{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bd06312f-a5d4-4a37-b9e3-dc6526a3071c",
   "metadata": {},
   "source": [
    "[Oregon Curriculum Network](http://4dsolutions.net/ocn/)<br/>\n",
    "[School of Tomorrow](School_of_Tomorrow.ipynb)\n",
    "\n",
    "\n",
    "# Koski Sums (\"Phi-dentities\")\n",
    "\n",
    "The sympy simplifier, on its own, just makes baroque expressions, but if confronted with $\\sqrt{2}$ and $\\sqrt{3}$ as equalities, [as suggested on StackOverFlow](https://stackoverflow.com/questions/37112738/sympy-comparing-expressions#37115190), the identities are confirmed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1c685ef6-3d88-44b0-8af9-324aa3f7ca7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1f6a6540-d63a-43dd-8ce4-04fb6a00fb4e",
   "metadata": {},
   "outputs": [],
   "source": [
    "ϕ = sym.Symbol('ϕ')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1127c6a9-0e64-439d-9b4e-9706b863d3e5",
   "metadata": {},
   "source": [
    "This will be $\\sqrt{2}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "434a3938-ca65-466c-8e40-c7bcdadea44d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{ϕ + \\frac{1}{ϕ^{2}}}$"
      ],
      "text/plain": [
       "sqrt(ϕ + ϕ**(-2))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = sym.sqrt(ϕ + ϕ**-2)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f44614f5-a851-4cfd-abe5-7e298c881e8e",
   "metadata": {},
   "source": [
    "This will be $\\sqrt{3}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "93090737-cc80-47ca-9120-0379f8487c22",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{ϕ^{2} + \\frac{1}{ϕ^{2}}}$"
      ],
      "text/plain": [
       "sqrt(ϕ**2 + ϕ**(-2))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 = sym.sqrt(ϕ**2 + ϕ**-2)\n",
    "e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7c087d8f-1bb9-4a22-a306-7c215137b556",
   "metadata": {},
   "outputs": [],
   "source": [
    "ϕ = (sym.sqrt(5)+1)/2  # real value time!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e22a90e1-4c5d-4597-8b8d-27641f3fdd6a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.6180339887498948482045868343656381177203091798058$"
      ],
      "text/plain": [
       "1.6180339887498948482045868343656381177203091798058"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕ.evalf(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04b8118b-e401-4812-8b26-66c2d6ed6afc",
   "metadata": {},
   "source": [
    "# Checking $\\sqrt{2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "314ab19e-dbb4-4e00-abf3-564125f1ef42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{\\frac{1}{\\left(\\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\right)^{2}} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}$"
      ],
      "text/plain": [
       "sqrt((1/2 + sqrt(5)/2)**(-2) + 1/2 + sqrt(5)/2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = sym.sqrt(ϕ + ϕ**-2)\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e2562ef0-7130-436a-b6cb-fd67d6dba3f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.4142135623730950488016887242096980785696718753769$"
      ],
      "text/plain": [
       "1.4142135623730950488016887242096980785696718753769"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.evalf(50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c5a38d37-c1b0-4256-9563-f3e487cfc973",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.4142135623730950488016887242096980785696718753769$"
      ],
      "text/plain": [
       "1.4142135623730950488016887242096980785696718753769"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.N(sym.sqrt(2), 50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "104c6dbc-36cd-46ab-a402-7cac2afbd53a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.equals(sym.sqrt(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92c55ed4-ccd2-42ef-bf1d-e33f3128c036",
   "metadata": {},
   "source": [
    "# Checking $\\sqrt{3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0cfb34a2-2a08-4df3-ba2c-457af947e0b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{\\frac{1}{\\left(\\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\right)^{2}} + \\left(\\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\right)^{2}}$"
      ],
      "text/plain": [
       "sqrt((1/2 + sqrt(5)/2)**(-2) + (1/2 + sqrt(5)/2)**2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 = sym.sqrt(ϕ**2 + ϕ**-2)\n",
    "e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e4761877-2532-477f-8221-0f15bca0fdca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.7320508075688772935274463415058723669428052538104$"
      ],
      "text/plain": [
       "1.7320508075688772935274463415058723669428052538104"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2.evalf(50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8f323c1c-8d22-402c-8270-c48e0a185e5d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.7320508075688772935274463415058723669428052538104$"
      ],
      "text/plain": [
       "1.7320508075688772935274463415058723669428052538104"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.N(sym.sqrt(3), 50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "d218c157-241e-475f-aec7-870900ef26aa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2.equals(sym.sqrt(3))"
   ]
  }
 ],
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