{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e6040bd4-7d0d-43f3-b73f-51b4bc5d3504",
   "metadata": {},
   "source": [
    "We use Python as a calculator."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9313a7fa-8474-4a43-b0c9-c3cd351e94fe",
   "metadata": {},
   "source": [
    "# 1. Integers and Floats"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ed8c2c3-a191-49db-a134-0d53bcb50900",
   "metadata": {},
   "source": [
    "The integers in Python can grow arbitrarily long.  The exponentiation in Python is executed with `**` not as `^`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "2103cd61-c821-4a0a-91c1-35cfcd22f1d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1267650600228229401496703205376"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2**10)**10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b76a268c-f8f1-41c4-be6b-016e6275d159",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2676506002282294e+30"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2.0**10)**10"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ec3f1db-6b20-49c0-a85c-c22ca77fe41e",
   "metadata": {},
   "source": [
    "Observe the scientific notation, `e+30` stands for `10**30`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "675b8820-2eea-4bfa-b299-d06e2eb4fd33",
   "metadata": {},
   "source": [
    "Observe: `2` is not the same as `2.0`.  The `2` is the integer number two, while `2.0` is the number two as a floating-point number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c92c923f-9338-432c-a99f-21dc8c05b26f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "int"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e729928f-0259-4ce6-9681-b4f21bf19be8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "float"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(2.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03f6db3b-1359-4079-8836-52b2eff94b2f",
   "metadata": {},
   "source": [
    "The machine precision `eps` is *the smallest positive number that can be added to 1.0 and results in a number different from 1.0*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "743fb64c-1591-4565-a058-13fb147f4da5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000002"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1.0 + 2**(-52)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "073ad3d6-f12b-4f4c-b874-136defa0f01a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1.0 + 2**(-53)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e0cfff8-8f8e-471e-a1e0-87910c200fac",
   "metadata": {},
   "source": [
    "The machine precision for the type `float` in Python is therefore `2**(-52)`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e16823f8-6db5-4627-bca1-60b629d016dc",
   "metadata": {},
   "source": [
    "## 2.1 Integer Division"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57b485e8-0bba-41c5-b2d6-14d7725ca64d",
   "metadata": {},
   "source": [
    "Python does by default exact integer arithmetic, but not rational arithmetic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "777df523-7f1a-4746-9a34-96db05df521f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3333333333333333"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "52f2f9ae-bd1d-4b1a-810a-209bb6f69f38",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1//3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c82edfaa-ea2c-45a8-8d01-8c97b978d802",
   "metadata": {},
   "source": [
    "Mathematically, both answers are wrong!  From a computer science perspective they are valid if we understand the logic of the typing.  The operator `/` on integers return a machine float.  The operator `//` return the quotient of the integer division."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51bc3aa2-44db-49d8-b939-1b6440bdf71b",
   "metadata": {},
   "source": [
    "# 2. Fractions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cde6d14-c041-4ded-9def-c4b4a63d1617",
   "metadata": {},
   "source": [
    "Rational arithmetic is available via the `fractions` module, which must be imported."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "1dfa8a23-55c1-4b6f-b616-86d60693d607",
   "metadata": {},
   "outputs": [],
   "source": [
    "from fractions import Fraction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "fd7c6a8a-d90d-4fdb-9873-b1ea6025d6f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "onethird = Fraction(1, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1ea6ccc-31e8-4acb-b13e-845515b9a2af",
   "metadata": {},
   "source": [
    "A fraction has a numerator and demoninator, which we select as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "322cd729-a885-4e1a-8bbf-8ba558874674",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "onethird.numerator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "4bc0fac2-dc3b-4ec8-b0a1-6d3e30254e10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "onethird.denominator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "4e550c7a-45f9-4016-a0dc-c5ebbf7c475b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(11, 15)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fraction(1, 3) + Fraction(2, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "452ac8c0-8692-4d91-9768-417fd3b6ec79",
   "metadata": {},
   "source": [
    "The `math` module also exports the greatest divisor as the function `gcd()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "62d949cb-ccf5-4176-9cd7-463895e51be3",
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import gcd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "a123e2a4-fbd2-4bbc-b163-945d1c7c6df5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(18, 12)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02e8f854-7af0-4561-b56c-c7414bef5ae2",
   "metadata": {},
   "source": [
    "# 3. ASCII Codes and Strings"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b271b95d-6d1b-4018-81f0-800402f060f6",
   "metadata": {},
   "source": [
    "The function `ord()` returns the ASCII code of a character."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "026e8dbf-d1e7-4ad6-85f7-3614a3f0afe0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "97"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ord('a')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "3b32e6d6-598c-4418-b041-f2fa8eec2695",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "98"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ord('b')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1dace9f-394a-475e-a3b4-d3ffc03c2302",
   "metadata": {},
   "source": [
    "The function `chr()` returns the character with the given ASCII code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "c857a3fb-028a-4e41-952b-c64867dd774e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'b'"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chr(98)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53783708-b235-41c2-965f-418fd9a81ff4",
   "metadata": {},
   "source": [
    "The quotes around `b` indicate that `'b'` is a string."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "603cd702-d37e-485c-8e25-157cc80aa8b1",
   "metadata": {},
   "source": [
    "Every number has a string representation, which can be computed with the `%` operator."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "425d9167-8e7b-4d7a-88c4-5234e3aa7f51",
   "metadata": {},
   "source": [
    "For example, the string representation of `math.pi` with 5 decimals after the `.` in scientific notation is computed as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "c64f0e55-d593-48da-a16e-2015821c7399",
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "4d154282-4b9d-4099-b223-d2126fbcc368",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'3.14159e+00'"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "'%.5e' % pi"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8418876c-21a4-4d70-8244-5b030ffe42e5",
   "metadata": {},
   "source": [
    "Observe the double use of `%`:\n",
    "\n",
    "1. once at the start of the format string; and\n",
    "\n",
    "2. once as the conversion operator before `pi`."
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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