{"id":4346,"date":"2023-06-08T03:37:25","date_gmt":"2023-06-08T03:37:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4346"},"modified":"2025-08-23T00:47:00","modified_gmt":"2025-08-23T00:47:00","slug":"positional-systems-and-bases-apply-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/positional-systems-and-bases-apply-it-1\/","title":{"raw":"Positional Systems and Bases: Apply It 1","rendered":"Positional Systems and Bases: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand different number systems<\/li>\r\n\t<li>Convert different number systems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Digital Defense League - Intro to Binary, Octal, and Hexadecimal Systems<\/h2>\r\n<p>Zuri, a college student who has a passion for cybersecurity, just started a new internship at a local cybersecurity firm. She has just been assigned to the \"Digital Defense League\", a special unit responsible for the protection of the digital space in her area. She has been given three critical tasks, each requiring understanding and application of a different number system: Binary (Base-[latex]2[\/latex]), Octal (Base-[latex]8[\/latex]), and Hexadecimal (Base-[latex]16[\/latex]). Let's take what you just learned about positional systems and bases and help Zuri with her new tasks.<\/p>\r\n<center>\r\n[caption id=\"attachment_6499\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-6499\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-300x200.jpg\" alt=\"Woman working on a computer.\" width=\"500\" height=\"334\" \/> Figure 1. Using positional systems and bases, help Zuri with her two tasks[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Before we dive in, let's take a moment to explore these number systems a bit more.<\/p>\r\n<ul>\r\n\t<li><strong>Binary (Base-2) System:<\/strong> The binary system is used in virtually all digital systems and is at the core of computing. It uses only two digits, [latex]0[\/latex] and [latex]1[\/latex]. Each digit position represents a power of [latex]2[\/latex].<\/li>\r\n\t<li><strong>Octal (Base-8) System:<\/strong> The octal system uses digits from [latex]0[\/latex] to [latex]7[\/latex]. It's often used in computer programming as it provides a shorthand way of representing binary numbers since each octal digit represents three binary digits.<\/li>\r\n\t<li><strong>Hexadecimal (Base-16) System:<\/strong> The hexadecimal system uses sixteen distinct symbols, [latex]0-9[\/latex] and [latex]A-F[\/latex], where [latex]A[\/latex] to [latex]F[\/latex] represent the numbers [latex]10[\/latex] to [latex]15[\/latex]. Hexadecimal is commonly used in programming and computer engineering because it's convenient for representing binary code and is human-friendly.<\/li>\r\n<\/ul>\r\n<section class=\"textbox proTip\">The base-[latex]10[\/latex] and base-[latex]16[\/latex] systems share the same numerical representation up to [latex]9[\/latex]. In both systems, the number [latex]9[\/latex] is represented as [latex]9[\/latex]. However, when we reach the number [latex]10[\/latex], the hexadecimal system introduces a new convention. Instead of using a single digit, base-[latex]16[\/latex] utilizes the letters [latex]A[\/latex] to [latex]F[\/latex] to represent the values [latex]10[\/latex] to [latex]15[\/latex], respectively. For example:\u00a0 [latex]A[\/latex] represents [latex]10[\/latex], [latex]B[\/latex] represents [latex]11[\/latex], and so on.<\/section>\r\n<p>Having introduced these systems, let's get back to helping Zuri.<\/p>\r\n<h3>Critical Task 1: Binary Bank Break-In<\/h3>\r\n<p>Zuri's first task is to help Binary Bank. Binary Bank uses a binary (Base-2) system to protect its vaults. A suspicious binary code from the bank's logs has been discovered. They suspect it might be a password attempt from an unauthorized user. In order to understand the potential threat to Binary Bank's system Zuri needs to convert the binary code from the login to base-[latex]10[\/latex].<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]9042[\/ohm2_question]<\/section>\r\n<p>By deciphering the potential password attempt, Zrui managed to stop a possible break-in at Binary Bank!<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand different number systems<\/li>\n<li>Convert different number systems<\/li>\n<\/ul>\n<\/section>\n<h2>Digital Defense League &#8211; Intro to Binary, Octal, and Hexadecimal Systems<\/h2>\n<p>Zuri, a college student who has a passion for cybersecurity, just started a new internship at a local cybersecurity firm. She has just been assigned to the &#8220;Digital Defense League&#8221;, a special unit responsible for the protection of the digital space in her area. She has been given three critical tasks, each requiring understanding and application of a different number system: Binary (Base-[latex]2[\/latex]), Octal (Base-[latex]8[\/latex]), and Hexadecimal (Base-[latex]16[\/latex]). Let&#8217;s take what you just learned about positional systems and bases and help Zuri with her new tasks.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_6499\" aria-describedby=\"caption-attachment-6499\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6499\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-300x200.jpg\" alt=\"Woman working on a computer.\" width=\"500\" height=\"334\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-300x200.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-1024x684.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-768x513.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-1536x1025.jpg 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-2048x1367.jpg 2048w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-1200x801.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-65x43.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-225x150.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/10204130\/christina-wocintechchat-com-L85a1k-XqH8-unsplash_50-350x234.jpg 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-6499\" class=\"wp-caption-text\">Figure 1. Using positional systems and bases, help Zuri with her two tasks<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Before we dive in, let&#8217;s take a moment to explore these number systems a bit more.<\/p>\n<ul>\n<li><strong>Binary (Base-2) System:<\/strong> The binary system is used in virtually all digital systems and is at the core of computing. It uses only two digits, [latex]0[\/latex] and [latex]1[\/latex]. Each digit position represents a power of [latex]2[\/latex].<\/li>\n<li><strong>Octal (Base-8) System:<\/strong> The octal system uses digits from [latex]0[\/latex] to [latex]7[\/latex]. It&#8217;s often used in computer programming as it provides a shorthand way of representing binary numbers since each octal digit represents three binary digits.<\/li>\n<li><strong>Hexadecimal (Base-16) System:<\/strong> The hexadecimal system uses sixteen distinct symbols, [latex]0-9[\/latex] and [latex]A-F[\/latex], where [latex]A[\/latex] to [latex]F[\/latex] represent the numbers [latex]10[\/latex] to [latex]15[\/latex]. Hexadecimal is commonly used in programming and computer engineering because it&#8217;s convenient for representing binary code and is human-friendly.<\/li>\n<\/ul>\n<section class=\"textbox proTip\">The base-[latex]10[\/latex] and base-[latex]16[\/latex] systems share the same numerical representation up to [latex]9[\/latex]. In both systems, the number [latex]9[\/latex] is represented as [latex]9[\/latex]. However, when we reach the number [latex]10[\/latex], the hexadecimal system introduces a new convention. Instead of using a single digit, base-[latex]16[\/latex] utilizes the letters [latex]A[\/latex] to [latex]F[\/latex] to represent the values [latex]10[\/latex] to [latex]15[\/latex], respectively. For example:\u00a0 [latex]A[\/latex] represents [latex]10[\/latex], [latex]B[\/latex] represents [latex]11[\/latex], and so on.<\/section>\n<p>Having introduced these systems, let&#8217;s get back to helping Zuri.<\/p>\n<h3>Critical Task 1: Binary Bank Break-In<\/h3>\n<p>Zuri&#8217;s first task is to help Binary Bank. Binary Bank uses a binary (Base-2) system to protect its vaults. A suspicious binary code from the bank&#8217;s logs has been discovered. They suspect it might be a password attempt from an unauthorized user. In order to understand the potential threat to Binary Bank&#8217;s system Zuri needs to convert the binary code from the login to base-[latex]10[\/latex].<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm9042\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=9042&theme=lumen&iframe_resize_id=ohm9042&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>By deciphering the potential password attempt, Zrui managed to stop a possible break-in at Binary Bank!<\/p>\n","protected":false},"author":23,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Image of woman with laptop\",\"author\":\"Christina \",\"organization\":\"Unsplash\",\"url\":\"https:\/\/unsplash.com\/photos\/L85a1k-XqH8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":40,"module-header":"apply_it","content_attributions":[{"type":"copyrighted_video","description":"Image of woman with laptop","author":"Christina ","organization":"Unsplash","url":"https:\/\/unsplash.com\/photos\/L85a1k-XqH8","project":"","license":"arr","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4346"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/23"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4346\/revisions"}],"predecessor-version":[{"id":15565,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4346\/revisions\/15565"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/40"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4346\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4346"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4346"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4346"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}