{"id":143739,"date":"2024-09-24T05:42:24","date_gmt":"2024-09-24T05:42:24","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=143739"},"modified":"2024-09-24T05:42:27","modified_gmt":"2024-09-24T05:42:27","slug":"how-many-bits-do-we-need-in-order-to-represent-the-first-15-numbers","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/09\/24\/how-many-bits-do-we-need-in-order-to-represent-the-first-15-numbers\/","title":{"rendered":"How many bits do we need in order to represent the first 15 numbers"},"content":{"rendered":"\n<p>How many bits do we need in order to represent the first 15 numbers? How about the first 16, 25 or n numbers?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine how many bits are required to represent a certain range of numbers, we need to understand the relationship between bits and numerical representation in binary format. In binary, each bit can have two possible values: 0 or 1. Therefore, the number of unique values that can be represented by ( b ) bits is given by the formula:<\/p>\n\n\n\n<p>[<br>\\text{Unique values} = 2^b<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. First 15 Numbers<\/h3>\n\n\n\n<p>The first 15 numbers are typically considered to be the integers from 0 to 14. To find the number of bits required to represent these numbers, we need to find the smallest ( b ) such that:<\/p>\n\n\n\n<p>[<br>2^b \\geq 15<br>]<\/p>\n\n\n\n<p>Calculating for ( b ):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( 2^0 = 1 ) (1 value, not enough)<\/li>\n\n\n\n<li>( 2^1 = 2 ) (2 values, not enough)<\/li>\n\n\n\n<li>( 2^2 = 4 ) (4 values, not enough)<\/li>\n\n\n\n<li>( 2^3 = 8 ) (8 values, not enough)<\/li>\n\n\n\n<li>( 2^4 = 16 ) (16 values, sufficient)<\/li>\n<\/ul>\n\n\n\n<p>Thus, we need <strong>4 bits<\/strong> to represent the first 15 numbers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. First 16 Numbers<\/h3>\n\n\n\n<p>For the first 16 numbers (0 to 15), we follow a similar process:<\/p>\n\n\n\n<p>[<br>2^b \\geq 16<br>]<\/p>\n\n\n\n<p>Here, ( 2^4 = 16 ) is sufficient, so <strong>4 bits<\/strong> are also required for the first 16 numbers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. First 25 Numbers<\/h3>\n\n\n\n<p>For the first 25 numbers (0 to 24):<\/p>\n\n\n\n<p>[<br>2^b \\geq 25<br>]<\/p>\n\n\n\n<p>Calculating:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( 2^4 = 16 ) (not enough)<\/li>\n\n\n\n<li>( 2^5 = 32 ) (sufficient)<\/li>\n<\/ul>\n\n\n\n<p>Therefore, we need <strong>5 bits<\/strong> to represent the first 25 numbers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. General Case for ( n ) Numbers<\/h3>\n\n\n\n<p>For any ( n ) numbers, where we want to represent numbers from 0 to ( n-1 ), we determine ( b ) as follows:<\/p>\n\n\n\n<p>[<br>2^b \\geq n<br>]<\/p>\n\n\n\n<p>Thus, the number of bits required can be expressed as:<\/p>\n\n\n\n<p>[<br>b = \\lceil \\log_2(n) \\rceil<br>]<\/p>\n\n\n\n<p>Where ( \\lceil x \\rceil ) is the ceiling function, which rounds up to the nearest integer. This means you will need enough bits to represent all numbers up to ( n-1 ), ensuring that you have sufficient unique combinations of 0s and 1s to cover the range.<\/p>\n\n\n\n<p>In summary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For the first 15 numbers: <strong>4 bits<\/strong><\/li>\n\n\n\n<li>For the first 16 numbers: <strong>4 bits<\/strong><\/li>\n\n\n\n<li>For the first 25 numbers: <strong>5 bits<\/strong><\/li>\n\n\n\n<li>For ( n ) numbers: ( b = \\lceil \\log_2(n) \\rceil ) bits.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>How many bits do we need in order to represent the first 15 numbers? How about the first 16, 25 or n numbers? The Correct answer and Explanation is: To determine how many bits are required to represent a certain range of numbers, we need to understand the relationship between bits and numerical representation in [&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-143739","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/143739","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=143739"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/143739\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=143739"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=143739"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=143739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}