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   "source": [
    "Lecture 2 of Summer 2023 MCS 320 Introduction to Symbolic Computation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computing with Symbols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SageMath is a computer algebra system, capable of working with symbols.\n",
    "For example, the transcendental constant $\\pi$ is known to SageMath as ``pi``.\n",
    "Applying the ``sin()`` function to $\\pi$ shows the true value of $\\sin(\\pi)$."
   ]
  },
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   "source": [
    "sin(pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The language of SageMath is based on Python.  Python is a dynamically typed language.  The type of each object determines the opererations that can be performed on the object.\n",
    "We query the type of an object with ``type()``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.symbolic.expression.Expression'>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that SageMath is case sensitive ``pi`` $\\not=$ ``Pi``."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Approximations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compute a numerical approximation for $\\pi$, with the application of the method ``n()`` on the object ``pi``, which is of type symbolic expression.\n",
    "In the argument of ``n()`` we can specify the number of decimal places with ``digits=``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.141592654\n",
      "<class 'sage.rings.real_mpfr.RealNumber'>\n"
     ]
    }
   ],
   "source": [
    "Pi10 = pi.n(digits=10)\n",
    "print(Pi10)\n",
    "print(type(Pi10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number we obtained is not a Python float, but of type ``RealNumber`` of the multiprecision  software GNU MPFR library.  Let us see by how much ``Pi10`` differs from $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pi - 3.141592654\n",
      "<class 'sage.symbolic.expression.Expression'>\n"
     ]
    }
   ],
   "source": [
    "delta = pi - Pi10\n",
    "print(delta)\n",
    "print(type(delta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The answer above is exact and correct, but it is not so helpful,\n",
    "we were hoping to see a number to verify that our ``Pi10`` approximates $\\pi$ really with 10 decimal places accuracy.  Let us compute a numerical approximation of ``delta``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0000000000\n",
      "3.6379788071e-12\n"
     ]
    }
   ],
   "source": [
    "print(delta.n(digits=10))\n",
    "print(delta.n(digits=11))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the 10-digit accurate rational approximation for pi: 833719/265381\n",
      "<class 'sage.rings.rational.Rational'>\n"
     ]
    }
   ],
   "source": [
    "Pi18q = QQ(Pi10)\n",
    "print('the 10-digit accurate rational approximation for pi:', Pi18q)\n",
    "print(type(Pi18q))"
   ]
  }
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