{"id":186931,"date":"2025-01-26T12:24:57","date_gmt":"2025-01-26T12:24:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186931"},"modified":"2025-01-26T12:24:59","modified_gmt":"2025-01-26T12:24:59","slug":"decimal-to-binary-conversion","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/26\/decimal-to-binary-conversion\/","title":{"rendered":"Decimal to Binary Conversion"},"content":{"rendered":"\n

Decimal to Binary Conversion<\/p>\n\n\n\n

Convert each decimal number to binary using the sum-of-weights method:<\/p>\n\n\n\n

(a) 23
(b) 57
(c) 45.5
Convert each decimal number to binary using the repeated division-by-2 method (repeated multiplication-by-2 for fractions):<\/p>\n\n\n\n

(a) 14
(b) 21
(c) 0.375<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Decimal to Binary Conversion Using Sum-of-Weights Method<\/h3>\n\n\n\n

Sum-of-Weights Method Explanation<\/h4>\n\n\n\n

The sum-of-weights method involves determining the binary representation of a decimal number by finding which powers of 2 (weights) add up to the decimal number. Starting from the largest power of 2 less than or equal to the number, subtract and continue until the remainder is zero.<\/p>\n\n\n\n

Solutions<\/h4>\n\n\n\n
    \n
  1. (a) 23<\/strong>\n
      \n
    • Largest power of 2 \u2264 23 is 24=162^4 = 16. Subtract: 23\u221216=723 – 16 = 7.<\/li>\n\n\n\n
    • Largest power of 2 \u2264 7 is 22=42^2 = 4. Subtract: 7\u22124=37 – 4 = 3.<\/li>\n\n\n\n
    • Largest power of 2 \u2264 3 is 21=22^1 = 2. Subtract: 3\u22122=13 – 2 = 1.<\/li>\n\n\n\n
    • Largest power of 2 \u2264 1 is 20=12^0 = 1. Subtract: 1\u22121=01 – 1 = 0.<\/li>\n\n\n\n
    • Binary: 23=10111223 = 10111_2.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • (b) 57<\/strong>\n
        \n
      • Largest power of 2 \u2264 57 is 25=322^5 = 32. Subtract: 57\u221232=2557 – 32 = 25.<\/li>\n\n\n\n
      • Largest power of 2 \u2264 25 is 24=162^4 = 16. Subtract: 25\u221216=925 – 16 = 9.<\/li>\n\n\n\n
      • Largest power of 2 \u2264 9 is 23=82^3 = 8. Subtract: 9\u22128=19 – 8 = 1.<\/li>\n\n\n\n
      • Largest power of 2 \u2264 1 is 20=12^0 = 1. Subtract: 1\u22121=01 – 1 = 0.<\/li>\n\n\n\n
      • Binary: 57=111001257 = 111001_2.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • (c) 45.5<\/strong>\n
          \n
        • Integer part: 4545. Use sum-of-weights: 45=101101245 = 101101_2.<\/li>\n\n\n\n
        • Fractional part: 0.5=2\u22121=0.120.5 = 2^{-1} = 0.1_2.<\/li>\n\n\n\n
        • Binary: 45.5=101101.1245.5 = 101101.1_2.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
          \n\n\n\n

          Decimal to Binary Conversion Using Division-by-2 Method<\/h3>\n\n\n\n

          Explanation<\/h4>\n\n\n\n

          For integers, repeatedly divide the number by 2, noting the remainders. The binary result is the remainders read from bottom to top. For fractions, repeatedly multiply by 2 and note the integer part. The result is the integer parts read from top to bottom.<\/p>\n\n\n\n

          Solutions<\/h4>\n\n\n\n
            \n
          1. (a) 14<\/strong>\n
              \n
            • Division-by-2:
              14\u00f72=714 \u00f7 2 = 7 remainder 00,
              7\u00f72=37 \u00f7 2 = 3 remainder 11,
              3\u00f72=13 \u00f7 2 = 1 remainder 11,
              1\u00f72=01 \u00f7 2 = 0 remainder 11.<\/li>\n\n\n\n
            • Binary: 14=1110214 = 1110_2.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
            • (b) 21<\/strong>\n
                \n
              • Division-by-2:
                21\u00f72=1021 \u00f7 2 = 10 remainder 11,
                10\u00f72=510 \u00f7 2 = 5 remainder 00,
                5\u00f72=25 \u00f7 2 = 2 remainder 11,
                2\u00f72=12 \u00f7 2 = 1 remainder 00,
                1\u00f72=01 \u00f7 2 = 0 remainder 11.<\/li>\n\n\n\n
              • Binary: 21=10101221 = 10101_2.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
              • (c) 0.375<\/strong>\n
                  \n
                • Multiply-by-2:
                  0.375\u00d72=0.750.375 \u00d7 2 = 0.75, integer part: 00,
                  0.75\u00d72=1.50.75 \u00d7 2 = 1.5, integer part: 11,
                  0.5\u00d72=1.00.5 \u00d7 2 = 1.0, integer part: 11.<\/li>\n\n\n\n
                • Binary: 0.375=0.01120.375 = 0.011_2.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
                  \n\n\n\n

                  Explanation of Methods (300 Words)<\/h3>\n\n\n\n

                  The sum-of-weights method is conceptually intuitive because it directly builds the binary representation by identifying which powers of 2 contribute to the number. It\u2019s particularly useful for understanding the breakdown of a decimal into binary and is helpful when manual calculation is required.<\/p>\n\n\n\n

                  The division-by-2 method is more systematic and algorithmic. For integers, repeatedly dividing by 2 and keeping track of the remainders ensures a reliable way to generate the binary representation. Starting from the largest binary place (most significant bit), this method guarantees all bits are calculated sequentially.<\/p>\n\n\n\n

                  For fractional values, multiplying by 2 is equally systematic. Each multiplication identifies the next binary digit by extracting the integer part of the product, building the fractional binary representation from left to right.<\/p>\n\n\n\n

                  Both methods achieve the same result but cater to different scenarios. The sum-of-weights method offers insight into binary decomposition, while the division\/multiplication methods are procedural, making them suitable for programming and repetitive calculations.<\/p>\n","protected":false},"excerpt":{"rendered":"

                  Decimal to Binary Conversion Convert each decimal number to binary using the sum-of-weights method: (a) 23(b) 57(c) 45.5Convert each decimal number to binary using the repeated division-by-2 method (repeated multiplication-by-2 for fractions): (a) 14(b) 21(c) 0.375 The correct answer and explanation is: Decimal to Binary Conversion Using Sum-of-Weights Method Sum-of-Weights Method Explanation The sum-of-weights method […]<\/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-186931","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186931","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=186931"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186931\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186931"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186931"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}