sage_tutorial.ipynb

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    "# Sage\n",
    "\n",
    "Sage (also known as SageMath) is a general purpose computer algebra system written on top of the python language. In Mathematica, Magma, and Maple, one writes code in the mathematica-language, the magma-language, or the maple-language. Sage is python.\n",
    "\n",
    "But no python background is necessary for the rest of today's guided tutorial. The purpose of today's tutorial is to give an indication about how one really *uses* sage, and what might be available to you if you want to try it out.\n",
    "\n",
    "I will spoil the surprise by telling you upfront the two main points I hope you'll take away from this tutorial.\n",
    "\n",
    "1. With tab-completion and documentation, you can do many things in sage without ever having done them before.\n",
    "2. The ecosystem of libraries and functionality available in sage is tremendous, and (usually) pretty easy to use."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lightning Preview\n",
    "\n",
    "Let's first get a small feel for sage by seeing some standard operations and what typical use looks like through a series of trivial, mostly unconnected examples."
   ]
  },
  {
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    {
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      "text/plain": [
       "5"
      ]
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     "execution_count": 1,
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     "output_type": "execute_result"
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   ],
   "source": [
    "# Fundamental manipulations work as you hope\n",
    "\n",
    "2+3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "You can also subtract, multiply, divide, exponentiate...\n",
    "\n",
    "    >>> 3-2\n",
    "    1\n",
    "    >>> 2*3\n",
    "    6\n",
    "    >>> 2^3\n",
    "    8\n",
    "    >>> 2**3 # (also exponentiation)\n",
    "    8\n",
    "    \n",
    "There is an order of operations, but these things work pretty much as you want them to work. You might try out several different operations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "Sage includes a lot of functionality, too. For instance,"
   ]
  },
  {
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   "execution_count": 2,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1 * 2^4 * 3^2 * 7"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(-1008)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(2, 4), (3, 2), (7, 1)]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(factor(1008))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the background, Sage is actually calling on pari/GP to do this factorization. Sage bundles lots of free and open source math software within it (which is why it's so large), and provides a common access point. The great thing here is that you can often use sage without needing to know much pari/GP (or other software)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage knows many functions and constants, and these are accessible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "e^2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage tries to internally keep expressions in exact form. To present approximations, use `N()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.38905609893065"
      ]
     },
     "execution_count": 6,
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     "output_type": "execute_result"
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   ],
   "source": [
    "N(exp(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14159265358979"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can ask for a number of digits in the approximation by giving a `digits` keyword to `N()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14159265358979323846264338327950288419716939937510582097494"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(pi, digits=60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are some benefits to having smart representations. In many computer algebra systems, computing (sqrt(2))^2 doesn't give you back 2, due to finite floating point arithmetic. But in sage, this problem is sometimes avoided."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(2)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(2)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, there are examples where floating point arithmetic gets in the way.\n",
    "\n",
    "In sage/python, integers have unlimited digit length. Real precision arithmetic is a bit more complicated, which is why sage tries to keep exact representations internally. We don't go into tracking digits of precision in sage, but it is usually possible to prescribe levels of precision."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage is written in python. You can use python functions in sage. [You can also import python libraries in sage, which extends sage's reach significantly]. The `range` function in python counts up to a given number, starting at 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "range(16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 0  1  2  3]\n",
       "[ 4  5  6  7]\n",
       "[ 8  9 10 11]\n",
       "[12 13 14 15]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = matrix(4,4, range(16))\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-5 -4 -3 -2]\n",
       "[-1  0  1  2]\n",
       "[ 3  4  5  6]\n",
       "[ 7  8  9 10]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = matrix(4,4, range(-5, 11))\n",
    "B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage can be smart about using the same operators in different contexts. (i.e. sage is very polymorphic). Multiplying, adding, subtracting, and dividing matrices works as expected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 26  32  38  44]\n",
       "[ 42  64  86 108]\n",
       "[ 58  96 134 172]\n",
       "[ 74 128 182 236]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A*B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are two sorts of ways that functions are called in sage. For some functions, you create the object (in this case, the matrix A), type a dot `.`, and then call the function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^4 - 30*x^3 - 80*x^2"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.charpoly()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are some top-level functions as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2 * (x^2 - 30*x - 80)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(A.charpoly())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sometimes you start with an object, such as a matrix, and you wonder what you can do with it. Sage has very good tab-completion and introspection in its notebook interface.\n",
    "\n",
    "Try typing \n",
    "\n",
    "    A.\n",
    "    \n",
    "and hit `<Tab>`. Sage should display a list of things it thinks it can do to the matrix A.\n",
    "\n",
    "### Warning\n",
    "\n",
    "Note that on CoCalc or external servers, tab completion sometimes has a small delay."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "scrolled": false
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   "outputs": [],
   "source": [
    "A."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Some of these are more meaningful than others, but you have a list of options. If you want to find out what an option does, like `A.eigenvalues()`, then type \n",
    "\n",
    "    A.eigenvalues?\n",
    "    \n",
    "and hit enter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "A.eigenvalues?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 0, -2.464249196572981?, 32.46424919657298?]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.eigenvalues()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "If you're really curious about what's going on, you can type\n",
    "\n",
    "    A.eigenvalues??\n",
    "    \n",
    "which will also show you the implementation of that functionality. (You usually don't need this)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "A.eigenvalues??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "There is a lot of domain-specific functionality within sage as well. We won't dwell too much on any particular functionality in this tutorial, but for example sage knows about elliptic curves."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = EllipticCurve([1,2,3,4,5])\n",
    "E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Tab complete me to see what's available\n",
    "E."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10351"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.conductor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.rank()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sage knows about complex numbers as well. Use `i` or `I` to mean a $\\sqrt{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*I + 1)*pi - (I - 2)*sqrt(5)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1+2*I) * (pi - sqrt(5)*I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "c = 1/(sqrt(3)*I + 3/4 + sqrt(29)*2/3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sage tries to keep computations in exact form. So `c` is stored with perfect representations of square roots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12/(8*sqrt(29) + 12*I*sqrt(3) + 9)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "But we can have sage give numerical estimates of objects by calling `N()` on them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.198754342458965 - 0.0793188720015053*I"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.19875 - 0.079319*I"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(c, 20) # Keep 20 \"bits\" of information"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "As one more general topic before we jump into a few deeper examples, sage is also very good at representing objects in latex. Use `latex(<object>)` to give a latex representation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\\frac{12}{8 \\, \\sqrt{29} + 12 i \\, \\sqrt{3} + 9}"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "y^2 + x y + 3 y = x^{3} + 2 x^{2} + 4 x + 5 "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 2 & 3 \\\\\n",
       "4 & 5 & 6 & 7 \\\\\n",
       "8 & 9 & 10 & 11 \\\\\n",
       "12 & 13 & 14 & 15\n",
       "\\end{array}\\right)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can have sage print the LaTeX version in the notebook by using `pretty_print`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 2 & 3 \\\\\n",
       "4 & 5 & 6 & 7 \\\\\n",
       "8 & 9 & 10 & 11 \\\\\n",
       "12 & 13 & 14 & 15\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[ 0  1  2  3]\n",
       "[ 4  5  6  7]\n",
       "[ 8  9 10 11]\n",
       "[12 13 14 15]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pretty_print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Basic Algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(),\n",
       " (1,6)(2,5)(3,4),\n",
       " (1,2,3,4,5,6),\n",
       " (1,5)(2,4),\n",
       " (2,6)(3,5),\n",
       " (1,3,5)(2,4,6),\n",
       " (1,4)(2,3)(5,6),\n",
       " (1,6,5,4,3,2),\n",
       " (1,4)(2,5)(3,6),\n",
       " (1,2)(3,6)(4,5),\n",
       " (1,5,3)(2,6,4),\n",
       " (1,3)(4,6)]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H = DihedralGroup(6)\n",
    "H.list()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1,6)(2,5)(3,4)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = H[1]\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1,2,3,4,5,6)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = H[2]\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2,6)(3,5)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a*b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1,6)(2,5)(3,4)\n",
      "(1,5)(2,4)\n",
      "(2,6)(3,5)\n",
      "(1,4)(2,3)(5,6)\n",
      "(1,4)(2,5)(3,6)\n",
      "(1,2)(3,6)(4,5)\n",
      "(1,3)(4,6)\n"
     ]
    }
   ],
   "source": [
    "for elem in H:\n",
    "    if elem.order() == 2:\n",
    "        print elem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1,6)(2,5)(3,4),\n",
       " (1,5)(2,4),\n",
       " (2,6)(3,5),\n",
       " (1,4)(2,3)(5,6),\n",
       " (1,4)(2,5)(3,6),\n",
       " (1,2)(3,6)(4,5),\n",
       " (1,3)(4,6)]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Or, in the \"pythonic\" way\n",
    "elements_of_order_2 = [elem for elem in H if elem.order() == 2]\n",
    "elements_of_order_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1,3)(4,6)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand_elem = H.random_element()\n",
    "rand_elem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Subgroup of (Dihedral group of order 12 as a permutation group) generated by [(1,3)(4,6)]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_sub = H.subgroup([rand_elem])\n",
    "G_sub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1,4)(2,3)(5,6)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Explicitly using elements of a group\n",
    "H(\"(1,2,3,4,5,6)\") * H(\"(1,5)(2,4)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "The real purpose of these exercises are to show you that it's often possible to use tab-completion to quickly find out what is and isn't possible to do within sage.\n",
    "\n",
    "1. What things does sage know about this subgroup? Can you find the cardinality of the subgroup? (Note that the subgroup is generated by a random element, and your subgroup might be different than your neighbor's). Can you list all subgroups of the dihedral group in sage?\n",
    "\n",
    "2. Sage knows other groups as well. Create a symmetric group on 5 elements. What does sage know about that? Can you verify that S5 isn't simple? Create some cyclic groups?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Changing the Field\n",
    "\n",
    "It's pretty easy to work over different fields in sage as well. I show a few examples of this"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# It may be necessary to use `reset('x')` if x has otherwise been defined\n",
    "K.<alpha> = NumberField(x**3 - 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Number Field in alpha with defining polynomial x^3 - 5"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "alpha"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*alpha^2 + 3*alpha + 6"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(alpha+1)**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "GF?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F7 = GF(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2/5"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 2/5\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "eqn = x**3 + sqrt(2)*x + 5 == 0\n",
    "a = solve(eqn, x)[0].rhs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/2*(I*sqrt(3) + 1)*(1/6*sqrt(8/3*sqrt(2) + 225) - 5/2)^(1/3) + 1/6*sqrt(2)*(-I*sqrt(3) + 1)/(1/6*sqrt(8/3*sqrt(2) + 225) - 5/2)^(1/3)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-\\frac{1}{2} \\, {\\left(i \\, \\sqrt{3} + 1\\right)} {\\left(\\frac{1}{6} \\, \\sqrt{\\frac{8}{3} \\, \\sqrt{2} + 225} - \\frac{5}{2}\\right)}^{\\frac{1}{3}} + \\frac{\\sqrt{2} {\\left(-i \\, \\sqrt{3} + 1\\right)}}{6 \\, {\\left(\\frac{1}{6} \\, \\sqrt{\\frac{8}{3} \\, \\sqrt{2} + 225} - \\frac{5}{2}\\right)}^{\\frac{1}{3}}}"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(i \\, \\sqrt{3} + 1\\right)} {\\left(\\frac{1}{6} \\, \\sqrt{\\frac{8}{3} \\, \\sqrt{2} + 225} - \\frac{5}{2}\\right)}^{\\frac{1}{3}} + \\frac{\\sqrt{2} {\\left(-i \\, \\sqrt{3} + 1\\right)}}{6 \\, {\\left(\\frac{1}{6} \\, \\sqrt{\\frac{8}{3} \\, \\sqrt{2} + 225} - \\frac{5}{2}\\right)}^{\\frac{1}{3}}}</script></html>"
      ],
      "text/plain": [
       "-1/2*(I*sqrt(3) + 1)*(1/6*sqrt(8/3*sqrt(2) + 225) - 5/2)^(1/3) + 1/6*sqrt(2)*(-I*sqrt(3) + 1)/(1/6*sqrt(8/3*sqrt(2) + 225) - 5/2)^(1/3)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pretty_print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Rational Field"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Also RR, CC\n",
    "QQ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "K.<b> = QQ[a]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Number Field in a with defining polynomial x^6 + 10*x^3 - 2*x^2 + 25"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^6 + 10*x^3 - 2*x^2 + 25"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.minpoly()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.class_number()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Sage tries to keep the same syntax even across different applications. Above, we factored a few integers. But we may also try to factor over a number field. You can factor 2 over the Gaussian integers by:\n",
    "\n",
    "1. Create the Gaussian integers. The constructor CC[I] works.\n",
    "2. Get the Gaussian integer 2 (which is programmatically different than the typical integer 2), by something like `CC[I](2)`.\n",
    "3. `factor` that 2.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some Calculus And Symbolic Manipulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "x + y\n"
     ]
    }
   ],
   "source": [
    "# Let's declare that we want x and y to mean symbolic variables\n",
    "x = 1\n",
    "y = 2\n",
    "print(x+y)\n",
    "\n",
    "reset('x')\n",
    "reset('y')\n",
    "var('x')\n",
    "var('y')\n",
    "\n",
    "print(x+y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x == -2, x == -1]"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x^2 + 3*x + 2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x == -1/2*y - 1/2*sqrt(y^2 - 8), x == -1/2*y + 1/2*sqrt(y^2 - 8)]"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x^2 + y*x + 2 == 0, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[p == 8, q == -26, x == (5/2), y == 1], [p == 8, q == 6, x == (-3/2), y == 1]]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Nonlinear systems with complicated solutions can be solved as well\n",
    "var('p,q')\n",
    "eq1 = p+1==9\n",
    "eq2 = q*y+p*x==-6\n",
    "eq3 = q*y**2+p*x**2==24\n",
    "s = solve([eq1, eq2, eq3, y==1], p,q,x,y)\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[p == 8, q == -26, x == (5/2), y == 1]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\\left[p = 8, q = \\left(-26\\right), x = \\left(\\frac{5}{2}\\right), y = 1\\right]"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(s[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\left[p = 8, q = \\left(-26\\right), x = \\left(\\frac{5}{2}\\right), y = 1\\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2 + 2*x + 1"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can also do some symbolic calculus\n",
    "f = x**2 + 2*x + 1\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*x + 2"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/3*x^3 + x^2 + x"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integral(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7/3"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = integral(f, x)\n",
    "F(x=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*x^2*cos(x^3)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x**3), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "81*x^8*sin(x^3) - 324*x^5*cos(x^3) - 180*x^2*sin(x^3)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the 4th derivative\n",
    "diff(sin(x**3), x, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/2*I*Ei(I*x) + 1/2*I*Ei(-I*x)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can try to foil sage by giving it a hard integral\n",
    "integral(sin(x)/x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(x^2)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sin(x**2)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/362880*x^18 - 1/5040*x^14 + 1/120*x^10 - 1/6*x^6 + x^2"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# And sage can give Taylor expansions\n",
    "f.taylor(x, 0, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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hqF607VLfDG5D0qwsyzCfzxGGITcQJnKMNHe1NiVVFYtNMZvstJ2mKdI03ei91IKkg4MD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\nWTjl1hzZF0oKur/hG75Be0hEtIYkSRBFEa5du6Y9lFp0u11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dpmkIhUKYOXPmpLtipL1I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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x,y)=y^2+1-x^3-x\n",
    "contour_plot(f, (x,-pi,pi), (y,-pi,pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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27dy5E/X19WhsbITneWhvbx/yGe3t7Zg+fXrZfocPomSiCgA33wz8/d+7uP/+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\n19f7+h2yL7yBVIqlFLKfuomiwLIsxOOhnUIQarq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pigJd16GqKlIpWg2SmVcrqiN33df7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MtlaQ59MU0TQ0NDlc3IZFg9/n2Odo21a9figgsuOOr3YYxh48aNmDt3buVjCxYsqPzvD3/4w1i4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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "contour_plot(f, (x,-pi,pi), (y,-pi,pi), colorbar=True, labels=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HpCZNpBEjTCcBABQTShJ8ra7OXYZkxAiJhyEBAPnEArfwtZdflj77THrsMdNJAADFhitJ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\ngg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Implicit plots\n",
    "f(x,y) = -x**3 + y**2 - y + x + 1\n",
    "implicit_plot(f(x,y)==0,(x,0,2*pi),(y,-pi,pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "1. Experiment with the above examples by trying out different functions and plots.\n",
    "\n",
    "2. Sage can do partial fractions for you as well. To do this, you first define your function you want to split up. Suppose you call it `f`. Then you use `f`.partial_fraction(x). Try this out\n",
    "\n",
    "3. Sage can also create 3d plots. Create one. Start by looking at the documentation for `plot3d`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of the various math software, sage+python provides my preferred plotting environment. I have used sage to create plots for notes, lectures, classes, experimentation, and publications. You can quickly create good-looking plots. For example, I used sage/python extensively in creating [this note for my students](http://davidlowryduda.com/an-intuitive-overview-of-taylor-series/) on Taylor Series (which is a classic \"hard topic\" that students have lots of questions about, at least in the US universities I'm familiar with. To this day, about 1/6 of the traffic to my website is to see that page).\n",
    "\n",
    "As a non-trivial example, I present the following interactive plot. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "635f914cd7924df9a9150f0bc1c61d79"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact\n",
    "def g(f=sin(x), c=0, n=(1..30),\n",
    "      xinterval=range_slider(-10, 10, 1, default=(-8,8), label=\"x-interval\"),\n",
    "      yinterval=range_slider(-50, 50, 1, default=(-3,3), label=\"y-interval\")):\n",
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    "    html('$f(x)\\;=\\;%s$'%latex(f))\n",
    "    html('$P_{%s}(x)\\;=\\;%s+R_{%s}(x)$'%(degree,latex(ft),degree))"
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    "### Additional Resources and Comments\n",
    "\n",
    "There are a variety of tutorials and resources for learning more about sage. I list several here.\n",
    "\n",
    "1. Sage provides some tutorials of its own. These include its [Guided Tour](https://doc.sagemath.org/html/en/tutorial/tour.html) and the [Standard Sage Tutorial](https://doc.sagemath.org/html/en/tutorial/index.html). The Standard Sage Tutorial is designed to take 2-4 hours to work through, and afterwards you should have a pretty good sense of the sage environment.\n",
    "2. [PREP Tutorials](http://doc.sagemath.org/html/en/prep/index.html) are a set of tutorials created in a program sponsored by the Mathematics Association of America, aimed at working with university students with sage. These tutorials are designed for people both new to sage and to programming.\n",
    "\n",
    "See also the [main sage website](http://www.sagemath.org/).\n",
    "\n",
    "For questions about specific things in sage, you can ask about these on [StackOverflow](https://stackoverflow.com/questions/tagged/sage) or [AskSage](https://ask.sagemath.org/questions/). You might also consider the sage-support or sage-edu [mailing lists](http://www.sagemath.org/help-groups.html).\n",
    "\n",
    "It isn't necessary to know python to use sage, but a heavy sage user will benefit significantly from learning some python. Conversely, sage is very easy to use if you know python."
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