{"id":186459,"date":"2025-01-25T07:57:46","date_gmt":"2025-01-25T07:57:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186459"},"modified":"2025-01-25T07:57:48","modified_gmt":"2025-01-25T07:57:48","slug":"given-our-discussion-of-positional-numbering-systems","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/25\/given-our-discussion-of-positional-numbering-systems\/","title":{"rendered":"Given our discussion of positional numbering systems"},"content":{"rendered":"\n<p>Given our discussion of positional numbering systems in Section 4.2.1, see whether you can determine the decimal value of the following number:<\/p>\n\n\n\n<p>a. 133 (base 4)<\/p>\n\n\n\n<p>b. 367 (base 8, also called octal)<\/p>\n\n\n\n<p>c. 1BA (base 16, also called hexadecimal. B is the digit that represents 11; A is the digit that represents 10.)<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>In Exercise 1(c), we use the letters A and B as digits of the base-16 number. Explain why that is necessary.<\/li>\n\n\n\n<li>Determine the decimal value of the following unsigned binary numbers:<\/li>\n<\/ol>\n\n\n\n<p>a. 11000 c. 1111111<\/p>\n\n\n\n<p>b. 110001 d. 1000000000<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s break this down step by step.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exercise 1<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">(a) 133 (base 4)<\/h4>\n\n\n\n<p>To convert this from base 4 to decimal, we multiply each digit by its corresponding power of 4, starting from the rightmost digit.<\/p>\n\n\n\n<p>[<br>133_{4} = (1 \\times 4^2) + (3 \\times 4^1) + (3 \\times 4^0)<br>]<br>[<br>= (1 \\times 16) + (3 \\times 4) + (3 \\times 1)<br>]<br>[<br>= 16 + 12 + 3 = 31<br>]<br>So, (133_4 = 31_{10}).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(b) 367 (base 8)<\/h4>\n\n\n\n<p>For base 8 (octal), we use powers of 8:<\/p>\n\n\n\n<p>[<br>367_{8} = (3 \\times 8^2) + (6 \\times 8^1) + (7 \\times 8^0)<br>]<br>[<br>= (3 \\times 64) + (6 \\times 8) + (7 \\times 1)<br>]<br>[<br>= 192 + 48 + 7 = 247<br>]<br>So, (367_8 = 247_{10}).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(c) 1BA (base 16)<\/h4>\n\n\n\n<p>In base 16 (hexadecimal), we use the digits A (10) and B (11). Now, let&#8217;s convert:<\/p>\n\n\n\n<p>[<br>1BA_{16} = (1 \\times 16^2) + (B \\times 16^1) + (A \\times 16^0)<br>]<br>[<br>= (1 \\times 256) + (11 \\times 16) + (10 \\times 1)<br>]<br>[<br>= 256 + 176 + 10 = 442<br>]<br>So, (1BA_{16} = 442_{10}).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exercise 2<\/h3>\n\n\n\n<p>In hexadecimal, the digits go beyond the standard 0\u20139 and use letters A\u2013F. This is because a base-16 system needs 16 distinct symbols to represent numbers. The digits A\u2013F correspond to the values 10\u201315. Specifically, A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.<\/p>\n\n\n\n<p>If we didn&#8217;t use these letters, we would be forced to represent numbers greater than 9 in some other way, potentially confusing the system. The letters allow for a clean, recognizable method to represent these larger values in the same structure of the positional numbering system.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exercise 3<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">(a) 11000 (binary)<\/h4>\n\n\n\n<p>In binary (base-2), each digit represents a power of 2:<\/p>\n\n\n\n<p>[<br>11000_2 = (1 \\times 2^4) + (1 \\times 2^3) + (0 \\times 2^2) + (0 \\times 2^1) + (0 \\times 2^0)<br>]<br>[<br>= (1 \\times 16) + (1 \\times 8) + (0 \\times 4) + (0 \\times 2) + (0 \\times 1)<br>]<br>[<br>= 16 + 8 = 24<br>]<br>So, (11000_2 = 24_{10}).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(b) 110001 (binary)<\/h4>\n\n\n\n<p>[<br>110001_2 = (1 \\times 2^5) + (1 \\times 2^4) + (0 \\times 2^3) + (0 \\times 2^2) + (0 \\times 2^1) + (1 \\times 2^0)<br>]<br>[<br>= (1 \\times 32) + (1 \\times 16) + (0 \\times 8) + (0 \\times 4) + (0 \\times 2) + (1 \\times 1)<br>]<br>[<br>= 32 + 16 + 1 = 49<br>]<br>So, (110001_2 = 49_{10}).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(c) 1111111 (binary)<\/h4>\n\n\n\n<p>[<br>1111111_2 = (1 \\times 2^6) + (1 \\times 2^5) + (1 \\times 2^4) + (1 \\times 2^3) + (1 \\times 2^2) + (1 \\times 2^1) + (1 \\times 2^0)<br>]<br>[<br>= (1 \\times 64) + (1 \\times 32) + (1 \\times 16) + (1 \\times 8) + (1 \\times 4) + (1 \\times 2) + (1 \\times 1)<br>]<br>[<br>= 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127<br>]<br>So, (1111111_2 = 127_{10}).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(d) 1000000000 (binary)<\/h4>\n\n\n\n<p>[<br>1000000000_2 = (1 \\times 2^9) + (0 \\times 2^8) + (0 \\times 2^7) + (0 \\times 2^6) + (0 \\times 2^5) + (0 \\times 2^4) + (0 \\times 2^3) + (0 \\times 2^2) + (0 \\times 2^1) + (0 \\times 2^0)<br>]<br>[<br>= (1 \\times 512) = 512<br>]<br>So, (1000000000_2 = 512_{10}).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>In summary, the decimal equivalents are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>(133_4 = 31_{10})<\/li>\n\n\n\n<li>(367_8 = 247_{10})<\/li>\n\n\n\n<li>(1BA_{16} = 442_{10})<\/li>\n\n\n\n<li>Binary conversions:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(11000_2 = 24_{10})<\/li>\n\n\n\n<li>(110001_2 = 49_{10})<\/li>\n\n\n\n<li>(1111111_2 = 127_{10})<\/li>\n\n\n\n<li>(1000000000_2 = 512_{10})<\/li>\n<\/ul>\n\n\n\n<p>The reason we use letters like A and B in hexadecimal is simply because base-16 requires more than 10 symbols, so letters A\u2013F represent values 10\u201315, keeping the system consistent and compact.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given our discussion of positional numbering systems in Section 4.2.1, see whether you can determine the decimal value of the following number: a. 133 (base 4) b. 367 (base 8, also called octal) c. 1BA (base 16, also called hexadecimal. B is the digit that represents 11; A is the digit that represents 10.) a. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-186459","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186459","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=186459"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186459\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186459"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186459"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186459"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}