ICBacon · August 29, 2015 14:25
diff --git a/MyFirstNotebook.yml b/MyFirstNotebook.yml
 url = https://api.github.com/gists/0c37e47f24531557decd/1670b009f4baa036dab816779165cfaeff0004c0

      "content": "{\n \"cells\": [\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"# Definition of a Field F in Mathematics (Field Axioms):\\n\",\n    \"\\n\",\n    \"* ### Closure of F under addition and multiplication (i.e. + and · are binary operations on F)\\n\",\n    \"\\n\",\n    \"    $\\\\forall$ a, b $\\\\epsilon$ F,\\n\",\n    \"    $$a + b\\\\; \\\\epsilon\\\\; F$$\\n\",\n    \"    $$a · b\\\\; \\\\epsilon\\\\; F$$\\n\",\n    \"\\n\",\n    \"* ### Commutativity of addition and multiplication\\n\",\n    \"\\n\",\n    \"     $\\\\forall$ a, b $\\\\epsilon$ F, the following equalities hold:\\n\",\n    \"     $$b + a = a + b$$\\n\",\n    \"     $$b · a = a · b$$\\n\",\n    \"\\n\",\n    \"* ### Associativity of addition and multiplication\\n\",\n    \"    \\n\",\n    \"    $\\\\forall$ a, b, and c $\\\\epsilon$ F, the following equalities hold:\\n\",\n    \"    $$a + (b + c) = (a + b) + c$$\\n\",\n    \"    $$a · (b · c) = (a · b) · c$$\\n\",\n    \"\\n\",\n    \"* ### Distributivity of multiplication over addition\\n\",\n    \"\\n\",\n    \"    $\\\\forall$ a, b, and c $\\\\epsilon$ F, the following equality holds:\\n\",\n    \"    \\n\",\n    \"    $$a · (b + c) = (a · b) + (a · c)$$\\n\",\n    \"\\n\",\n    \"* ### Existence of additive and multiplicative identity elements\\n\",\n    \"\\n\",\n    \"     $\\\\forall$ a $\\\\epsilon$ F, $\\\\exists$ 0 $\\\\epsilon$ F, called the additive identity element, s.t.\\n\",\n    \"    \\n\",\n    \"    $$a + 0 = a$$\\n\",\n    \"    \\n\",\n    \"    Likewise, $\\\\exists$ 1 $\\\\epsilon$ F, called the multiplicative identity element, s.t. $\\\\forall$ a $\\\\epsilon$ F,\\n\",\n    \"    \\n\",\n    \"    $$a · 1 = a$$\\n\",\n    \"\\n\",\n    \"* ### Existence of additive inverses and multiplicative inverses\\n\",\n    \"\\n\",\n    \"    $\\\\forall$ a $\\\\epsilon$ F, $\\\\exists$ (−a) $\\\\epsilon$ F, s.t.\\n\",\n    \"    \\n\",\n    \"    $$a + (−a) = 0$$\\n\",\n    \"    \\n\",\n    \"    Similarly, $\\\\forall$ a $\\\\epsilon$ F, with a $\\\\neq$ 0, $\\\\exists$ $a^{−1}$ $\\\\epsilon$ F, s.t.\\n\",\n    \"    \\n\",\n    \"    $$a · a^{−1} = 1$$\\n\",\n    \"    \\n\",\n    \"    (The elements a + (−b) and a · $b^{−1}$ are also denoted a − b and $\\\\dfrac{a}{b}$, respectively.) In other words, subtraction and division operations exist.\\n\",\n    \"\\n\",\n    \"* ### Nontrivial Field\\n\",\n    \"\\n\",\n    \"    To exclude the trivial ring, let's require the additive identity and the multiplicative identity to be distinct:\\n\",\n    \"    \\n\",\n    \"    $$0 \\\\neq 1$$\\n\",\n    \"\\n\",\n    \"* ### Summary\\n\",\n    \"\\n\",\n    \"    A *field* is therefore an algebraic structure〈F, +, ·, −, −1, 0, 1〉; of type〈2, 2, 1, 1, 0, 0〉, consisting of two abelian groups:\\n\",\n    \"    \\n\",\n    \"    * F under +, −, and 0\\n\",\n    \"    * F ∖ {0} under ·, −1, and 1, where 0 $\\\\neq$ 1, along with · distributing over +\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"### Example: Show that the set of all real numbers $\\\\mathbb{R}$ is a field\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Recall the set $\\\\mathbb{R}$ = {...,-3,-2,-1,0,1,2,3,...}\\n\",\n    \"\\n\",\n    \"Suppose $\\\\exists$ a, b, and c $\\\\epsilon$ $\\\\mathbb{R}$. To show that $\\\\mathbb{R}$ is a field by using the definition above, let's show that all of the field axioms hold true.\\n\",\n    \"\\n\",\n    \"#### Axiom 1 (Closure).  $\\\\forall$ $a, b$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n    \"$$a + b\\\\; \\\\epsilon\\\\; \\\\mathbb{R}$$\\n\",\n    \"$$a · b\\\\; \\\\epsilon\\\\; \\\\mathbb{R}$$\\n\",\n    \"\\n\",\n    \"#### Axiom 2 (Commutativity).  $\\\\forall$ $a, b$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n    \"$$b + a = a + b$$\\n\",\n    \"$$b · a = a · b$$\\n\",\n    \"\\n\",\n    \"#### Axiom 3 (Associativity). $\\\\forall$ $a,b,c$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n    \"$$a + (b + c) = (a + b) + c$$\\n\",\n    \"$$a · (b · c) = (a · b) · c$$\\n\",\n    \"\\n\",\n    \"#### Axiom 4 (Distributivity). $\\\\forall$ $a,b,c$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n    \"$$a · (b + c) = (a · b) + (a · c)$$\\n\",\n    \"\\n\",\n    \"#### Axiom 5 (Identities). $\\\\forall$ $b$ $\\\\epsilon$ $\\\\mathbb{R}$, $\\\\exists$ $0,1$ $\\\\epsilon$ $\\\\mathbb{R}$, $s.t.$\\n\",\n    \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad b + 0 = 0 + b = b\\\\qquad\\\\qquad(Additive\\\\, Identity)$$\\n\",\n    \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad b · 1 = 1 · b = b\\\\qquad\\\\quad\\\\;\\\\,(Multiplicative\\\\, Identity)$$\\n\",\n    \"\\n\",\n    \"#### Axiom 6 (Inverses). $\\\\forall$ $a$ $\\\\epsilon$ $\\\\mathbb{R}$, $\\\\exists$ $b,c$ $\\\\epsilon$ $\\\\mathbb{R}$, $and\\\\,\\\\, c\\\\neq0$ $s.t.$\\n\",\n    \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad a + b = b + a = 0\\\\qquad\\\\qquad (Additive\\\\, Inverse)$$\\n\",\n    \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad a · c = c · a = 1\\\\qquad\\\\quad\\\\;\\\\,(Multiplicative\\\\, Inverse)$$\\n\",\n    \"\\n\",\n    \"#### Axiom 7 (Non-Trivial Field). $$0 \\\\neq 1$$\\n\",\n    \"\\n\",\n    \"####Axioms (1-7) imply that $\\\\mathbb{R}$ is a field.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Let's take three real numbers a,b,c $\\\\epsilon$ $\\\\mathbb{R}$ to show this result. Suppose we take a,b,c to be:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 25,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"a = 2\\n\",\n    \"b = -4\\n\",\n    \"c = 5\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 26,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"2\"\n      ]\n     },\n     \"execution_count\": 26,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 27,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-4\"\n      ]\n     },\n     \"execution_count\": 27,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"b\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 28,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"5\"\n      ]\n     },\n     \"execution_count\": 28,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"c\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 29,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-2\"\n      ]\n     },\n     \"execution_count\": 29,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a + b\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 30,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-8\"\n      ]\n     },\n     \"execution_count\": 30,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a * b\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 31,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"3\"\n      ]\n     },\n     \"execution_count\": 31,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a + (b + c)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 32,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"3\"\n      ]\n     },\n     \"execution_count\": 32,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"(a + b) + c\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 33,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-40\"\n      ]\n     },\n     \"execution_count\": 33,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a * (b * c)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 34,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-40\"\n      ]\n     },\n     \"execution_count\": 34,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"(a * b) * c\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 35,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-2\"\n      ]\n     },\n     \"execution_count\": 35,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"b + a\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 36,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"-8\"\n      ]\n     },\n     \"execution_count\": 36,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"b * a\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 37,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"2\"\n      ]\n     },\n     \"execution_count\": 37,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a + 0\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 38,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"2\"\n      ]\n     },\n     \"execution_count\": 38,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a * 1\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 39,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"0\"\n      ]\n     },\n     \"execution_count\": 39,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a + (-a)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 40,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"1.0\"\n      ]\n     },\n     \"execution_count\": 40,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a * (a ** (-1))\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 41,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"2\"\n      ]\n     },\n     \"execution_count\": 41,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"a * (b + c)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 42,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"2\"\n      ]\n     },\n     \"execution_count\": 42,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"(a * b) + (a * c)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": null,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": []\n  }\n ],\n \"metadata\": {\n  \"kernelspec\": {\n   \"display_name\": \"Python 2\",\n   \"language\": \"python\",\n   \"name\": \"python2\"\n  },\n  \"language_info\": {\n   \"codemirror_mode\": {\n    \"name\": \"ipython\",\n    \"version\": 2\n   },\n   \"file_extension\": \".py\",\n   \"mimetype\": \"text/x-python\",\n   \"name\": \"python\",\n   \"nbconvert_exporter\": \"python\",\n   \"pygments_lexer\": \"ipython2\",\n   \"version\": \"2.7.8\"\n  }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n"
	url = https://api.github.com/gists/0c37e47f24531557decd/1670b009f4baa036dab816779165cfaeff0004c0

	"content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Definition of a Field F in Mathematics (Field Axioms):\\n\",\n \"\\n\",\n \"* ### Closure of F under addition and multiplication (i.e. + and · are binary operations on F)\\n\",\n \"\\n\",\n \" $\\\\forall$ a, b $\\\\epsilon$ F,\\n\",\n \" $$a + b\\\\; \\\\epsilon\\\\; F$$\\n\",\n \" $$a · b\\\\; \\\\epsilon\\\\; F$$\\n\",\n \"\\n\",\n \"* ### Commutativity of addition and multiplication\\n\",\n \"\\n\",\n \" $\\\\forall$ a, b $\\\\epsilon$ F, the following equalities hold:\\n\",\n \" $$b + a = a + b$$\\n\",\n \" $$b · a = a · b$$\\n\",\n \"\\n\",\n \"* ### Associativity of addition and multiplication\\n\",\n \" \\n\",\n \" $\\\\forall$ a, b, and c $\\\\epsilon$ F, the following equalities hold:\\n\",\n \" $$a + (b + c) = (a + b) + c$$\\n\",\n \" $$a · (b · c) = (a · b) · c$$\\n\",\n \"\\n\",\n \"* ### Distributivity of multiplication over addition\\n\",\n \"\\n\",\n \" $\\\\forall$ a, b, and c $\\\\epsilon$ F, the following equality holds:\\n\",\n \" \\n\",\n \" $$a · (b + c) = (a · b) + (a · c)$$\\n\",\n \"\\n\",\n \"* ### Existence of additive and multiplicative identity elements\\n\",\n \"\\n\",\n \" $\\\\forall$ a $\\\\epsilon$ F, $\\\\exists$ 0 $\\\\epsilon$ F, called the additive identity element, s.t.\\n\",\n \" \\n\",\n \" $$a + 0 = a$$\\n\",\n \" \\n\",\n \" Likewise, $\\\\exists$ 1 $\\\\epsilon$ F, called the multiplicative identity element, s.t. $\\\\forall$ a $\\\\epsilon$ F,\\n\",\n \" \\n\",\n \" $$a · 1 = a$$\\n\",\n \"\\n\",\n \"* ### Existence of additive inverses and multiplicative inverses\\n\",\n \"\\n\",\n \" $\\\\forall$ a $\\\\epsilon$ F, $\\\\exists$ (−a) $\\\\epsilon$ F, s.t.\\n\",\n \" \\n\",\n \" $$a + (−a) = 0$$\\n\",\n \" \\n\",\n \" Similarly, $\\\\forall$ a $\\\\epsilon$ F, with a $\\\\neq$ 0, $\\\\exists$ $a^{−1}$ $\\\\epsilon$ F, s.t.\\n\",\n \" \\n\",\n \" $$a · a^{−1} = 1$$\\n\",\n \" \\n\",\n \" (The elements a + (−b) and a · $b^{−1}$ are also denoted a − b and $\\\\dfrac{a}{b}$, respectively.) In other words, subtraction and division operations exist.\\n\",\n \"\\n\",\n \"* ### Nontrivial Field\\n\",\n \"\\n\",\n \" To exclude the trivial ring, let's require the additive identity and the multiplicative identity to be distinct:\\n\",\n \" \\n\",\n \" $$0 \\\\neq 1$$\\n\",\n \"\\n\",\n \"* ### Summary\\n\",\n \"\\n\",\n \" A field is therefore an algebraic structure〈F, +, ·, −, −1, 0, 1〉; of type〈2, 2, 1, 1, 0, 0〉, consisting of two abelian groups:\\n\",\n \" \\n\",\n \" * F under +, −, and 0\\n\",\n \" * F ∖ {0} under ·, −1, and 1, where 0 $\\\\neq$ 1, along with · distributing over +\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Example: Show that the set of all real numbers $\\\\mathbb{R}$ is a field\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Recall the set $\\\\mathbb{R}$ = {...,-3,-2,-1,0,1,2,3,...}\\n\",\n \"\\n\",\n \"Suppose $\\\\exists$ a, b, and c $\\\\epsilon$ $\\\\mathbb{R}$. To show that $\\\\mathbb{R}$ is a field by using the definition above, let's show that all of the field axioms hold true.\\n\",\n \"\\n\",\n \"#### Axiom 1 (Closure). $\\\\forall$ $a, b$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n \"$$a + b\\\\; \\\\epsilon\\\\; \\\\mathbb{R}$$\\n\",\n \"$$a · b\\\\; \\\\epsilon\\\\; \\\\mathbb{R}$$\\n\",\n \"\\n\",\n \"#### Axiom 2 (Commutativity). $\\\\forall$ $a, b$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n \"$$b + a = a + b$$\\n\",\n \"$$b · a = a · b$$\\n\",\n \"\\n\",\n \"#### Axiom 3 (Associativity). $\\\\forall$ $a,b,c$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n \"$$a + (b + c) = (a + b) + c$$\\n\",\n \"$$a · (b · c) = (a · b) · c$$\\n\",\n \"\\n\",\n \"#### Axiom 4 (Distributivity). $\\\\forall$ $a,b,c$ $\\\\epsilon$ $\\\\mathbb{R}$,\\n\",\n \"$$a · (b + c) = (a · b) + (a · c)$$\\n\",\n \"\\n\",\n \"#### Axiom 5 (Identities). $\\\\forall$ $b$ $\\\\epsilon$ $\\\\mathbb{R}$, $\\\\exists$ $0,1$ $\\\\epsilon$ $\\\\mathbb{R}$, $s.t.$\\n\",\n \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad b + 0 = 0 + b = b\\\\qquad\\\\qquad(Additive\\\\, Identity)$$\\n\",\n \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad b · 1 = 1 · b = b\\\\qquad\\\\quad\\\\;\\\\,(Multiplicative\\\\, Identity)$$\\n\",\n \"\\n\",\n \"#### Axiom 6 (Inverses). $\\\\forall$ $a$ $\\\\epsilon$ $\\\\mathbb{R}$, $\\\\exists$ $b,c$ $\\\\epsilon$ $\\\\mathbb{R}$, $and\\\\,\\\\, c\\\\neq0$ $s.t.$\\n\",\n \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad a + b = b + a = 0\\\\qquad\\\\qquad (Additive\\\\, Inverse)$$\\n\",\n \"$$\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad\\\\qquad a · c = c · a = 1\\\\qquad\\\\quad\\\\;\\\\,(Multiplicative\\\\, Inverse)$$\\n\",\n \"\\n\",\n \"#### Axiom 7 (Non-Trivial Field). $$0 \\\\neq 1$$\\n\",\n \"\\n\",\n \"####Axioms (1-7) imply that $\\\\mathbb{R}$ is a field.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Let's take three real numbers a,b,c $\\\\epsilon$ $\\\\mathbb{R}$ to show this result. Suppose we take a,b,c to be:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 25,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"a = 2\\n\",\n \"b = -4\\n\",\n \"c = 5\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 26,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"2\"\n ]\n },\n \"execution_count\": 26,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 27,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-4\"\n ]\n },\n \"execution_count\": 27,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"b\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 28,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"5\"\n ]\n },\n \"execution_count\": 28,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"c\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 29,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-2\"\n ]\n },\n \"execution_count\": 29,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a + b\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 30,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-8\"\n ]\n },\n \"execution_count\": 30,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a * b\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 31,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"3\"\n ]\n },\n \"execution_count\": 31,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a + (b + c)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 32,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"3\"\n ]\n },\n \"execution_count\": 32,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"(a + b) + c\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 33,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-40\"\n ]\n },\n \"execution_count\": 33,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a * (b * c)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 34,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-40\"\n ]\n },\n \"execution_count\": 34,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"(a * b) * c\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 35,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-2\"\n ]\n },\n \"execution_count\": 35,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"b + a\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 36,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"-8\"\n ]\n },\n \"execution_count\": 36,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"b * a\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 37,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"2\"\n ]\n },\n \"execution_count\": 37,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a + 0\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 38,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"2\"\n ]\n },\n \"execution_count\": 38,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a * 1\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 39,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"0\"\n ]\n },\n \"execution_count\": 39,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a + (-a)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 40,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"1.0\"\n ]\n },\n \"execution_count\": 40,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a * (a ** (-1))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 41,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"2\"\n ]\n },\n \"execution_count\": 41,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a * (b + c)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 42,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"2\"\n ]\n },\n \"execution_count\": 42,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"(a * b) + (a * c)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n"