{"id":13581,"date":"2024-12-03T00:59:36","date_gmt":"2024-12-03T03:59:36","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=13581"},"modified":"2024-12-05T20:47:27","modified_gmt":"2024-12-05T23:47:27","slug":"crazy-representations-of-half-a-million-natural-numbers","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=13581","title":{"rendered":"Crazy Representations of Half a Million Natural Numbers"},"content":{"rendered":"\n<p>In 2013-2014, author (<a href=\"http:\/\/bit.ly\/2wnZq6g\" target=\"_blank\" rel=\"noreferrer noopener\">click here<\/a>) wrote natural numbers in increasing and decreasing orders of 1 to 9 using only&nbsp;<strong>basic operations<\/strong>. See below some examples:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"837\" height=\"783\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/08\/Image-1.png\" alt=\"\" class=\"wp-image-12211\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/08\/Image-1.png 837w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/08\/Image-1-300x281.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/08\/Image-1-768x718.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/08\/Image-1-535x500.png 535w\" sizes=\"(max-width: 837px) 100vw, 837px\" \/><\/figure>\n<\/div>\n\n\n<p>For more details and extension work by other see the following link:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inder J. Taneja, The Crazy Representations and 10958 Problem,&nbsp; \n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2018\/11\/16\/crazy-representations-of-natural-numbers-the-10958-problem\/\">Crazy Representations of Natural Numbers \u2013 The 10958 Problem \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=893\">Crazy Representations of Natural Numbers \u2013 The 10958 Problem \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>The above study include the natural numbers till 300000. In some case&nbsp;<strong>factorials&nbsp;<\/strong>and or&nbsp;<strong>square-roots<\/strong>&nbsp;are used. There are only four numbers where we used <strong>square-root<\/strong>. All other numbers are basic operation incluing <strong>factorials<\/strong>. See below these four numbers:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"585\" height=\"218\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-1a.png\" alt=\"\" class=\"wp-image-13583\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-1a.png 585w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-1a-300x112.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-1a-535x199.png 535w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/figure>\n<\/div>\n\n\n<p>To see the whole work up to 300.000 numbers refer the following links:<\/p>\n\n\n\n<ul class=\"wp-block-list has-blush-light-purple-gradient-background has-background\">\n<li><strong>Inder J. Taneja, The Crazy Representations and 10958 Problem<\/strong>,&nbsp;\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2018\/11\/16\/crazy-representations-of-natural-numbers-the-10958-problem\/\">Crazy Representations of Natural Numbers \u2013 The 10958 Problem \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=893\">Crazy Representations of Natural Numbers \u2013 The 10958 Problem \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Crazy Representations of Natural Numbers Using Factorial: From 20001-100000<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2021\/12\/07\/crazy-representations-of-natural-numbers-using-factorial-from-20-001-100-000\/\">Crazy Representations of Natural Numbers Using Factorial: From 20001-100000 \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=12194\">Crazy Representations of Natural Numbers Using Factorial: From 20001-100000 \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Crazy Representations of Natural Numbers Using Factorial: From 100001-200000<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2021\/12\/11\/crazy-representations-of-natural-numbers-using-factorial-from-100001-200000\/\">Crazy Representations of Natural Numbers Using Factorial: From 100001-200000 \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=311\">Crazy Representations of Natural Numbers Using Factorial: From 100001-200000 \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Crazy Representations of Natural Numbers Using Factorial From 200001 to 300000<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2022\/01\/09\/crazy-representations-of-natural-numbers-using-factorial-from-200001-to-300000\/\">Crazy Representations of Natural Numbers Using Factorial From 200001 to 300000 \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=459\">Crazy Representations of Natural Numbers Using Factorial From 200001 to 300000 \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Representation of Numbers from 1 to 20000 in Terms of Palindromic Digits 1357-9-7531<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2022\/01\/06\/representation-of-numbers-from-1-to-20000-in-terms-of-palindromic-digits-1357-9-7531\/\">Representation of Numbers from 1 to 20000 in Terms of Palindromic Digits 1357-9-7531 \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/numbers-magic.com\/?p=449\">Representation of Numbers from 1 to 20000 in Terms of Palindromic Digits 1357-9-7531 \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Representation of Numbers from 1 to 10000 in Terms of Palindromic Digits 2022-2202,&nbsp;<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/inderjtaneja.wordpress.com\/2022\/01\/02\/representation-of-numbers-from-1-to-10000-in-terms-of-palindromic-digits-2022-2202\/\">Representation of Numbers from 1 to 10000 in Terms of Palindromic Digits 2022-2202 \u2013 Numbers Magic (wordpress.com)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja, Multiple Choices for Crazy Representations of Natural Numbers<\/strong>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/numbers-magic.com\/?p=11640\">Multiple Choices for Crazy Representations of Natural Numbers \u2013 Recreating Numbers and Magic Squares (numbers-magic.com)<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(255,245,203) 0%,rgb(174,196,79) 12%,rgb(182,227,212) 50%,rgb(255,255,255) 65%,rgb(51,167,181) 100%)\">Crazy Representations: Extention to Half a Million Natural Numbers<\/h3>\n\n\n\n<p>Recently, the author extended the above work up to half-million, i.e., 500000 numbers in increasing and decreating orders of 1 to 9 and 9 to 1. Most of numbers are by use of factorials. There are only few numbers where the square-root is  also applied.  See below:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Increasing Order<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1228\" height=\"904\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a.png\" alt=\"\" class=\"wp-image-13584\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a.png 1228w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a-300x221.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a-1024x754.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a-768x565.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-2a-535x394.png 535w\" sizes=\"(max-width: 1228px) 100vw, 1228px\" \/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Decreasing Order <\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"529\" height=\"281\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-3a.png\" alt=\"\" class=\"wp-image-13585\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-3a.png 529w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-3a-300x159.png 300w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Missing Numbers<\/strong>: Still there are 7 missing numbers. Out of them 4 are in the increasing case and 3 are in the decreasing case. See below these numbers:\n<ul class=\"wp-block-list\">\n<li><strong>Increasing order missing:<\/strong> 469786, 469883, 474686, 499370<\/li>\n\n\n\n<li><strong>Decreasing order missing<\/strong>: 431588, 464854, 468310.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>These numbers can be writtern in terms of <strong>Triangular numbers<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Increasing order<\/strong><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>469786<\/strong>:=1-T(2+3)+T(4+5)<em>T(6<\/em>(7+8+9));<\/li>\n\n\n\n<li><strong>469883<\/strong>:=-1-T(2+T(T(T(3))))+(T(T(4))-T(T(T(5))))(T(T(6))-T(7+8+9)); <\/li>\n\n\n\n<li><strong>474686<\/strong>:=(1+T(2+3))(T(T(4))-5+T(T(T(6))))+T(T(7+8+9));<\/li>\n\n\n\n<li><strong>499370<\/strong>:=-1+T(T(T(2+T(3))))+4<em>(T(5)+T(T(6))<\/em>T(7+8+9));<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Decreasing order<\/strong><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>431588<\/strong>:=-9+T(T(T(8)+7)+6+5)-T(4)-T(T(T(3+2+1)));<\/li>\n\n\n\n<li><strong>464854<\/strong>:=-9+8-T(T(7))+6+T(T(5<em>4))<\/em>T(3+2+1);<\/li>\n\n\n\n<li><strong>468310<\/strong>:=-9-T(T(8))+T(T(7))<em>T(T(6))<\/em>5+T(4+3+2+1);<\/li>\n<\/ul>\n\n\n\n<p>Among 300.001 to 500.000 there are 200.000 numbers. The whole work we have divided in folloing five parts:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Inder J. Taneja, <\/strong>Crazy Representations of Half a Million Natural Numbers: <strong>Part 1<\/strong> &#8211; 300.001 to 340.000, <strong>Zenodo<\/strong>, December 05, 2024, pp. 1-1042, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14267963\">https:\/\/doi.org\/10.5281\/zenodo.14267963<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja, <\/strong>Crazy Representations of Half a Million Natural Numbers: <strong>Part 2<\/strong> &#8211; 340.001 to 380.000, <strong>Zenodo<\/strong>, December 05, 2024, pp. 1-1035, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14277644\">https:\/\/doi.org\/10.5281\/zenodo.14277644<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja, <\/strong>Crazy Representations of Half a Million Natural Numbers: <strong>Part 3<\/strong> &#8211; 380.001 to 420.000, <strong>Zenodo<\/strong>, December 05, 2024, pp. 1-1040, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14277646\">https:\/\/doi.org\/10.5281\/zenodo.14277646<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja, <\/strong>Crazy Representations of Half a Million Natural Numbers: <strong>Part 4<\/strong> &#8211; 420.001 to 460.000,<strong> Zenodo<\/strong>, December 05, 2024, pp. 1-1041, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14277648\">https:\/\/doi.org\/10.5281\/zenodo.14277648<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja, <\/strong>Crazy Representations of Half a Million Natural Numbers: <strong>Part 5<\/strong> &#8211; 460.001 to 500.000, <strong>Zenodo<\/strong>, December 05, 2024, pp. 1-1045, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14277650\">https:\/\/doi.org\/10.5281\/zenodo.14277650<\/a>.<\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 2013-2014, author (click here) wrote natural numbers in increasing and decreasing orders of 1 to 9 using only&nbsp;basic operations. See below some examples: For more details and extension work by other see the following link: The above study include the natural numbers till 300000. In some case&nbsp;factorials&nbsp;and or&nbsp;square-roots&nbsp;are used. There are only four numbers [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":13585,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4],"tags":[],"class_list":["post-13581","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-crazy-numbers"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/12\/image-3a.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13581","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13581"}],"version-history":[{"count":6,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13581\/revisions"}],"predecessor-version":[{"id":13601,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13581\/revisions\/13601"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/13585"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13581"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13581"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}