Sage introduction.ipynb

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   "source": [
    "# T-317-CAST\n",
    "# Introduction to Sage\n",
    "\n",
    "In this course we will use the [SageMath&trade;](http://www.sagemath.org) mathematics system based \n",
    "on the Python 2.7.10 programming language. This document aims to give a brief introduction to Sage for the uninitiated."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculator\n",
    "Sage can be used as a basic calculator."
   ]
  },
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    "2+4-8"
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   "source": [
    "5.5+2.8"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage follows the standard order of operations: $(),\\times,\\div,+,-$. Parenthesis change order of calculation."
   ]
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    "5-2*8+6"
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   "cell_type": "markdown",
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   "source": [
    "We can also use some trigonometric functions. These all take their inputs in radians (not degrees)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(90)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(90)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to get a value you may need to wrap something with float."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "scrolled": true
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   "source": [
    "float(sin(90))"
   ]
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   "cell_type": "code",
   "execution_count": 10,
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     "data": {
      "text/plain": [
       "1"
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     "execution_count": 10,
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     "output_type": "execute_result"
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   ],
   "source": [
    "sin(pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
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   "outputs": [
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     "execution_count": 11,
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   ],
   "source": [
    "cos(pi/2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Program control"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Variables\n",
    "You can store variables using the `=` operator and use them in any other expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "k = 12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variables can be reassigned at any point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = 17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "29"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lists are a special kind of variable that can hold many different values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_list = [1,2,3]\n",
    "my_list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to access a list we can index it using square brackets `[]`. List indexes start at 0, you can also use negative indices, the last element is `-1`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "print my_list[0]\n",
    "print my_list[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3, 4]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_list.append(4)\n",
    "my_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_list.pop()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another kind of special object is a dictionary, this maps a 'key' to a 'value'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'apple': 80, 'banana': 100, 'pineapple': 300}"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fruit_costs = {\"banana\":100, \"apple\":80, \"pineapple\":300}\n",
    "fruit_costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fruit_costs['banana']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('pineapple', 300), ('banana', 100), ('apple', 80)]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fruit_costs.items()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['pineapple', 'banana', 'apple']"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fruit_costs.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[300, 100, 80]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fruit_costs.values()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Boolean expressions\n",
    "\n",
    "Sage (and Python) has two boolean values `True` and `False`, as well as the standard boolean operators of `and`, `or` and `not`. \n",
    "\n",
    "These boolean operators short circuit, meaning that they stop evaluating as soon as they are either satisfied or cannot be satisfied.\n",
    "\n",
    "All comparisons return a boolean value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "not True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "False or True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "True and True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "5 > 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An empty list has the truth value of `False`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "not []"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Boolean expressions allow us to form conditional expressions using the `if` statement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is small\n",
      "2\n"
     ]
    }
   ],
   "source": [
    "x = randint(0,10)\n",
    "if x < 3:\n",
    "    print \"x is small\"\n",
    "elif x > 7:\n",
    "    print \"x is big\"\n",
    "else:\n",
    "    print \"x is in the middle\"\n",
    "print x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Loops\n",
    "\n",
    "Loops in Sage are denoted by the `for ... in ...` statement.\n",
    "\n",
    "We can generate a range of numbers by using the `range(stop)` or `range(start, stop[, step])` built-in functions.\n",
    "For larger ranges we can use `xrange` which saves memory.\n",
    "\n",
    "Range always includes the `start`, but does not include the `stop`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "range(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 4, 5, 6, 7]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "range(3,8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "range(10,-1,-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "xrange(9)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xrange(9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The current number is 0\n",
      "The current number is 1\n",
      "The current number is 2\n",
      "The current number is 3\n",
      "The current number is 4\n"
     ]
    }
   ],
   "source": [
    "for x in range(5):\n",
    "    print \"The current number is \" +str(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also loop through arbitrary lists of things. Earlier we saw that the `Dictionary.keys()` function returned a list, let's use that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A pineapple is tasty\n",
      "A banana is tasty\n",
      "A apple is tasty\n"
     ]
    }
   ],
   "source": [
    "for fruit in fruit_costs.keys():\n",
    "    print \"A \" + fruit + \" is tasty\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also loop through the keys of a dictionary without using the `keys` method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A pineapple is tasty\n",
      "A banana is tasty\n",
      "A apple is tasty\n"
     ]
    }
   ],
   "source": [
    "for fruit in fruit_costs:\n",
    "    print \"A \" + fruit + \" is tasty\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The items method of dictionaries allow us to make even easier loops."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A pineapple costs 300kr.\n",
      "A banana costs 100kr.\n",
      "A apple costs 80kr.\n"
     ]
    }
   ],
   "source": [
    "for fruit, cost in fruit_costs.items():\n",
    "    print \"A \" + fruit + \" costs \" + str(cost) + \"kr.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions\n",
    "A function in python is defined using the `def` keyword.\n",
    "\n",
    "Functions can return a value usign the `return` keyword"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def pythagoras_theorem(a, b):\n",
    "    \"\"\"\n",
    "    Calculates the length of the hypotenuse of a right-angle triangle\n",
    "    given the length of the two other sides\n",
    "    \"\"\"\n",
    "    a_square = a*a\n",
    "    b_square = b*b\n",
    "    c_square = a_square + b_square\n",
    "    c = sqrt(c_square)\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pythagoras_theorem(12,9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variables in a function do not exist outside of the function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'c' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-42-2cd6ee2c70b0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'c' is not defined"
     ]
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But functions can change variables that you give to them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "new_list = []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def add_random(base_list):\n",
    "    base_list.append(randint(1,100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "add_random(new_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[61]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that `add_random` did not return any value."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Errors\n",
    "\n",
    "Python is very descriptive when it gives you an error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid syntax (<ipython-input-47-93caf34f137d>, line 1)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-47-93caf34f137d>\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m    def my_function(pa)\u001b[0m\n\u001b[0m                       ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
     ]
    }
   ],
   "source": [
    "def my_function(pa)\n",
    "    print pa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The \"traceback\" tells you where there is an error in your function, here we are missing a colon at the end of the line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def my_looper(inlist):\n",
    "    for item in inlist:\n",
    "        print item + \" banana\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "one banana\n",
      "two banana\n"
     ]
    },
    {
     "ename": "TypeError",
     "evalue": "unsupported operand parent(s) for '+': 'Integer Ring' and '<type 'str'>'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-49-b94e3536066f>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mmy_looper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m\"one\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m\"two\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m<ipython-input-48-c3b08cc7b42e>\u001b[0m in \u001b[0;36mmy_looper\u001b[0;34m(inlist)\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mmy_looper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minlist\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mitem\u001b[0m \u001b[0;32min\u001b[0m \u001b[0minlist\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m         \u001b[0;32mprint\u001b[0m \u001b[0mitem\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\" banana\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Applications/SageMath-7.2.app/Contents/Resources/sage/src/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.RingElement.__add__ (/Applications/SageMath-7.2.app/Contents/Resources/sage/src/build/cythonized/sage/structure/element.c:15651)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1642\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mint\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1643\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mRingElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_add_long\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPyInt_AS_LONG\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1644\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mcoercion_model\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbin_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0madd\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1645\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1646\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mRingElement\u001b[0m \u001b[0m_add_long\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlong\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Applications/SageMath-7.2.app/Contents/Resources/sage/src/sage/structure/coerce.pyx\u001b[0m in \u001b[0;36msage.structure.coerce.CoercionModel_cache_maps.bin_op (/Applications/SageMath-7.2.app/Contents/Resources/sage/src/build/cythonized/sage/structure/coerce.c:9915)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1077\u001b[0m         \u001b[0;31m# We should really include the underlying error.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1078\u001b[0m         \u001b[0;31m# This causes so much headache.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1079\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marith_error_message\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mop\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1080\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1081\u001b[0m     \u001b[0mcpdef\u001b[0m \u001b[0mcanonical_coercion\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: unsupported operand parent(s) for '+': 'Integer Ring' and '<type 'str'>'"
     ]
    }
   ],
   "source": [
    "my_looper([\"one\", \"two\", 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we have a type error on the third line of this function, the traceback tells\n",
    "us that we are not able to add a integer and a string. Sometimes the tracebacks are complicated, the last line usually tells you the real problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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