{"id":235584,"date":"2025-06-15T14:00:49","date_gmt":"2025-06-15T14:00:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=235584"},"modified":"2025-06-15T14:00:51","modified_gmt":"2025-06-15T14:00:51","slug":"convert-the-following-positive-binary-numbers-to-their-decimal-number-equivalents-showing-all-steps","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/15\/convert-the-following-positive-binary-numbers-to-their-decimal-number-equivalents-showing-all-steps\/","title":{"rendered":"Convert the following positive binary numbers to their decimal number equivalents, showing all steps"},"content":{"rendered":"\n

Convert the following positive binary numbers to their decimal number equivalents, showing all steps. a) 111111 b) 1110.11 c) 101111.101 d) 10111110.00001 e) 110011.011<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Conversion of Binary to Decimal:<\/h3>\n\n\n\n
\n\n\n\n

a) 111111<\/strong><\/p>\n\n\n\n

Each binary digit (bit) is multiplied by 2 raised to the power of its position from right to left, starting at 0.1\u00d725+1\u00d724+1\u00d723+1\u00d722+1\u00d721+1\u00d720=32+16+8+4+2+1=631 \\times 2^5 + 1 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 1 \\times 2^1 + 1 \\times 2^0 \\\\ = 32 + 16 + 8 + 4 + 2 + 1 = \\boxed{63}1\u00d725+1\u00d724+1\u00d723+1\u00d722+1\u00d721+1\u00d720=32+16+8+4+2+1=63\u200b<\/p>\n\n\n\n

\n\n\n\n

b) 1110.11<\/strong><\/p>\n\n\n\n

Integer part:1\u00d723+1\u00d722+1\u00d721+0\u00d720=8+4+2+0=141 \\times 2^3 + 1 \\times 2^2 + 1 \\times 2^1 + 0 \\times 2^0 = 8 + 4 + 2 + 0 = 141\u00d723+1\u00d722+1\u00d721+0\u00d720=8+4+2+0=14<\/p>\n\n\n\n

Fractional part:1\u00d72\u22121+1\u00d72\u22122=0.5+0.25=0.751 \\times 2^{-1} + 1 \\times 2^{-2} = 0.5 + 0.25 = 0.751\u00d72\u22121+1\u00d72\u22122=0.5+0.25=0.75<\/p>\n\n\n\n

Combined:14+0.75=14.7514 + 0.75 = \\boxed{14.75}14+0.75=14.75\u200b<\/p>\n\n\n\n

\n\n\n\n

c) 101111.101<\/strong><\/p>\n\n\n\n

Integer part:1\u00d725+0\u00d724+1\u00d723+1\u00d722+1\u00d721+1\u00d720=32+0+8+4+2+1=471 \\times 2^5 + 0 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 1 \\times 2^1 + 1 \\times 2^0 = 32 + 0 + 8 + 4 + 2 + 1 = 471\u00d725+0\u00d724+1\u00d723+1\u00d722+1\u00d721+1\u00d720=32+0+8+4+2+1=47<\/p>\n\n\n\n

Fractional part:1\u00d72\u22121+0\u00d72\u22122+1\u00d72\u22123=0.5+0+0.125=0.6251 \\times 2^{-1} + 0 \\times 2^{-2} + 1 \\times 2^{-3} = 0.5 + 0 + 0.125 = 0.6251\u00d72\u22121+0\u00d72\u22122+1\u00d72\u22123=0.5+0+0.125=0.625<\/p>\n\n\n\n

Combined:47+0.625=47.62547 + 0.625 = \\boxed{47.625}47+0.625=47.625\u200b<\/p>\n\n\n\n

\n\n\n\n

d) 10111110.00001<\/strong><\/p>\n\n\n\n

Integer part:1\u00d727+0\u00d726+1\u00d725+1\u00d724+1\u00d723+1\u00d722+1\u00d721+0\u00d720=128+0+32+16+8+4+2+0=1901 \\times 2^7 + 0 \\times 2^6 + 1 \\times 2^5 + 1 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 1 \\times 2^1 + 0 \\times 2^0 \\\\ = 128 + 0 + 32 + 16 + 8 + 4 + 2 + 0 = 1901\u00d727+0\u00d726+1\u00d725+1\u00d724+1\u00d723+1\u00d722+1\u00d721+0\u00d720=128+0+32+16+8+4+2+0=190<\/p>\n\n\n\n

Fractional part:0\u00d72\u22121+0\u00d72\u22122+0\u00d72\u22123+0\u00d72\u22124+1\u00d72\u22125=0.031250 \\times 2^{-1} + 0 \\times 2^{-2} + 0 \\times 2^{-3} + 0 \\times 2^{-4} + 1 \\times 2^{-5} = 0.031250\u00d72\u22121+0\u00d72\u22122+0\u00d72\u22123+0\u00d72\u22124+1\u00d72\u22125=0.03125<\/p>\n\n\n\n

Combined:190+0.03125=190.03125190 + 0.03125 = \\boxed{190.03125}190+0.03125=190.03125\u200b<\/p>\n\n\n\n

\n\n\n\n

e) 110011.011<\/strong><\/p>\n\n\n\n

Integer part:1\u00d725+1\u00d724+0\u00d723+0\u00d722+1\u00d721+1\u00d720=32+16+0+0+2+1=511 \\times 2^5 + 1 \\times 2^4 + 0 \\times 2^3 + 0 \\times 2^2 + 1 \\times 2^1 + 1 \\times 2^0 = 32 + 16 + 0 + 0 + 2 + 1 = 511\u00d725+1\u00d724+0\u00d723+0\u00d722+1\u00d721+1\u00d720=32+16+0+0+2+1=51<\/p>\n\n\n\n

Fractional part:0\u00d72\u22121+1\u00d72\u22122+1\u00d72\u22123=0+0.25+0.125=0.3750 \\times 2^{-1} + 1 \\times 2^{-2} + 1 \\times 2^{-3} = 0 + 0.25 + 0.125 = 0.3750\u00d72\u22121+1\u00d72\u22122+1\u00d72\u22123=0+0.25+0.125=0.375<\/p>\n\n\n\n

Combined:51+0.375=51.37551 + 0.375 = \\boxed{51.375}51+0.375=51.375\u200b<\/p>\n\n\n\n

\n\n\n\n

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

Binary numbers operate on base 2, meaning each digit represents a power of 2. Converting from binary to decimal involves summing the value of each binary digit after multiplying it by 2 raised to its positional index. For integer parts, the index starts from 0 on the right and increases leftward. For fractional parts, the index begins at -1 moving rightward.<\/p>\n\n\n\n

In example (a), all bits are 1, so each power of 2 from 2\u2070 to 2\u2075 contributes to the total. This results in a sum of 63.<\/p>\n\n\n\n

In (b), both integer and fractional parts are evaluated separately. The integer part (1110) translates to 14, and the fractional part (0.11) adds 0.75, resulting in 14.75.<\/p>\n\n\n\n

In (c), a similar process is followed. The integer portion (101111) equals 47, and the fractional part (0.101) adds 0.625, yielding a combined total of 47.625.<\/p>\n\n\n\n

In (d), the longer binary number still follows the same structure. The integer part (10111110) gives 190, while the fractional component (0.00001) contributes a small value of 0.03125. Together, they total 190.03125.<\/p>\n\n\n\n

Lastly, (e) contains a mixture of 1s and 0s on both sides of the decimal. The integer portion sums to 51, and the fractional part adds 0.375, giving a final result of 51.375.<\/p>\n\n\n\n

This process ensures accurate conversion using place-value rules of binary positional notation.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Convert the following positive binary numbers to their decimal number equivalents, showing all steps. a) 111111 b) 1110.11 c) 101111.101 d) 10111110.00001 e) 110011.011 The Correct Answer and Explanation is: Conversion of Binary to Decimal: a) 111111 Each binary digit (bit) is multiplied by 2 raised to the power of its position from right to […]<\/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-235584","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235584","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=235584"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235584\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=235584"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=235584"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=235584"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}