{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4d757b3c",
   "metadata": {},
   "source": [
    "# July 21, 23 2021\n",
    "\n",
    "Lets review some of what we've studied, while also introducing some new stuff."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "eaaf6d5c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "43ce906d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['__abs__', '__add__', '__all__', '__and__', '__builtins__', '__cached__', '__concat__', '__contains__', '__delitem__', '__doc__', '__eq__', '__file__', '__floordiv__', '__ge__', '__getitem__', '__gt__', '__iadd__', '__iand__', '__iconcat__', '__ifloordiv__', '__ilshift__', '__imatmul__', '__imod__', '__imul__', '__index__', '__inv__', '__invert__', '__ior__', '__ipow__', '__irshift__', '__isub__', '__itruediv__', '__ixor__', '__le__', '__loader__', '__lshift__', '__lt__', '__matmul__', '__mod__', '__mul__', '__name__', '__ne__', '__neg__', '__not__', '__or__', '__package__', '__pos__', '__pow__', '__rshift__', '__setitem__', '__spec__', '__sub__', '__truediv__', '__xor__', '_abs', 'abs', 'add', 'and_', 'attrgetter', 'concat', 'contains', 'countOf', 'delitem', 'eq', 'floordiv', 'ge', 'getitem', 'gt', 'iadd', 'iand', 'iconcat', 'ifloordiv', 'ilshift', 'imatmul', 'imod', 'imul', 'index', 'indexOf', 'inv', 'invert', 'ior', 'ipow', 'irshift', 'is_', 'is_not', 'isub', 'itemgetter', 'itruediv', 'ixor', 'le', 'length_hint', 'lshift', 'lt', 'matmul', 'methodcaller', 'mod', 'mul', 'ne', 'neg', 'not_', 'or_', 'pos', 'pow', 'rshift', 'setitem', 'sub', 'truediv', 'truth', 'xor']\n"
     ]
    }
   ],
   "source": [
    "print(dir(operator))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d81f6825",
   "metadata": {},
   "outputs": [],
   "source": [
    "from operator import mul, add"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "685ca9e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mul(3, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "232e869f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "08ef5d99",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "155367767558744989035099915780681312"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mul(52938745927349857293,  2934859238485984)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a8dcbbd0",
   "metadata": {},
   "outputs": [],
   "source": [
    "from functools import reduce"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8c4154b5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Everything went smoothly captain\n",
      "Close a file, log a message -- even if still exceptions\n",
      "We only get this far if any exceptions were handled\n"
     ]
    }
   ],
   "source": [
    "try: \n",
    "    assert 23**560 % 561 == 1 # perfectly OK\n",
    "except TypeError:\n",
    "    print(\"That's a type error\")\n",
    "except AssertionError:\n",
    "    print(\"I have a problem with that.\")\n",
    "    raise # where you catch an exception, and re-raise it\n",
    "else:\n",
    "    print(\"Everything went smoothly captain\")\n",
    "finally:\n",
    "    print(\"Close a file, log a message -- even if still exceptions\")\n",
    "    \n",
    "print(\"We only get this far if any exceptions were handled\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b718d427",
   "metadata": {},
   "outputs": [],
   "source": [
    "from primes import factors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c69c4f1f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 3, 11, 17)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factors(561)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "da2ade15",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reduce(add, (4, 3, 2, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "eca4cdb7",
   "metadata": {},
   "outputs": [],
   "source": [
    "def factorial(n):\n",
    "    if n==0:\n",
    "        return 1\n",
    "    return reduce(mul, range(1, n+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ca44af8c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "52de186a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import primes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "361a9f25",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['PrimeNumbers',\n",
       " '__builtins__',\n",
       " '__cached__',\n",
       " '__doc__',\n",
       " '__file__',\n",
       " '__loader__',\n",
       " '__name__',\n",
       " '__package__',\n",
       " '__path__',\n",
       " '__spec__',\n",
       " 'all_factors',\n",
       " 'eratosthenes',\n",
       " 'euler_test',\n",
       " 'factors',\n",
       " 'invmod',\n",
       " 'isprime',\n",
       " 'primes_gen',\n",
       " 'primesplay',\n",
       " 'xgcd']"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(primes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "2470c2c0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eliminating multiples of: 2\n",
      "Eliminating multiples of: 3\n",
      "Eliminating multiples of: 5\n",
      "Eliminating multiples of: 7\n",
      "Eliminating multiples of: 11\n",
      "Eliminating multiples of: 13\n",
      "Eliminating multiples of: 17\n",
      "Eliminating multiples of: 19\n",
      "Eliminating multiples of: 23\n",
      "Eliminating multiples of: 29\n",
      "Eliminating multiples of: 31\n",
      "[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]\n"
     ]
    }
   ],
   "source": [
    "print(primes.eratosthenes(1000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "133b884d",
   "metadata": {},
   "source": [
    "## A Tour with Stops\n",
    "\n",
    "Lets check out some other exhibits regarding the high school math topics at hand.\n",
    "\n",
    "* [Number Theory](https://nbviewer.jupyter.org/github/4dsolutions/School_of_Tomorrow/blob/master/NumberTheory.ipynb) (School of Tomorrow)\n",
    "* [Group Theory](https://github.com/4dsolutions/School_of_Tomorrow/blob/master/GroupTheory.ipynb) (School of Tomorrow)\n",
    "* [Numeracy Series](http://4dsolutions.net/ocn/numeracy0.html) (Oregon Curriculum Network)\n",
    "* [Tabular Python](TabularPython.ipynb) (this school)\n",
    "* [Trigonometry](Trig.ipynb) (this school)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "9911b2b2",
   "metadata": {},
   "outputs": [],
   "source": [
    "class Mod:\n",
    "    \n",
    "    _modulus = 12\n",
    "    \n",
    "    def __init__(self, n):\n",
    "        self.value = n % self._modulus\n",
    "        \n",
    "    def __eq__(self, other):\n",
    "        return self.value == other.value % self._modulus\n",
    "    \n",
    "    def __repr__(self):\n",
    "        return \"({} mod {})\".format(self.value, self._modulus)\n",
    "    \n",
    "    def __add__(self, other):\n",
    "        return type(self)(self.value + other.value)\n",
    "    \n",
    "    def __mul__(self, other):\n",
    "        return type(self)(self.value * other.value)\n",
    "    \n",
    "    def __pow__(self, n):\n",
    "        return type(self)(pow(self.value, n, self._modulus))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "c42f34a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(3 * 4) % 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "9c620323",
   "metadata": {},
   "outputs": [],
   "source": [
    "m1 = Mod(1); m2 = Mod(2); m3 = Mod(3); m4 = Mod(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "bd4e0a2f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3 mod 12)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b7a4fab0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0 mod 12)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m3.__mul__(m4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "9c56f84a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3 mod 12)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m3 * m3 * m3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "fe6c6fff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0 mod 12)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m4 * m3 * m3 * m2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "b0339c4c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(4 * 3 * 2 * 1) % 12"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "790fe0d1",
   "metadata": {},
   "source": [
    "## Introducing Group Theory by Multiplying Totatives of N Modulo N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "1940a544",
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import gcd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "52dcd75f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def totient(n):\n",
    "    totatives = [x for x in range(1, n) if gcd(x, n)==1]\n",
    "    return len(totatives), totatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "acd94d63",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4, [1, 5, 7, 11])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "totient(12)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51a8a21a",
   "metadata": {},
   "source": [
    "Getting totatives of 12 as Mod type numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "7588bf2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "m1 = Mod(1); m5 = Mod(5); m7 = Mod(7); m11 = Mod(11)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e512d7ae",
   "metadata": {},
   "source": [
    "#### Showing Closure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "62ac6d46",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(7 mod 12)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m1 * m7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "de7dc437",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(11 mod 12)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m11 * m1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77284311",
   "metadata": {},
   "source": [
    "#### Showing inverses:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "b3157102",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 mod 12)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m5 * m5  # m5 is its own inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "d3b05292",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(7 mod 12)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m7 * m11 * m11 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "4c3603ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 mod 12)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m11 * m11   # m11 is its own inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "0b8f3f0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 mod 12)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m5 * m5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "e607758b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 mod 12)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m7 * m7   # m7 is its own inverse"
   ]
  }
 ],
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