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   "cell_type": "markdown",
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   "source": [
    "[Home](Home.ipynb)\n",
    "\n",
    "# Complex Numbers\n",
    "\n",
    "The invention called the complex plane lets us dial in any place on the unit circle, by raising $e$ to a power.  \n",
    "\n",
    "Setting $\\theta$ equal to $\\pi$ in the equation below, leaves us half way around the circle, at (x, y) = (-1, 0).\n",
    "\n",
    "$$\n",
    "e^{ \\pm i\\theta } = \\cos \\theta \\pm i\\sin \\theta\n",
    "$$\n",
    "\n",
    "$$\n",
    "e^{ i\\pi} = -1\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "090870a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "from cmath import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9ef350df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.718281828459045"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "312ab91f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import cmath"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "167be3b5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['__doc__', '__file__', '__loader__', '__name__', '__package__', '__spec__', 'acos', 'acosh', 'asin', 'asinh', 'atan', 'atanh', 'cos', 'cosh', 'e', 'exp', 'inf', 'infj', 'isclose', 'isfinite', 'isinf', 'isnan', 'log', 'log10', 'nan', 'nanj', 'phase', 'pi', 'polar', 'rect', 'sin', 'sinh', 'sqrt', 'tan', 'tanh', 'tau']\n"
     ]
    }
   ],
   "source": [
    "print(dir(cmath))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2bc03177",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on built-in function log in module cmath:\n",
      "\n",
      "log(x, y_obj=None, /)\n",
      "    The logarithm of z to the given base.\n",
      "    \n",
      "    If the base not specified, returns the natural logarithm (base e) of z.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(log)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "96fcb23a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5707963267948966j"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(0+1j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "458cb096",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793j"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "6b53ffcc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0, 3.141592653589793)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polar(-1+0j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "9be05fea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0, -1.5707963267948966)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polar(-1j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4e72f24c",
   "metadata": {},
   "outputs": [],
   "source": [
    "r, theta = polar(-1j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "7a0c28f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6.123233995736766e-17+0j)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(theta)# + sin(theta)*j"
   ]
  }
 ],
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