eulers_number.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "source": [
    "# Euler's number"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "General imports:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import seaborn as sns\n",
    "sns.set_style(\"darkgrid\")\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = (10, 8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Some facts about $e$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Excerpt from a nice thread on $e$: https://www.quora.com/Whats-so-special-about-the-Euler-number-e\n",
    "\n",
    "> There exists a number, $2.718281828...$ and if you take that to the power of anything, represented by x, its derivative will have the exact same curve. It will have all the same properties. And the derivative of the derivative of the derivative... will have the same properties. You could say that the value of the function at a specific point will be the same value as its growth at that point. It is in one way insane. You start to ask questions like: Why that number? Why those digits? It sort of doesn't make any sense. Having for example $2.7^x$ or $3^x$ I won't get the same awesome results but having this weird number I do.\n",
    "\n",
    "> I think it is one of the coolest but still in a way annoying things in the universe. Why couldn't it just have been a more simple number?\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.718281828459045"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.exp(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`math.exp(1)` is the same as $e^1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* $e$ is known as _Euler's Number_ and is the base of **Natural Logarithms**, meaning $e$ is the unique number whose natural logarithm is equal to one"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$log(e)=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.log(math.exp(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The function $f(x) = e^x$ is called the (natural) exponential function, and is the unique exponential function __equal to its own derivative__. It's the only function on Earth that will do that!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def e_x(x):\n",
    "    return math.exp(x)\n",
    "\n",
    "vals_ex = [e_x(x) for x in range(1,100)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10909ec50>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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IigEAWs/OLvMSdgGA1nNcMPMSdgGA1rvaZ1d0YTZWDADQevrsMi9hFwBovcFBza7owmys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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10902e590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(ncols=1, nrows=1)\n",
    "ax.plot(vals_ex,label='e^x')\n",
    "ax.legend(loc='center right', bbox_to_anchor=(1.20, 0.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How to calculate $e$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are several ways to do so, but non of them gives an exact answer because $e$ is an _irrational_ number. For example:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "\\begin{equation*}\n",
    "\\Bigl(1 + \\frac{1}{n}\\Bigr)^n\n",
    "\\end{equation*}\n",
    "\n",
    "or\n",
    "\n",
    "\\begin{equation*}\n",
    "\\sum_{k=0}^n\\frac{1}{k!}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This expressions approach $e$ as $n$ gets bigger and bigger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculation_1(n):\n",
    "    return (1+1./n)**n\n",
    "\n",
    "def calculation_2(n):\n",
    "    return sum([1./math.factorial(k) for k in range(0,n+1)])\n",
    "\n",
    "vals_0 = [math.exp(1) for n in range(1,100)]\n",
    "vals_1 = [calculation_1(n) for n in range(1,100)]\n",
    "vals_2 = [calculation_2(n) for n in range(1,100)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10b9a8a90>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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t+rWnls4zuTjFsdEjDFeHNv36kiRJuj6FCrxZrb5lN6w9Pf0MgNVdSZKkHlOo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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b9288d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(ncols=1, nrows=1)\n",
    "ax.plot(vals_0,'--',label='e')\n",
    "ax.plot(vals_1,label='1',alpha=0.7)\n",
    "ax.plot(vals_2,label='2',alpha=0.7)\n",
    "ax.legend(loc='center right', bbox_to_anchor=(1.20, 0.5))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}