{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This review contains some representative problems to prepare for the first midterm exam."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider $S = \\sqrt{2}$\n",
    "\n",
    "1. Construct a nearby rational approximation for $S$ where the size of the denominator is not larger than 1000.\n",
    "\n",
    "2. Construct a sequence of 10 nearby rational approximations, where the denominator of the $k$-th approximation is smaller than $10^k$.\n",
    "\n",
    "3. Verify the accuracy of the rational approximations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a function ``whatis`` which stores definitions.\n",
    "\n",
    "Here is a session with this function:\n",
    "\n",
    "1. ``whatis('computer algebra')`` returns\n",
    "\n",
    "   ``call again with the definition for computer algebra``\n",
    "   \n",
    "   \n",
    "2. ``whatis('computer algebra', 'the study of algorithms in symbolic computation')`` \n",
    "\n",
    "   stores the definition for ``'computer algebra'``\n",
    "   \n",
    "   \n",
    "3. ``whatis('computer algebra')`` returns\n",
    "\n",
    "   ``the study of algorithms in symbolic computation``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the polynomial $p = x^4 + x^2 + 1$\n",
    "over the field ${\\mathbb Z}_2$, the field of bits, 0 and 1.\n",
    "   \n",
    "   1. Is $p$ irreducible?  If not, then what are its factors?\n",
    "   \n",
    "   2. Extend the field ${\\mathbb Z}_2$ with sufficiently many formal roots\n",
    "      so $p$ factors as a product of linear factors.\n",
    "      \n",
    "      Write the factorization of $p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a Python function ``python_sum`` which takes on input a positive integer $n$ and which returns\n",
    "the floating-point value of\n",
    "\n",
    "$$\n",
    "    \\frac{\\pi}{n} \\sum_{i=1}^{n-1} \\cos\\left( - \\frac{\\pi}{2} + i \\frac{\\pi}{n} \\right).\n",
    "$$\n",
    "\n",
    "Write a more efficient version ``numpy_sum`` using vectorization.\n",
    "\n",
    "Time the two versions to illustrate the efficiency."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Explain what is the fully factored normal form of rational polynomials.\n",
    "\n",
    "Why does this automatic simplification not happen automatically?\n",
    "\n",
    "Illustrate with a good example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Type \n",
    "``reset(); x, y = var('x,y'); q = (x^2 - y)/(y^2 - x)``\n",
    "and draw the expression tree of ``q.``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rewrite\n",
    "\n",
    "$$\n",
    "   x + \\frac{(x - y)^5}{(x + y)^5}\n",
    "$$\n",
    "\n",
    "into\n",
    "\n",
    "$$\n",
    "   \\frac{(x + y)^5 \\cdot x + (x - y)^5}{(x + y)^5}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# question 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider $p = -y^2 z^3 - y z^4 + 3 x y^3 + x^3 + z^2$\n",
    "as a polynomial with integer coefficients in the variables $x$, $y$, and $z$.\n",
    "   \n",
    "The monomials of $p$ are sorted in the degree lexicographical order.\n",
    "   \n",
    "Convert $p$ (*without retyping $p$!*) into the pure lexicographical term order."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.3",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}