égalité est vraie ou pas\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "id": "N5RQfUKXC_9o" }, "source": [ "🌵 Des fois oui et des fois non....\n", "\n", "#### Quelle est la probabilité, pour un entier $n$, que : $n \\times \\dfrac{1}{10} = \\dfrac{n}{10}$ ? \n", " \n" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "PdkqUsDxC_9o", "outputId": "fed2a055-ea15-46ab-e837-9cd68cbcdf66" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "la probabilité que n/10 soit exactement égal à n * (1/10) est de : 64.914 % environ\n" ] } ], "source": [ "# On va choisir au hasard 100000 valeurs de n (entre 0 et 1000) et compter combien de fois on à l'egalité vraie :\n", "\n", "nb_True = 0\n", "nTirages = 100000\n", "\n", "for i in range(nTirages) :\n", " n = randint(1, 1000)\n", " if fct_etrange(n) :\n", " nb_True = nb_True + 1\n", "\n", "print('la probabilité que n/10 soit exactement égal à n * (1/10) est de :', nb_True/nTirages*100,'% environ')" ] }, { "cell_type": "markdown", "metadata": { "id": "YM1RyRrfwA94" }, "source": [ "#### Précisons un peu :\n", " Quelle est la distance moyenne que voit Python, lorsqu'il constate une différence, entre : $n \\times \\dfrac{1}{10}$ et $ \\dfrac{n}{10}$ ? \n", "En maths, nous avons vu que la distance entre deux nombres $a$ et $b$ était égale à $|b-a|$. En python, cela s'écrit `abs(b-a)`." ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "fztQAhYHukEQ", "outputId": "9efcc16c-7229-4e9d-ec56-323d0da76c97" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La probabilité que n/10 soit exactement égal à n * (1/10) est de : 64.546 % environ\n", "L'écart moyen dans les cas de non égalité constatée est : 7.950069277297281e-15\n" ] } ], "source": [ "# On va choisir au hasard 100000 valeurs de n et compter combien de fois on a l'egalité vraie. :\n", "# Si l'égalité est fausse, on va calculer l'écart entre (1/10)*n et n/10.\n", "# On affichera la moyenne de ces écarts.\n", "\n", "from statistics import mean\n", "nb_True = 0\n", "nTirages = 100000\n", "\n", "liste_ecarts = [] # constitution de la liste des écarts\n", "\n", "for i in range(nTirages) :\n", " n = randint(1, 1000)\n", " if fct_etrange(n) :\n", " nb_True = nb_True + 1\n", " if not fct_etrange(n):\n", " liste_ecarts.append(abs((1/10)*n - n/10))\n", "moyenne_ecarts = mean(liste_ecarts)\n", "\n", "\n", "print('La probabilité que n/10 soit exactement égal à n * (1/10) est de :',nb_True/nTirages*100,'% environ')\n", "print(\"L'écart moyen dans les cas de non égalité constatée est : \", moyenne_ecarts)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Comment fait-on pour tester l'égalité alors ?
" ] }, { "cell_type": "markdown", "metadata": { "id": "sAXbc_RG0o7-" }, "source": [ "💡 Il y a plusieurs possibilités :\n", "- Si c'est possible, on travaille avec des entiers.\n", "- On peut convenir, si c'est acceptable, que deux nombres sont égaux si la distance entre les deux est inférieure à une certaine valeur très petite, par exemple à $10^{-14}$. \n", "- Pour les très grands nombres, on peut aussi estimer qu'ils sont égaux, si leur distance relative est inférieure à une certaine valeur très petite." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Regardons l'égalité de nombres \"pas trop grands\"
" ] }, { "cell_type": "markdown", "metadata": { "id": "Ul-wWD-r2UNn" }, "source": [ "Reprenons l'exemple de la fonction étrange modifiée comme expliqué ci-dessus." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "id": "suxejI6h2edm" }, "outputs": [], "source": [ "def new_fct_etrange(n) :\n", " if abs((1/10)*n - n/10) < 10**(-14):\n", " return True\n", " else:\n", " return False # True si 1/10 * n est égal à n/10 et False sinon" ] }, { "cell_type": "markdown", "metadata": { "id": "Cskci1-53COU" }, "source": [ "Reprenons le test précédant :" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "ODQhx6-h3F5N", "outputId": "bfa68bdd-998e-4fd6-ff91-547dc22d94d0" }, "outputs": [], "source": [ "nb_True = 0\n", "nTirages = 100000\n", "\n", "for i in range(nTirages) :\n", " n = randint(1, 1000)\n", " if new_fct_etrange(n) : # on utilise la nouvelle fonction\n", " nb_True = nb_True + 1\n", "\n", "print('la probabilité que n/10 soit exactement égal à n * (1/10) est de :', nb_True/nTirages*100, '% environ')" ] }, { "cell_type": "markdown", "metadata": { "id": "WSUQwJ9m3hjJ" }, "source": [ "Manifestement le choix de $10^{-14}$ n'est pas bon. Reprenons avec $10^{-13}$." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "JTv6Slwh30fD", "outputId": "724a8212-0531-4461-e44c-0e64c4f3e288" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "la probabilité que n/10 soit exactement égal à n * (1/10) est de : 100.0 % environ\n" ] } ], "source": [ "def new_fct_etrange(n) :\n", " if abs((1/10)*n - n/10) < 10**(-13):\n", " return True\n", " else:\n", " return False # True si 1/10 * n est égal à n/10 et False sinon\n", "\n", "nb_True = 0\n", "nTirages = 100000\n", "\n", "for i in range(nTirages) :\n", " n = randint(1, 1000)\n", " if new_fct_etrange(n) : # on utilise la nouvelle fonction\n", " nb_True = nb_True + 1\n", "\n", "print('la probabilité que n/10 soit exactement égal à n * (1/10) est de :', nb_True/nTirages*100, '% environ')" ] }, { "cell_type": "markdown", "metadata": { "id": "1N2W8swZ375r" }, "source": [ "😀 Cela semble mieux ... \n" ] }, { "cell_type": "markdown", "metadata": { "id": "dNvv_ZxopNBF" }, "source": [ "Regardons l'égalité de nombres \"grands\"
\n", " " ] }, { "cell_type": "markdown", "metadata": { "id": "0iiltNOoplYQ" }, "source": [ "Nous allons tester si Python trouve que $3 \\times 10^{-1} \\times 10^{16}$ est égal à $3 \\times 10^{15}$" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "63eAhZEUpR8L", "outputId": "68a6304a-72bd-4243-8ba9-f7896ae33734" }, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "3*(10**(-1))*(10**16) == 3.0*10**15" ] }, { "cell_type": "markdown", "metadata": { "id": "s59ZmMQ7qPyK" }, "source": [ "Cherchons la distance absolue que Python trouve entre ces deux nombres :" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "65BLQ6IHqJXv", "outputId": "645eb0b1-a1ea-42f3-b588-5f338a887a06" }, "outputs": [ { "data": { "text/plain": [ "0.5" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "abs(3*(10**(-1))*(10**16) - 3.0*10**15)" ] }, { "cell_type": "markdown", "metadata": { "id": "WYx4_aVFqaiG" }, "source": [ "😟 \n", "- Cette distance est très supérieure à la distance $10^{-13}$ que nous avions choisie dans le cas précédant des nombres \"pas trop grands\". Elle est même assez \"énorme\" pour considérer que les deux nombres sont identiques... \n", "- Nous allons maintenant regarder la **distance \"relative\"** entre ces deux nombres : \n", "Posons $a=3 \\times 10^{-1} \\times 10^{16}$ et $b= 3 \\times 10^{15}$. \n", "La distance relative entre $a$ et $b$ est égale à $\\dfrac{|b-a|}{a}$ \n", "Dans notre cas elle est égale à :" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "w6mT39TX2HjV", "outputId": "ae0d57bc-5fe6-421b-cf4b-a322e70f4677" }, "outputs": [ { "data": { "text/plain": [ "1.6666666666666663e-16" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "0.5/(3*10**(-1) * 10**(16))\n" ] }, { "cell_type": "markdown", "metadata": { "id": "EZqvvm562X9T" }, "source": [ "🙂 Nous obtenons une distance relative entre $a$ et $b$ de l'ordre de $10^{-16}$. C'est tout à fait raisonnable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "En bref ...
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n",
"❓ Bienvenue dans le monde de l'informatique, où voir si deux nombres sont \"égaux\" n'est pas aussi simple qu'en mathématiques.
\n", " 👉 On regardera leur distance \"absolue\" pour des \"petits\" nombres, et leur distance \"relative\" pour les \"grands\" nombres ...\n", "
\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n", " 👉 On regardera leur distance \"absolue\" pour des \"petits\" nombres, et leur distance \"relative\" pour les \"grands\" nombres ...\n", "
\n", " Partie 4 : Est-ce important de connaître ces différences entre les nombres \"théoriques\", et ceux qui sont utilisés dans nos ordinateurs ?\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "id": "LXxwmei96_xd" }, "source": [ "😢 Il y a quelques anecdotes tragiques qui prouvent qu'il ne faut pas ignorer ce problème. \n", "Mais restons dans le domaine des mathématiques. \n", "Posons $u=\\frac{1}{3}$ \n", "Calculons $4 \\times \\frac{1}{3} - 1$. Cela donne $\\frac{4}{3} - \\frac{3}{3}$, c'est à dire $\\dfrac{1}{3}$ \n", "Vérifions en Python :\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "CVZaJ8fW8KT2", "outputId": "8a09e6e1-47dd-4f52-f90a-067361509685" }, "outputs": [], "source": [ "u = 1/3\n", "u = 4*u - 1\n", "print(u)" ] }, { "cell_type": "markdown", "metadata": { "id": "ZG5yQGn78fEa" }, "source": [ "Nous aurions imaginé un résultat égal à 0.33333333333333333, ou à 0.33333333333333334. \n", "Regardons si nous faisons ce processus deux fois. Théoriquement, le résultat au bout d'une fois étant égal exectement à $\\dfrac{1}{3}$, le résultat au bout de deux fois sera aussi égal exectement à $\\dfrac{1}{3}$. On peut recommencer indéfiniment, on devrait toujours obtenir $\\dfrac{1}{3}$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "zDgaRDBz8z-z", "outputId": "d68b4629-6315-4a94-e201-795232515acb" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.33333333333333304\n" ] } ], "source": [ "u = 1/3\n", "for i in range(2):\n", " u = 4*u - 1\n", "print(u)" ] }, { "cell_type": "markdown", "metadata": { "id": "M2gsX3X7Hzhe" }, "source": [ "Cela ne semble pas si mal... \n", "Et si nous faisions 27 fois de suite ce processus ?" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "A7ARy4c8ICLU", "outputId": "9d4f9df2-1470-4a07-805b-d39c63153218" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.0\n" ] } ], "source": [ "u = 1/3\n", "for i in range(27):\n", " u = 4*u - 1\n", "print(u)" ] }, { "cell_type": "markdown", "metadata": { "id": "GaQ0d9JgIGKf" }, "source": [ "🌵 Aïe ! C'est quand même très différent de $\\dfrac{1}{3}$! \n", "Essayons 28 fois de suite ...\n" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "HP7HmUmaIacA", "outputId": "393584f2-ecf6-47fe-faca-6ea68627e634" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-1.0\n" ] } ], "source": [ "u = 1/3\n", "for i in range(28):\n", " u = 4*u - 1\n", "print(u)" ] }, { "cell_type": "markdown", "metadata": { "id": "mkXPMXdzIg2_" }, "source": [ "🌵🌵 Aïe ! Aïe ! Aïe ! Aucun rapport avec le résultat $\\dfrac{1}{3}$ attendu... \n", "Regardons pour 50 tours de boucles, puis pour 100 tours de boucles ..." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "Gs60NQtMloyL", "outputId": "4d0bbd9d-af56-472d-cc5b-b56b650e5239" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "après 50 tours de boucles u = -23456248059221.0\n", "après 100 tours de boucles u = -2.9734326931374163e+43\n" ] } ], "source": [ "u = 1/3\n", "for i in range(50):\n", " u = 4*u - 1\n", "print(\"après 50 tours de boucles u = \", u)\n", "\n", "u = 1/3\n", "for i in range(100):\n", " u = 4*u - 1\n", "print(\"après 100 tours de boucles u = \", u)" ] }, { "cell_type": "markdown", "metadata": { "id": "-o3h1HOfmSSs" }, "source": [ "🌵🌵🌵 C'est catastrophique ! \n", "Après 100 tours de boucles, notre nombre qui devrait être égal à $\\dfrac{1}{3}$ est de l'ordre de $-3 \\times 10^{43}$..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Si vous avez déjà étudié en mathématiques les suites de nombres
" ] }, { "cell_type": "markdown", "metadata": { "id": "EZoffh3Jn1J8" }, "source": [ "Utilisez le menu des suites de votre calculatrice, pour faire afficher le tableau des 35 premiers termes de la suite constante que nous venons de découvrir : \n", "$\\left\\{\\begin{matrix}\n", "u_{0}=\\frac{1}{3}\\\\ \n", "u_{n+1}=4u_{n}-1\n", "\\end{matrix}\\right.$ \n", "Le résultat risque de ne pas être celui que vous attendiez ..." ] }, { "cell_type": "markdown", "metadata": { "id": "XBV-DUNpEDS1" }, "source": [ "
\n", " Partie 5 : D'autres \"nombres\" de types float\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "id": "J_SZdXNTEUrJ" }, "source": [ "🙋 Regardons ces exemples : A quoi correspondent-ils ?" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "a1010FHRJcBK", "outputId": "51d7eaa0-5d77-4316-eef5-1ff26512f9be" }, "outputs": [ { "data": { "text/plain": [ "inf" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "float(\"inf\")" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "0DXGNuGkJh5T", "outputId": "dd65994a-181e-4831-b5d1-bb6f319d5bc5" }, "outputs": [ { "data": { "text/plain": [ "nan" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "float(\"Nan\")" ] }, { "cell_type": "markdown", "metadata": { "id": "gL_vuGBlKDDq" }, "source": [ "Remarquons que ces \"nombres\" `inf` et `nan` se trouvent dans la bibliothèque `math`" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "id": "hVhyjCOOEaIi" }, "outputs": [], "source": [ "from math import inf, nan\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "id": "DcOiku7NK_tP" }, "source": [ "`inf` représente \"+ infini\", `-inf` représente \"- infini\" et `nan` représente \"not a number\". \n", "Nous retrouvons en particulier les **formes indéterminées** étudiées en mathématiques :" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "H05i4Hh9H5vV", "outputId": "d1ca8a39-f7de-4dce-d340-da53223a5552" }, "outputs": [ { "data": { "text/plain": [ "nan" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "inf + (-inf)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "0aGKexLIFv4k", "outputId": "104fee3d-0bda-44c9-ec8f-35abb0643957" }, "outputs": [ { "data": { "text/plain": [ "nan" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "inf * 0" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "Y7cd9WdWHkHB", "outputId": "7f006b05-aa10-4bf4-d7ea-564ee6e39859" }, "outputs": [ { "data": { "text/plain": [ "nan" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "inf / inf" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Eplications sur `nan` :
" ] }, { "cell_type": "markdown", "metadata": { "id": "qSv_3VaEG8_s" }, "source": [ "nan sur wikipedia\n", "
" ] }, { "cell_type": "markdown", "metadata": { "id": "UrHwTvOZVX5-" }, "source": [ "\n", "
![\"Creative](\"https://s2.qwant.com/thumbr/700x0/f/9/436dcce5e02185c0b2e1b0e7a969aa81d7b0438d0a7f2eaee89abb9661b70b/cc-by-nc-sa.svg_.png?u=http%3A%2F%2Fopensiddur.org%2Fwp-content%2Fuploads%2F2011%2F03%2Fcc-by-nc-sa.svg_.png&q=0&b=1&p=0&a=1\")
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. " ] } ], "metadata": { "colab": { "collapsed_sections": [], "name": "les_nombres_et_python.ipynb", "provenance": [] }, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 1 }