{"id":10435,"date":"2022-04-09T22:27:36","date_gmt":"2022-04-09T16:57:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10435"},"modified":"2022-04-09T22:32:53","modified_gmt":"2022-04-09T17:02:53","slug":"the-following-real-numbers-have-decimal-expansions-as-given-below","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/the-following-real-numbers-have-decimal-expansions-as-given-below\/","title":{"rendered":"The following real numbers have decimal expansions as given below."},"content":{"rendered":"

Question<\/strong> : The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form \\(p\\over q\\), what can you say about the prime factors of q ?<\/p>\n

(i)  43.123456789<\/p>\n

(ii)  0.120120012000120000…..<\/p>\n

(iii)  43.\\(\\overline{123456789}\\)<\/p>\n

Solution<\/strong> :<\/p>\n

(i)<\/strong>  43.123456789 is terminating<\/strong>.<\/p>\n

So, it represents a rational number<\/strong>.<\/p>\n

Thus, 43.123456789 = \\(43123456789\\over 1000000000\\) = \\(p\\over q\\). Thus, q = \\(10^9\\).<\/p>\n

(ii)<\/strong>  0.120120012000120000….. is non-terminating and non-repeating. So, it is an irrational.<\/strong><\/p>\n

(iii)<\/strong>  43.123456789 is non-terminating but repeating. So, it is a rational.<\/strong><\/p>\n

Thus, 43.\\(\\overline{123456789}\\) = \\(4312345646\\over 999999999\\) = \\(p\\over q\\)<\/p>\n

Thus, q = 999999999<\/p>\n","protected":false},"excerpt":{"rendered":"

Question : The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form \\(p\\over q\\), what can you say about the prime factors of q ? (i)  43.123456789 (ii)  0.120120012000120000….. (iii)  43.\\(\\overline{123456789}\\) Solution : (i)  43.123456789 is terminating. …<\/p>\n

The following real numbers have decimal expansions as given below.<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,909],"tags":[],"yoast_head":"\nThe following real numbers have decimal expansions as given below. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/the-following-real-numbers-have-decimal-expansions-as-given-below\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The following real numbers have decimal expansions as given below. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : The following real numbers have decimal expansions as given below. 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