The binary number 0.101101101101101101\u2026. is equal to the decimal number (without any approximation)<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Let’s analyze the binary number:<\/p>\n\n\n\n 0.101101101101101101…<\/strong> (repeating pattern “101” continuously)<\/p>\n\n\n\n The binary number is a repeating fractional binary with the repeating sequence:<\/p>\n\n\n\n 101<\/strong><\/p>\n\n\n\n This pattern repeats infinitely after the decimal point.<\/p>\n\n\n\n The binary number can be written as: 0.101101101101…=0.101+0.000101+0.000000101+\u20260.101101101101… = 0.101 + 0.000101 + 0.000000101 + \\dots<\/p>\n\n\n\n Each “101” block is shifted 3 bits further to the right each time.<\/p>\n\n\n\n The first block “101” is the bits after the decimal point at positions 2\u221212^{-1}, 2\u221222^{-2}, and 2\u221232^{-3}:<\/p>\n\n\n\n Sum of one block “101”: 0.5+0+0.125=0.6250.5 + 0 + 0.125 = 0.625<\/p>\n\n\n\n The first term a=0.625a = 0.625<\/p>\n\n\n\n Each subsequent term is shifted right by 3 bits, which means multiplied by 2\u22123=182^{-3} = \\frac{1}{8}.<\/p>\n\n\n\n So the number is: x=0.625+0.625\u00d718+0.625\u00d7(18)2+\u22efx = 0.625 + 0.625 \\times \\frac{1}{8} + 0.625 \\times \\left(\\frac{1}{8}\\right)^2 + \\cdots<\/p>\n\n\n\n The sum of an infinite geometric series is: S=a1\u2212rS = \\frac{a}{1 – r}<\/p>\n\n\n\n where<\/p>\n\n\n\n So x=0.6251\u221218=0.62578=0.625\u00d787=58\u00d787=57x = \\frac{0.625}{1 – \\frac{1}{8}} = \\frac{0.625}{\\frac{7}{8}} = 0.625 \\times \\frac{8}{7} = \\frac{5}{8} \\times \\frac{8}{7} = \\frac{5}{7}<\/p>\n\n\n\n 0.101101101101…2=57(in decimal)\\boxed{ 0.101101101101…_2 = \\frac{5}{7} \\quad \\text{(in decimal)} }<\/p>\n\n\n\n The given binary number is a repeating binary fraction with the pattern “101” repeating indefinitely after the decimal point. To convert such a repeating binary fraction into a decimal fraction, we first express the number as an infinite geometric series.<\/p>\n\n\n\n Each group of three bits “101” corresponds to a decimal value when interpreted in binary fractional form. Specifically, the “1” in the first bit after the decimal point represents 2\u22121=0.52^{-1} = 0.5, the “0” in the next bit represents 0, and the last “1” represents 2\u22123=0.1252^{-3} = 0.125. Adding these gives 0.5+0+0.125=0.6250.5 + 0 + 0.125 = 0.625.<\/p>\n\n\n\n Because this pattern repeats every three bits, each subsequent “101” is shifted three places to the right, which in decimal terms means dividing by 23=82^3 = 8. This gives the ratio r=18r = \\frac{1}{8} for the geometric series.<\/p>\n\n\n\n The number xx is thus the sum of the infinite series: x=0.625+0.625\u00d718+0.625\u00d7(18)2+\u2026x = 0.625 + 0.625 \\times \\frac{1}{8} + 0.625 \\times \\left(\\frac{1}{8}\\right)^2 + \\dots<\/p>\n\n\n\n This is a geometric series with first term a=0.625a = 0.625 and common ratio r=18r = \\frac{1}{8}. Using the sum formula for infinite geometric series: S=a1\u2212rS = \\frac{a}{1-r}<\/p>\n\n\n\n Plugging in values: x=0.6251\u221218=0.62578=0.625\u00d787=57x = \\frac{0.625}{1 – \\frac{1}{8}} = \\frac{0.625}{\\frac{7}{8}} = 0.625 \\times \\frac{8}{7} = \\frac{5}{7}<\/p>\n\n\n\n Hence, the repeating binary fraction 0.101101101…0.101101101… is exactly equal to the rational decimal number 57\\frac{5}{7} without any approximation. This exact fraction shows how repeating patterns in binary fractions correspond to rational numbers in decimal form.<\/p>\n","protected":false},"excerpt":{"rendered":" The binary number 0.101101101101101101\u2026. is equal to the decimal number (without any approximation) The correct answer and explanation is: Let’s analyze the binary number: 0.101101101101101101… (repeating pattern “101” continuously) Step 1: Identify the repeating pattern The binary number is a repeating fractional binary with the repeating sequence: 101 This pattern repeats infinitely after the decimal […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-21999","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=21999"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21999\/revisions"}],"predecessor-version":[{"id":22000,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21999\/revisions\/22000"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=21999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=21999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=21999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Identify the repeating pattern<\/h3>\n\n\n\n
\n\n\n\nStep 2: Express as an infinite geometric series<\/h3>\n\n\n\n
\n\n\n\nStep 3: Convert one block “101” to decimal fraction<\/h3>\n\n\n\n
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\n\n\n\nStep 4: Write the full number as a geometric series<\/h3>\n\n\n\n
\n\n\n\nStep 5: Sum the infinite geometric series<\/h3>\n\n\n\n
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\n\n\n\nFinal answer:<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation (300 words):<\/h3>\n\n\n\n