{"id":535,"date":"2008-08-26T16:00:17","date_gmt":"2008-08-26T23:00:17","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=535"},"modified":"2008-08-25T17:52:29","modified_gmt":"2008-08-26T00:52:29","slug":"a-little-mathematical-game","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/a-little-mathematical-game\/","title":{"rendered":"A little mathematical game"},"content":{"rendered":"<p>Given five things:<\/p>\n<ul>\n<li>The number 0, denoted 0<\/li>\n<li>The ability to find e<sup>x<\/sup> for any number x, denoted {x}<\/li>\n<li>The ability to find the principle natural logarithm log(x) for any number x, denoted [x]<\/li>\n<li>The ability to find the additive inverse -x for any number x, denoted &lt;x&gt;<\/li>\n<li>The ability to find the sum x+y for any pair of numbers x and y, denoted (x+y)<\/li>\n<\/ul>\n<p>What numbers can you make and how long are their denotations? This gives some sort of metric to how \"complicated\" a number is. Write the length of the smallest possible denotation for number x as L(x). Then:<\/p>\n<ul>\n<li>Subtraction: a-b is denoted as (a+&lt;b&gt;) and L(a-b) &lt;= 5+L(a)+L(b)<\/li>\n<li>Multiplication: ab is denoted as {([a]+[b])} and L(ab) &lt;= 9+L(a)+L(b)<\/li>\n<li>Division: a\/b is denoted as {([a]+&lt;[b]&gt;)} and L(a\/b) &lt;= 11+L(a)+L(b)<\/li>\n<li>Exponentiation: b<sup>a<\/sup> is denoted as {{([a]+[[b]])}} and L(b<sup>a<\/sup>) &lt;= 13+L(a)+L(b)<\/li>\n<\/ul>\n<p>Some interesting numbers, with their complexities:<\/p>\n<table>\n<tr>\n<td>1<\/td>\n<td>{0}<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>e<\/td>\n<td>{{0}}<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>-1<\/td>\n<td>&lt;{0}&gt;<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>({0}+{0})<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>i<\/td>\n<td>{{([[&lt;{0}&gt;]]+&lt;[({0}+{0})]&gt;)}}<\/td>\n<td>29<\/td>\n<\/tr>\n<tr>\n<td>&pi;<\/td>\n<td><{([[&lt;{0}&gt;]]+[{{([[&lt;{0}&gt;]]+&lt;[({0}+{0})]&gt;)}}])}&gt;<\/td>\n<td>47<\/td>\n<\/tr>\n<\/table>\n<p>Some interesting questions:<\/p>\n<ul>\n<li>How does the complexity function L grow with its argument?<\/li>\n<li>What interesting numbers do not have finite complexity?<\/li>\n<li>How could the game be changed to include them?<\/li>\n<\/ul>\n<p>Related: <a href=\"https:\/\/www.reenigne.org\/blog\/fine-structure-constant-update\">Fine structure constant update<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given five things: The number 0, denoted 0 The ability to find ex for any number x, denoted {x} The ability to find the principle natural logarithm log(x) for any number x, denoted [x] The ability to find the additive inverse -x for any number x, denoted &lt;x&gt; The ability to find the sum x+y [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-535","post","type-post","status-publish","format-standard","hentry","category-maths"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=535"}],"version-history":[{"count":2,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/535\/revisions"}],"predecessor-version":[{"id":674,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/535\/revisions\/674"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}