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   "source": [
    "Quiz 1 MCS 471, on Friday 26 August 2021, due 10:50am."
   ]
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   "source": [
    "Consider a floating-point system with five places in the fraction\n",
    "and base 10, with exponents ranging between $-10$ and $+10$.\n",
    "\n",
    "1. What is the machine precision in this number system?\n",
    "\n",
    "2. Compute $386.04 - 6.8907$ in this number system.  Show all steps in your calculation."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## answer to question 1"
   ]
  },
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   "metadata": {},
   "source": [
    "The machine precision is $10^{-5}$."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## answer to question 2"
   ]
  },
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   "source": [
    "We proceed first by representing the numbers.\n",
    "\n",
    "1. $386.04 = +.38604 \\times 10^{+3}$ and $6.8907 = +.68907 \\times 10^{+1}$\n",
    "   \n",
    "2. Before the subtraction, we denormalize the smallest number.\n",
    "   \n",
    "   $$\n",
    "          \\begin{array}{clcl}\n",
    "              & +.38604   & \\times & 10^{+3} \\\\\n",
    "            - & +.0068907 & \\times & 10^{+3} \\\\ \\hline\n",
    "              & +.3791493 & \\times & 10^{+3}\n",
    "          \\end{array}\n",
    "   $$\n",
    "      \n",
    "3. The final result is rounded: $+.3791493 \\times 10^{+3} = +.37915 \\times 10^{+3} = 379.15$."
   ]
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   "source": [
    "The cell below verifies the subtraction of the fractions."
   ]
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       "3791493"
      ]
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     "execution_count": 1,
     "metadata": {},
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   "source": [
    "3860400 - 68907"
   ]
  }
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