notebooks/sage-6d.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sage Complex Numbers\n",
    "\n",
    "Floating point numbers or real numbers are represented using the ring CC."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constructor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = CC(3 + 4*i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.rings.complex_number.ComplexNumber'>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A list of n complex numbers as powers of a given complex number a can be obtained using a.powers(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.00000000000000, 3.00000000000000 + 4.00000000000000*I, -7.00000000000000 + 24.0000000000000*I, -117.000000000000 + 44.0000000000000*I, -527.000000000000 - 336.000000000000*I]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<class 'list'>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1 = a.powers(5)\n",
    "print(l1)\n",
    "type(l1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inquiry functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can ask for the real and imaginary part of a complex number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.00000000000000\n",
      "4.00000000000000\n"
     ]
    }
   ],
   "source": [
    "print(a.real())\n",
    "print(a.imag())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can check if a complex number has only real part."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.is_real()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can inquire if a complex number has only the imaginary part."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = CC(12*i)\n",
    "b.is_imaginary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Precision error of real numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.22044604925031e-16"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC.epsilon()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Precision of self using RR.prec() or RR.precision()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "53"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC.prec()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC.is_field()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions on CC"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conjugate of a complex number is readily available."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.00000000000000 - 4.00000000000000*I"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.conjugate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The distance between two complex numbers c1 and c2 is available using CC.dist(c1, c2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.41421356237310"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1 = CC(3 + 4*i)\n",
    "c2 = CC(2 + 3*i)\n",
    "CC.dist(c1, c2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize a complex number to unity using the function CC.one()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.00000000000000\n"
     ]
    }
   ],
   "source": [
    "c1 = CC.one()\n",
    "print(c1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can initialize to zero too, using the function CC.zero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000000000000000\n"
     ]
    }
   ],
   "source": [
    "c1 = CC.zero()\n",
    "print(c1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of pi to the precision as required is available also in the complex ring using the function CC.pi()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14159265358979"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC.pi()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can create a complex field of a specific precision (256 bit, here) and then ask for the value of pi."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793238462643383279502884197169399375105820974944592307816406286"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = ComplexField(256)\n",
    "C.pi()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sum of all complex numbers in a list can be found using CC.sum(list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.0000000000000 + 14.0000000000000*I"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1 = [CC(1+2*i), CC(2+3*i), CC(3+4*i), CC(4+5*i)]\n",
    "CC.sum(l1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, product of all complex numbers in a list can be found using CC.prod(list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-185.000000000000 - 180.000000000000*I"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC.prod(l1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions on Complex Numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.785398163397448 + 0.785398163397448*I\n"
     ]
    }
   ],
   "source": [
    "c1 = CC( CC.pi() / 4 + i*CC.pi()/4 )\n",
    "print(c1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trigonometric functions are all defined on Complex Numbers. Other functions such as abs, exp, log are also defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.936640069431430 + 0.614243127486596*I 0.936640069431430 - 0.614243127486595*I 0.398536815338387 + 0.917152335667274*I\n"
     ]
    }
   ],
   "source": [
    "print(c1.sin(), c1.cos(), c1.tan())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.105009115009482 + 0.785398163397448*I 1.55088319691803 + 1.55088319691803*I\n"
     ]
    }
   ],
   "source": [
    "print(c1.log(), c1.exp())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Magnitude (norm) of a complex number c1 is given by c1.norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.23370055013617"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1.norm()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The nth root of a complex number c1 is obtained using c1.nth_root(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.00865095233196 + 0.159754616567907*I\n"
     ]
    }
   ],
   "source": [
    "c2 = c1.nth_root(5)\n",
    "print(c2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify this by raising the power to the same number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.785398163397449 + 0.785398163397449*I"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2 ** 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Within the precision of representation, the difference between $c_2^5$ and $c_1$ is zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.55111512312578e-16 + 5.55111512312578e-16*I"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2 ** 5 - c1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can visualize the complex numbers as points in a complex plane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can superpose the plots to visualize multiple points with legends as you wish."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = c1.plot(color='blue', legend_label='c1')\n",
    "p2 = c2.plot(color='red', legend_label='c2')\n",
    "(p1+p2).show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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