WhatIsMultiplication.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports and boilerplate to make graphs look better\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import numpy as np\n",
    "import scipy\n",
    "\n",
    "def setup_graph(title='', x_label='', y_label='', fig_size=None, grid=True):\n",
    "    fig = plt.figure()\n",
    "    if fig_size != None:\n",
    "        fig.set_size_inches(fig_size[0], fig_size[1])\n",
    "    ax = fig.add_subplot(111)\n",
    "    ax.set_title(title)\n",
    "    ax.set_xlabel(x_label)\n",
    "    ax.set_ylabel(y_label)\n",
    "    plt.grid(grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# What is multiplication?\n",
    "\n",
    "And how does it differ from addition?\n",
    "\n",
    "Take the number 6. You can get it by multiplying 2*3. You can also get 6 by adding 3+3, or 2+4, or 1+5. But you can also get from addition 6 in other ways if you use negative numbers such as -2+8. Additionally, you can get 6 from addition in an infinite number of ways of you use consider all real numbers and not just integers... "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "setup_graph(fig_size=(5,5))\n",
    "x = np.linspace(-10, 10)\n",
    "y = 6-x\n",
    "_= plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You could also visualize this if you consider 6 as the distance from the origin like this. Then it could be described as:\n",
    "\n",
    "|x|+|y| = 6\n",
    "\n",
    "which would be a circle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also get 6 in many more ways with multiplication if you aren't just looking at real numbers..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "setup_graph(fig_size=(7,7))\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = 6/x\n",
    "_= plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Complex multiplication\n",
    "\n",
    "Now let's introduce complex multiplication. Multiplying a real number by i produces a 90 degree rotation on the complex plane\n",
    "\n",
    "Here is 3*i - note the 90 degree rotation..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x10b181c18>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def complex_to_euclidean(c):\n",
    "    return (c.real, c.imag)\n",
    "\n",
    "setup_graph(title='3 * i = 3i', x_label='real', y_label='imag', fig_size=(6,6))\n",
    "a = (3+0j)\n",
    "b = (0+1j)\n",
    "a_times_b = a*b\n",
    "plt.scatter(*complex_to_euclidean(a), color='blue', s=100)\n",
    "plt.scatter(*complex_to_euclidean(b), color='green', s=100)\n",
    "plt.scatter(*complex_to_euclidean(a_times_b), color='red', s=100)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Taken in this light, multiplying a real number by a negative real number produces a 180 degree rotation. The strictly real-number line is a 0-sphere, which is composed of just 2 non-connected points, which is why it just \"jumps\" when multiplying by a negative.\n",
    "\n",
    "Of course, this also explains why multipying by 2 negatives equals a positive - because you are doing 2 180 degree transformations, which ends up back at the starting place.\n",
    "\n",
    "Whereas strictly real numbers form a 0-sphere, the complex number plane represents a 1-sphere, which is connected. So we can actually continuously raise to higher and higher powers (which is simply multiplying by i in a continuous fasion) non-integrer powers of i to produce a circle..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "setup_graph(title='(1+0i) * i^x for x=0 to 4', x_label='real', y_label='imag', fig_size=(6,6))\n",
    "real_num = (1+0j)\n",
    "\n",
    "# Values of the exponent of i from 0 to 3, which will include i^0, i^1, i^2 (-1), and i^3 (-i)\n",
    "imag_values = np.linspace(0, 3, 100)\n",
    "xy = [real_num*((0+1j)**imag_exp) for imag_exp in imag_values]\n",
    "for c in xy:\n",
    "    plt.scatter(c.real, c.imag, color='blue', s=10)\n",
    "\n",
    "# Values of the exponent of i from 3 to 4, which will range from i^3 (-i) to i^4 (+1)\n",
    "xy2 = [real_num*((0+1j)**imag_exp) for imag_exp in np.linspace(3, 4, 25)]\n",
    "for c in xy2:\n",
    "    plt.scatter(c.real, c.imag, color='red', s=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplication spiral\n",
    "\n",
    "So here's an intersting thought... there should be some spiral such that:\n",
    "* 2**0 should equal 1\n",
    "* 2**1 should equal 2\n",
    "* 2**2 should equal 4\n",
    "* 2**3 should equal 8\n",
    "\n",
    "In other words, scaling should happen along with rotating, which sounds like a spiral. In fact, this reminds me of the Archimedes spiral. I hypothesize that this spiral should also hold true for 2**(x+xi). It should also work out for other numbers too. Let's see if I can find the right parameters to produce this spiral...\n",
    "\n",
    "According to wikipedia, the Archimedes spiral has the equation r = a + b*θ. So let's try to find a and be...\n",
    "\n",
    "for 2**0:\n",
    "* 1 = a + b*0, (since θ should be 0 because it should be pointing down the positive side of the real axis)\n",
    "* 2 = a + b*(2π) (since θ should equal 2π because a one full rotation should have happened at this point)\n",
    "* 4 = a + b*(4π) (since θ should equal 4π because a two full rotation should have happened at this point)\n",
    "\n",
    "So just looking at the first equation, it looks like a should equal 1 since 1 = 1 + 0. So then, looking at the second equation:\n",
    "    2 = 1 + b*(2π)\n",
    "So\n",
    "    1/(2π) = b\n",
    "\n",
    "So the polar equation should be:\n",
    "\n",
    "r = 1 + (1/(2π)) * θ\n",
    "\n",
    "Okay, but how do I convert from this polar equation into the complex plane?\n",
    "\n",
    "Here's some things I know:\n",
    "* r is the magnitude, which is sqrt(real**2 + imag**2).\n",
    "* θ should be 0 radians at i^0\n",
    "* θ should be π/2 at i^1\n",
    "* θ should be π at i^2 = -1\n",
    "* θ should be 2π at i^4 = 1\n",
    "\n",
    "So in the above graph which produces a circle with a radius of 1, the equation was:\n",
    "\n",
    "(1+0i) * i^x, for x=0 through 4.\n",
    "\n",
    "So how can I modify this equation to produce a spiral which crosses the real axis at these points?\n",
    "* θ=0, real=1, imag=0\n",
    "* θ=2π, real=2, imag=0\n",
    "* θ=4π, real=4, imag=0\n",
    "\n",
    "I think I want (2+0j) * (1+0j)^x * (0+1j)^x\n",
    "* x = 0 -> (2+0j) * (1+0j)^0 = (2+0j)\n",
    "* x = 1 -> (2+0j) * (1+j)^1 = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "setup_graph(title='archimedes spiral?', x_label='real', y_label='imag', fig_size=(6,6))\n",
    "real_num = (2+0j)\n",
    "\n",
    "imag_values = np.linspace(0, 8, 500)\n",
    "xy = [ real_num * (1+0j)**x * (0+1j)**x for imag_exp in imag_values]\n",
    "for c in xy:\n",
    "    plt.scatter(c.real, c.imag, color='blue', s=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}