{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "16b94382",
   "metadata": {},
   "source": [
    "# Exercices sur les nombres\n",
    "\n",
    "## Fonctions mathématiques\n",
    "\n",
    "* Est ce que la fonction log est le logarithme décimal ou népérien ?\n",
    "* Calculer $x = \\sqrt{2}$ puis calculer $x^2$. Que se passe-t-il ?\n",
    "* Calculer $\\arccos{\\frac{\\sqrt{2}}{2}}$ et comparer à sa valeur théorique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d75da916",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.302585092994046\n"
     ]
    }
   ],
   "source": [
    "from math import log\n",
    "\n",
    "print(log(10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "bba315c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0000000000000004"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import sqrt\n",
    "sqrt(2)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "478ab27c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7853981633974483\n",
      "0.7853981633974483\n"
     ]
    }
   ],
   "source": [
    "from math import acos, pi\n",
    "\n",
    "print(acos(sqrt(2)/2))\n",
    "print(pi/4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb6c743d",
   "metadata": {},
   "source": [
    "## Constante de structure fine\n",
    "\n",
    "La constante de structure fine est définie en physique comme étant égale à \n",
    "\n",
    "$$ \n",
    "\\alpha = \\frac{e^2}{2\\epsilon_0 h c}\n",
    "$$ \n",
    "\n",
    "où \n",
    "\n",
    "* $e$ est la charge de l'électron et vaut $1.602176634 \\times 10^{-19} C$\n",
    "* $h$ est la constante de Planck et vaut $6.626\\,070\\,15 \\times 10^{-34} J s$ \n",
    "* $\\epsilon_0$ la permitivité du vide et vaut $8.8541878128 \\times 10^{-12} F/m$\n",
    "* $c$ la célérité de la lumière dans le vide, $c=299792458 m/s$\n",
    "\n",
    "Définissez en Python les variables ``e``, ``hbar``, ``epsilon_0`` et ``c``. Calculez $\\alpha$ et $1/\\alpha$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "ad63ad85",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "137.0359990841083\n"
     ]
    }
   ],
   "source": [
    "from math import pi\n",
    "\n",
    "e = 1.602176634E-19\n",
    "h = 6.62607015E-34\n",
    "epsilon_0 = 8.8541878128E-12\n",
    "c = 299792458\n",
    "\n",
    "\n",
    "alpha = e**2/(2*epsilon_0*h*c)\n",
    "print(1/alpha)\n",
    "              "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e16c06e",
   "metadata": {},
   "source": [
    "## Précision des nombres\n",
    "\n",
    "* Soit $x=1$ et $\\epsilon = 10^{-15}$. Calculez $y=x + \\epsilon$ et ensuite $y - x$. \n",
    "* Pourquoi le résultat est différent de $10^{-15}$. \n",
    "* Que vaut cette valeur ? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "c78f8cc2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.1102230246251565e-15\n"
     ]
    }
   ],
   "source": [
    "x = 1\n",
    "epsilon = 1E-15\n",
    "y = x + epsilon\n",
    "print(y - x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "a790a63a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.1102230246251565e-15\n"
     ]
    }
   ],
   "source": [
    "print(5*2**-52)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93b96efe",
   "metadata": {},
   "source": [
    "## Calcul d'une dérivée\n",
    "\n",
    "On considère une fonction $f(x)$. On rappelle que la dérivée peut se définir comme\n",
    "\n",
    "$$\n",
    "f^\\prime(x) = \\lim_{\\epsilon\\rightarrow0}\\frac{f(x+\\epsilon) - f(x)}{\\epsilon}\n",
    "$$\n",
    "\n",
    "Pour calculer numériquement une dérivée, il faut évaluer la limite en prenant une valeur 'petite' de $\\epsilon$. \n",
    "\n",
    "On prendra comme exemple $f(x) = \\sin(x)$. \n",
    "\n",
    "* Calculer numériquement la dérivée de $f$ en $\\pi/4$ en utilisant la formule pour $\\epsilon = 10^{-6}$.\n",
    "* Comparer à la valeur théorique $\\cos(x)$ pour différentes valeurs de $\\epsilon$ que l'on prendra comme puissance de 10 ($\\epsilon = 10^{-n}$). Que se passe-t-il si $\\epsilon$ est trop petit ? trop grand ?\n",
    "* Ecrire la fonction ``sin_prime(x, epsilon)`` qui calcule la dérivée de sin en $x$\n",
    "* Ecrire une fonction qui prend une fonction quelconque et renvoie la fonction dérivée."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "5ee9a942",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.365427460879424e-06\n",
      "n= 1 -0.03650380828255784\n",
      "n= 2 -0.0035472894973379576\n",
      "n= 3 -0.0003536712121802177\n",
      "n= 4 -3.535651724428934e-05\n",
      "n= 5 -3.5355413900983734e-06\n",
      "n= 6 -3.5344236126721995e-07\n",
      "n= 7 -3.5807553921962665e-08\n",
      "n= 8 3.050251939917814e-09\n",
      "n= 9 3.635694267867251e-08\n",
      "n= 10 9.245353623787977e-07\n",
      "n= 11 5.365427460879424e-06\n",
      "n= 12 -5.7368027853721415e-06\n",
      "n= 13 0.00010528549967714351\n",
      "n= 14 0.003435954573552613\n"
     ]
    }
   ],
   "source": [
    "from math import sin, pi, cos\n",
    "\n",
    "x = pi/4\n",
    "epsilon = 1E-11\n",
    "\n",
    "d = (sin(x + epsilon) - sin(x))/epsilon - cos(x)\n",
    "print(d)\n",
    "\n",
    "for n in range(1, 15):\n",
    "    epsilon = 10**(-n)\n",
    "    d = (sin(x + epsilon) - sin(x))/epsilon - cos(x)\n",
    "    print('n=', n, d)\n",
    "\n",
    "# Lorsque epsilon est trop petit, il y a des erreurs d'arrondi. Lorsque epsilon est trop grand, nous\n",
    "# sommes loin de la limite"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "4b8390eb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.050251939917814e-09"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def sin_prime(x, epsilon):\n",
    "    return (sin(x + epsilon) - sin(x))/epsilon\n",
    "\n",
    "def derivee(f, epsilon):\n",
    "    def f_prime(x):\n",
    "        return (f(x+epsilon) - f(x))/epsilon\n",
    "    return f_prime\n",
    "\n",
    "derivee(sin, epsilon=1E-8)(x) - cos(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "baacad7a",
   "metadata": {},
   "source": [
    "## Nombre complexe\n",
    "\n",
    "* Ecrire une fonction qui calcule le module d'un nombre complexe $z$\n",
    "* Ecrire une fonction qui à partir de $r$ et $\\theta$ renvoie le nombre $z = re^{i\\theta} = r\\cos(\\theta) + ir\\sin(\\theta)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "78f719b3",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.8.8"
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}