{"id":195701,"date":"2025-02-28T15:34:47","date_gmt":"2025-02-28T15:34:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195701"},"modified":"2025-02-28T15:34:49","modified_gmt":"2025-02-28T15:34:49","slug":"the-binary-system","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/28\/the-binary-system\/","title":{"rendered":"THE BINARY SYSTEM"},"content":{"rendered":"\n<p>1.1 THE BINARY SYSTEM 3 Decimal Binary Decimal Binary 0 9 10 1 2 3 02 12 102 112 100 1012 1102 1112 1000 11 12 13 10012 1010 10112 1100 1101 11102 11112 10000; 4 5 14 15 16 7 8 Table 1.1 00178950 EXERCISES 1.1 1. Continue the counting started in Table 1.1; that is, count in binary from 17 to 32. 2. Write the following decimal numbers in expanded notation: a. 142 b. 7045 3. Convert the following decimal numbers to binary: a. 38 b. 71 e. 142 d. 279 4. Write the following binary numbers in expanded notation: a. 10011012 b. 10100012 5. Convert the following binary numbers to decimal: a. 1011012 b. 10101102 c. 1110011012 6. Add the following binary numbers: a. 10110102 + 1011102 b. 101102 + 10100, + 111002 c. 111112 + 100012 d. 1010101, + 11001112 7. Subtract the following binary numbers: a. 101102 &#8211; 1012 b. 10110112 1101112 c. 111002 &#8211; 100112 d. 1111112 &#8211; 101012 8. Multiply the following binary numbers: a. 1101, x 1102 b. 1010,<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s go step-by-step through the exercises.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. Counting in binary from 17 to 32:<\/h3>\n\n\n\n<p>Starting from 17 in decimal and converting it to binary, we get:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>17 \u2192 10001<\/li>\n\n\n\n<li>18 \u2192 10010<\/li>\n\n\n\n<li>19 \u2192 10011<\/li>\n\n\n\n<li>20 \u2192 10100<\/li>\n\n\n\n<li>21 \u2192 10101<\/li>\n\n\n\n<li>22 \u2192 10110<\/li>\n\n\n\n<li>23 \u2192 10111<\/li>\n\n\n\n<li>24 \u2192 11000<\/li>\n\n\n\n<li>25 \u2192 11001<\/li>\n\n\n\n<li>26 \u2192 11010<\/li>\n\n\n\n<li>27 \u2192 11011<\/li>\n\n\n\n<li>28 \u2192 11100<\/li>\n\n\n\n<li>29 \u2192 11101<\/li>\n\n\n\n<li>30 \u2192 11110<\/li>\n\n\n\n<li>31 \u2192 11111<\/li>\n\n\n\n<li>32 \u2192 100000<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. Writing decimal numbers in expanded notation:<\/h3>\n\n\n\n<p>a. <strong>142 in expanded notation<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>142 = (1 \\times 10^2 + 4 \\times 10^1 + 2 \\times 10^0)<\/li>\n<\/ul>\n\n\n\n<p>b. <strong>7045 in expanded notation<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>7045 = (7 \\times 10^3 + 0 \\times 10^2 + 4 \\times 10^1 + 5 \\times 10^0)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. Converting decimal numbers to binary:<\/h3>\n\n\n\n<p>a. <strong>38 to binary<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>38 \u00f7 2 = 19 remainder 0<\/li>\n\n\n\n<li>19 \u00f7 2 = 9 remainder 1<\/li>\n\n\n\n<li>9 \u00f7 2 = 4 remainder 1<\/li>\n\n\n\n<li>4 \u00f7 2 = 2 remainder 0<\/li>\n\n\n\n<li>2 \u00f7 2 = 1 remainder 0<\/li>\n\n\n\n<li>1 \u00f7 2 = 0 remainder 1<\/li>\n<\/ul>\n\n\n\n<p>So, 38 in binary is <strong>100110<\/strong>.<\/p>\n\n\n\n<p>b. <strong>71 to binary<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>71 \u00f7 2 = 35 remainder 1<\/li>\n\n\n\n<li>35 \u00f7 2 = 17 remainder 1<\/li>\n\n\n\n<li>17 \u00f7 2 = 8 remainder 1<\/li>\n\n\n\n<li>8 \u00f7 2 = 4 remainder 0<\/li>\n\n\n\n<li>4 \u00f7 2 = 2 remainder 0<\/li>\n\n\n\n<li>2 \u00f7 2 = 1 remainder 0<\/li>\n\n\n\n<li>1 \u00f7 2 = 0 remainder 1<\/li>\n<\/ul>\n\n\n\n<p>So, 71 in binary is <strong>1000111<\/strong>.<\/p>\n\n\n\n<p>c. <strong>142 to binary<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>142 \u00f7 2 = 71 remainder 0<\/li>\n\n\n\n<li>71 \u00f7 2 = 35 remainder 1<\/li>\n\n\n\n<li>35 \u00f7 2 = 17 remainder 1<\/li>\n\n\n\n<li>17 \u00f7 2 = 8 remainder 1<\/li>\n\n\n\n<li>8 \u00f7 2 = 4 remainder 0<\/li>\n\n\n\n<li>4 \u00f7 2 = 2 remainder 0<\/li>\n\n\n\n<li>2 \u00f7 2 = 1 remainder 0<\/li>\n\n\n\n<li>1 \u00f7 2 = 0 remainder 1<\/li>\n<\/ul>\n\n\n\n<p>So, 142 in binary is <strong>10001110<\/strong>.<\/p>\n\n\n\n<p>d. <strong>279 to binary<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>279 \u00f7 2 = 139 remainder 1<\/li>\n\n\n\n<li>139 \u00f7 2 = 69 remainder 1<\/li>\n\n\n\n<li>69 \u00f7 2 = 34 remainder 1<\/li>\n\n\n\n<li>34 \u00f7 2 = 17 remainder 0<\/li>\n\n\n\n<li>17 \u00f7 2 = 8 remainder 1<\/li>\n\n\n\n<li>8 \u00f7 2 = 4 remainder 0<\/li>\n\n\n\n<li>4 \u00f7 2 = 2 remainder 0<\/li>\n\n\n\n<li>2 \u00f7 2 = 1 remainder 0<\/li>\n\n\n\n<li>1 \u00f7 2 = 0 remainder 1<\/li>\n<\/ul>\n\n\n\n<p>So, 279 in binary is <strong>100010111<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. Writing binary numbers in expanded notation:<\/h3>\n\n\n\n<p>a. <strong>1001101\u2082<\/strong> in expanded notation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1001101\u2082 = (1 \\times 2^6 + 0 \\times 2^5 + 0 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 0 \\times 2^1 + 1 \\times 2^0)<\/li>\n\n\n\n<li>This gives: 1 \u00d7 64 + 0 \u00d7 32 + 0 \u00d7 16 + 1 \u00d7 8 + 1 \u00d7 4 + 0 \u00d7 2 + 1 \u00d7 1<\/li>\n\n\n\n<li>Which equals: <strong>64 + 8 + 4 + 1 = 77<\/strong><\/li>\n<\/ul>\n\n\n\n<p>b. <strong>1010001\u2082<\/strong> in expanded notation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1010001\u2082 = (1 \\times 2^6 + 0 \\times 2^5 + 1 \\times 2^4 + 0 \\times 2^3 + 0 \\times 2^2 + 0 \\times 2^1 + 1 \\times 2^0)<\/li>\n\n\n\n<li>This gives: 1 \u00d7 64 + 0 \u00d7 32 + 1 \u00d7 16 + 0 \u00d7 8 + 0 \u00d7 4 + 0 \u00d7 2 + 1 \u00d7 1<\/li>\n\n\n\n<li>Which equals: <strong>64 + 16 + 1 = 81<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">5. Converting binary numbers to decimal:<\/h3>\n\n\n\n<p>a. <strong>101101\u2082<\/strong> to decimal:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>101101\u2082 = (1 \\times 2^5 + 0 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 0 \\times 2^1 + 1 \\times 2^0)<\/li>\n\n\n\n<li>This gives: 1 \u00d7 32 + 0 \u00d7 16 + 1 \u00d7 8 + 1 \u00d7 4 + 0 \u00d7 2 + 1 \u00d7 1<\/li>\n\n\n\n<li>Which equals: <strong>32 + 8 + 4 + 1 = 45<\/strong><\/li>\n<\/ul>\n\n\n\n<p>b. <strong>1010110\u2082<\/strong> to decimal:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1010110\u2082 = (1 \\times 2^6 + 0 \\times 2^5 + 1 \\times 2^4 + 0 \\times 2^3 + 1 \\times 2^2 + 1 \\times 2^1 + 0 \\times 2^0)<\/li>\n\n\n\n<li>This gives: 1 \u00d7 64 + 0 \u00d7 32 + 1 \u00d7 16 + 0 \u00d7 8 + 1 \u00d7 4 + 1 \u00d7 2 + 0 \u00d7 1<\/li>\n\n\n\n<li>Which equals: <strong>64 + 16 + 4 + 2 = 86<\/strong><\/li>\n<\/ul>\n\n\n\n<p>c. <strong>111001101\u2082<\/strong> to decimal:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>111001101\u2082 = (1 \\times 2^8 + 1 \\times 2^7 + 1 \\times 2^6 + 0 \\times 2^5 + 0 \\times 2^4 + 1 \\times 2^3 + 1 \\times 2^2 + 0 \\times 2^1 + 1 \\times 2^0)<\/li>\n\n\n\n<li>This gives: 1 \u00d7 256 + 1 \u00d7 128 + 1 \u00d7 64 + 0 \u00d7 32 + 0 \u00d7 16 + 1 \u00d7 8 + 1 \u00d7 4 + 0 \u00d7 2 + 1 \u00d7 1<\/li>\n\n\n\n<li>Which equals: <strong>256 + 128 + 64 + 8 + 4 + 1 = 461<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">6. Binary addition:<\/h3>\n\n\n\n<p>a. <strong>1011010\u2082 + 101110\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   1011010\n+    101110\n-----------\n   10010100<\/code><\/pre>\n\n\n\n<p>Result: <strong>10010100\u2082<\/strong> (148 in decimal)<\/p>\n\n\n\n<p>b. <strong>10110\u2082 + 10100\u2082 + 11100\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>    10110\n+    10100\n+    11100\n-----------\n    111100<\/code><\/pre>\n\n\n\n<p>Result: <strong>111100\u2082<\/strong> (60 in decimal)<\/p>\n\n\n\n<p>c. <strong>11111\u2082 + 10001\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>    11111\n+    10001\n-----------\n   101000<\/code><\/pre>\n\n\n\n<p>Result: <strong>101000\u2082<\/strong> (40 in decimal)<\/p>\n\n\n\n<p>d. <strong>1010101\u2082 + 1100111\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   1010101\n+  1100111\n-----------\n   10111000<\/code><\/pre>\n\n\n\n<p>Result: <strong>10111000\u2082<\/strong> (184 in decimal)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">7. Binary subtraction:<\/h3>\n\n\n\n<p>a. <strong>10110\u2082 &#8211; 101\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   10110\n-    101\n-----------\n    10001<\/code><\/pre>\n\n\n\n<p>Result: <strong>10001\u2082<\/strong> (17 in decimal)<\/p>\n\n\n\n<p>b. <strong>1011011\u2082 &#8211; 110111\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   1011011\n-   110111\n-----------\n    110100<\/code><\/pre>\n\n\n\n<p>Result: <strong>110100\u2082<\/strong> (52 in decimal)<\/p>\n\n\n\n<p>c. <strong>11100\u2082 &#8211; 10011\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   11100\n-   10011\n-----------\n    01001<\/code><\/pre>\n\n\n\n<p>Result: <strong>01001\u2082<\/strong> (9 in decimal)<\/p>\n\n\n\n<p>d. <strong>111111\u2082 &#8211; 10101\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   111111\n-   10101\n-----------\n    101010<\/code><\/pre>\n\n\n\n<p>Result: <strong>101010\u2082<\/strong> (42 in decimal)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">8. Binary multiplication:<\/h3>\n\n\n\n<p>a. <strong>1101\u2082 \u00d7 110\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>    1101\n\u00d7    110\n-----------\n    0000\n+   1101\n+  11010\n-----------\n   100110<\/code><\/pre>\n\n\n\n<p>Result: <strong>100110\u2082<\/strong> (38 in decimal)<\/p>\n\n\n\n<p>b. <strong>1010\u2082 \u00d7 1010\u2082<\/strong>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>    1010\n\u00d7    1010\n-----------\n    0000\n+   1010\n+  10100\n-----------\n   110100<\/code><\/pre>\n\n\n\n<p>Result: <strong>110100\u2082<\/strong> (52 in decimal)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>Binary numbers work on a base-2 system, where each digit is a power of 2. In expanded notation, binary numbers are written as a sum of powers of 2, similar to how decimal numbers are written as sums of powers of 10. To convert decimal numbers to binary, you divide by 2 repeatedly and record the remainders. Binary arithmetic (addition, subtraction, multiplication) follows the same basic rules as decimal, but it uses base-2 and simple carry\/borrow techniques.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1.1 THE BINARY SYSTEM 3 Decimal Binary Decimal Binary 0 9 10 1 2 3 02 12 102 112 100 1012 1102 1112 1000 11 12 13 10012 1010 10112 1100 1101 11102 11112 10000; 4 5 14 15 16 7 8 Table 1.1 00178950 EXERCISES 1.1 1. Continue the counting started in Table 1.1; [&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-195701","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195701","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=195701"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195701\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}