Symbols-Fa22.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "2b847051",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "46c3fc97",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "33d6de53",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.symbols('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1f6e9e0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "79c463ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "[a , b, c] = sympy.symbols('a b c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "af6c5a59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a$"
      ],
      "text/plain": [
       "a"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c6a89876",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a + b$"
      ],
      "text/plain": [
       "a + b"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "8d428b55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a + b + 30$"
      ],
      "text/plain": [
       "a + b + 30"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b + 30 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "a600d244",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4}$"
      ],
      "text/plain": [
       "a**4"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a ** 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "4ce759ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-sqrt(10)*I, sqrt(10)*I]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(a ** 2  + 10 , a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "02eaae3d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "9eef274a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x$"
      ],
      "text/plain": [
       "2*x"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "26792607",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * x == 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4f111ee6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * x == x + x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "21a4d138",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(2 * x + 10 == 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "39a0a7c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * x + 10 == 10 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "508adc6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-5*sqrt(2), 5*sqrt(2)]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.Eq(2 * x**2 + 10,110))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "51e6b2b4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "7cd87904",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.sin(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "f95498a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "dd38062c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-I*log(1 + sqrt(2)),\n",
       " I*log(1 + sqrt(2)),\n",
       " pi - I*log(1 + sqrt(2)),\n",
       " pi + I*log(1 + sqrt(2))]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x)**2+1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "a1a1aa27",
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = sympy.S('1/10')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "dc5048c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{10}$"
      ],
      "text/plain": [
       "1/10"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "6ed4992a",
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = sympy.S('x / 20')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "32a92ef8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x}{20} + \\frac{1}{10}$"
      ],
      "text/plain": [
       "x/20 + 1/10"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1 + f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "0389846f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x + 2}{20}$"
      ],
      "text/plain": [
       "(x + 2)/20"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.together(f1 + f2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "25335039",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\left(x + 1\\right) + x^{2}$"
      ],
      "text/plain": [
       "a*(x + 1) + x**2"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x ** 2 + a * x + a,a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2f7e474a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fcf28400040>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(sympy.log(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "999ce975",
   "metadata": {},
   "outputs": [],
   "source": [
    "y  = sympy.S('y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "4853c100",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: 55/14, y: 5/7}"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve( [ sympy.Eq(2 * x + 3 * y, 10), sympy.Eq(4 * x - 8 * y , 10)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "90813120",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{x: 5*y/3}]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve( sympy.Eq( 2 * x + 3 * y , 5 * x - 2 * y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "303a5d50",
   "metadata": {},
   "outputs": [],
   "source": [
    "def howManyZeros(eqn):\n",
    "    return len(sympy.solve(sympy.S(eqn)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "04c29583",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x - 10$"
      ],
      "text/plain": [
       "x**2 + 2*x - 10"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.S('2 * x + x ^ 2 - 10')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "887759e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1 + sqrt(11), -sqrt(11) - 1]"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.S('2 * x + x ^ 2 - 10'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "50365311",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "howManyZeros('x ** 2 - 10 * x + 5')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5c35bac0",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = [1,3,5,7,8,9,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "81e8a2b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[9]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[5::2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b6ffafe4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 7, 8]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[2:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "096d8996",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(119, 10)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num = 10\n",
    "def foo(x):\n",
    "    #global num\n",
    "    num = 20\n",
    "    return x + num\n",
    "foo(99), num"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f8b53113",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(119, 20)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num = 10\n",
    "def foo(x):\n",
    "    global num \n",
    "    num = 20\n",
    "    return x + num\n",
    "foo(99), num"
   ]
  },
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