DBremen/complex numbers.ipynb

## complex numbers.ipynb
{
  "cells": [
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "import numpy as np\nimport sympy as sym\nfrom IPython.display import Math,display",
      "execution_count": 15,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# j is in python the imaginary operator\nprint(np.sqrt(-1))\nprint(np.sqrt(-1,dtype='complex'))\nprint(sym.I)",
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": "nan\n1j\nI\n",
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": "C:\\Users\\Dirk\\Anaconda3\\lib\\site-packages\\ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in sqrt\n  \n",
          "name": "stderr"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Two ways to create complex numbers"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "real_part = 1\nimaginary_part = -5\ncn1 = np.complex(real_part,imaginary_part)\ncn2 = real_part + 1j*imaginary_part\nprint(cn1)\nprint(cn2)",
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(1-5j)\n(1-5j)\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Add and subtract complex numbers"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z1 = np.complex(4,5)\nz2 = np.complex(3,2)\nprint(z1)\nprint(np.real(z1))\nprint(np.imag(z1))\n\nz1 - z2\n",
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(4+5j)\n4.0\n5.0\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 4,
          "data": {
            "text/plain": "(1+3j)"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "w = np.complex(2,4)\nz = np.complex(5,6)\nprint(w + z)\nprint(np.complex(np.real(w) + np.real(z), np.imag(w) + np.imag(z)))",
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(7+10j)\n(7+10j)\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Conjungate and multiplying numbers"
    },
    {
      "metadata": {
        "trusted": false
      },
      "cell_type": "markdown",
      "source": "Conjungate = sames number but the sign of the imaginary part is flipped"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z1 = np.complex(4,5)\nz2 = np.complex(6,-2)\nprint(z1*z2)",
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(34+22j)\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "print(np.conj(z1))",
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 7,
          "data": {
            "text/plain": "(4-5j)"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "display(Math('z\\\\times z^* = a^2 + b^2'))",
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle z\\times z^* = a^2 + b^2$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "a,b = sym.symbols('a b',real=True)\nz = a + b * sym.I\nconjz = sym.conjugate(z)\nsym.expand(z * conjz)",
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 13,
          "data": {
            "text/plain": "a**2 + b**2",
            "text/latex": "$\\displaystyle a^{2} + b^{2}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Dividing complex numbers"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z = np.complex(4,2)\ndisplay(Math('\\\\frac{%s}{2} = %s' %(z,z/2)))",
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\frac{(4+2j)}{2} = (2+1j)$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z1 = np.complex(4,2)\nz2 = np.complex(2,-3)\ndisplay(Math('\\\\frac{%s}{%s} = \\\\frac{%s\\\\times %s}{%s\\\\times %s} = %s'\n             %(z1,z2,\n               z1,np.conj(z2),z2,np.conj(z2),\n               z1/z2\n              )))",
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\frac{(4+2j)}{(2-3j)} = \\frac{(4+2j)\\times (2+3j)}{(2-3j)\\times (2+3j)} = (0.15384615384615383+1.2307692307692308j)$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Graphing complex numbers"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "import matplotlib.pyplot as plt",
      "execution_count": 23,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z = np.complex(2,3)\nplt.plot(np.real(z), np.imag(z),'ro')\nplt.plot([0,np.real(z)],[0,np.imag(z)],'r')\nplt.xlabel('real')\nplt.ylabel('imaginary')\nplt.grid()\nplt.axis([-4,4,-4,4])",
      "execution_count": 31,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 31,
          "data": {
            "text/plain": "[-4, 4, -4, 4]"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z1 = np.complex(-3,1)\nz2 = np.complex(-1,1)\ncalc = z1 + z2\nplt.plot([0,np.real(z1)],[0,np.imag(z1)],label='z1')\nplt.plot([0,np.real(z2)],[0,np.imag(z2)],label='z2')\nplt.plot([0,np.real(calc)],[0,np.imag(calc)],label='z1+z2')\nplt.xlabel('real')\nplt.ylabel('imaginary')\nplt.axis('square')\nplt.legend()\nplt.grid()\nplt.axis([-5,5,-5,5])",
      "execution_count": 35,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 35,
          "data": {
            "text/plain": "[-5, 5, -5, 5]"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "z1 = np.complex(-3,1)\nz2 = np.complex(-1,1)\ncalc = z1 * z2\nplt.plot([0,np.real(z1)],[0,np.imag(z1)],label='z1')\nplt.plot([0,np.real(z2)],[0,np.imag(z2)],label='z2')\nplt.plot([0,np.real(calc)],[0,np.imag(calc)],label='z1*z2')\nplt.xlabel('real')\nplt.ylabel('imaginary')\nplt.axis('square')\nplt.legend()\nplt.grid()\nplt.axis([-5,5,-5,5])",
      "execution_count": 36,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 36,
          "data": {
            "text/plain": "[-5, 5, -5, 5]"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "",
      "execution_count": null,
      "outputs": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3",
      "language": "python"
    },
    "language_info": {
      "name": "python",
      "version": "3.7.4",
      "mimetype": "text/x-python",
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "pygments_lexer": "ipython3",
      "nbconvert_exporter": "python",
      "file_extension": ".py"
    },
    "toc": {
      "nav_menu": {},
      "number_sections": true,
      "sideBar": true,
      "skip_h1_title": false,
      "base_numbering": 1,
      "title_cell": "Table of Contents",
      "title_sidebar": "Contents",
      "toc_cell": false,
      "toc_position": {},
      "toc_section_display": true,
      "toc_window_display": false
    },
    "gist": {
      "id": "d1fed338583dfba348c9b433fd7e29a4",
      "data": {
        "description": "Complex numbers",
        "public": true
      }
    },
    "_draft": {
      "nbviewer_url": "https://gist.github.com/d1fed338583dfba348c9b433fd7e29a4"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 2
}