Suppose that the infinity of decimal numbers between zero and one is the same as the infinity of counting numbers. Then all the decimal numbers can be denumerated in a list.<\/p>\n\n\n\n
1 \u2192 d1<\/sub> = 0.d11<\/sub>d12<\/sub>d13<\/sub>d14<\/sub> ……. 3 \u2192 d3<\/sub> = 0.d31<\/sub>d32<\/sub>d33<\/sub>d34<\/sub> …….<\/p>\n\n\n\n 4 \u2192 d4<\/sub> = 0.d41<\/sub>d42<\/sub>d43<\/sub>d44<\/sub> …….<\/p>\n\n\n\n .<\/strong> Suppose that the infinity of decimal numbers between zero and one is the same as the infinity of counting numbers. Then all the decimal numbers can be denumerated in a list. 1 \u2192 d1 = 0.d11d12d13d14 .......2 \u2192 d2 = 0.d21d22d23d24 ....... 3 \u2192 d3 = 0.d31d32d33d34 ....... 4 \u2192 d4 = 0.d41d42d43d44 ....... ...n ...more about Cantor’s Diagonalization Argument<\/a><\/p>\n","protected":false},"author":115,"featured_media":0,"parent":124,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-415","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/415","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/users\/115"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/comments?post=415"}],"version-history":[{"count":3,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/415\/revisions"}],"predecessor-version":[{"id":2113,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/415\/revisions\/2113"}],"up":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/124"}],"wp:attachment":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/media?parent=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
2 \u2192 d2<\/sub> = 0.d21<\/sub>d22<\/sub>d23<\/sub>d24<\/sub> …….<\/p>\n\n\n\n
.<\/strong>
.<\/strong>
n \u2192 dn<\/sub> = 0.dn1<\/sub>dn2<\/sub>dn3<\/sub>dn4<\/sub> …….
.<\/strong>
.<\/strong>
.<\/strong>
Consider the decimal number x = 0.x1<\/sub>x2<\/sub>x3<\/sub>x4<\/sub>x5<\/sub> ……. , where x1<\/sub> is any digit other than d11<\/sub>; x2<\/sub> is different from d22<\/sub>; x3<\/sub> is not equal to d33<\/sub>; x4<\/sub> is not d44<\/sub>; and so on. Now, x is a decimal number, and x is less than one, so it must be in our list. But where? x can’t be first, since x’s first digit differs from d1<\/sub>‘s first digit. x can’t be second in the list, because x and d2<\/sub> have different hundredths place digits. In general, x is not equal to dn<\/sub>, since their nth digits are not the same.
x is nowhere to be found in the list. In other words, we have exhibited a decimal number that ought to be in the list but isn’t. No matter how we try to list the decimal numbers, at least one will be left out. Therefore, “listing” the decimal numbers is impossible, so the infinity of decimal numbers is greater than the infinity of counting numbers.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"