{"id":3196,"date":"2025-06-05T02:45:17","date_gmt":"2025-06-05T02:45:17","guid":{"rendered":"https:\/\/diznr.com\/?p=3196"},"modified":"2025-06-05T02:45:17","modified_gmt":"2025-06-05T02:45:17","slug":"basic-of-digital-logic-number-systemcodesarithmetic-operationssigned-representation-number","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/basic-of-digital-logic-number-systemcodesarithmetic-operationssigned-representation-number\/","title":{"rendered":"Basic of digital logic- Number system,Codes,Arithmetic Operations,Signed Number Representation."},"content":{"rendered":"<p>Basic of digital logic- Number system,Codes,Arithmetic Operations,Signed Number Representation.<\/p>\n<p>[fvplayer id=&#8221;294&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"31\"><strong data-start=\"2\" data-end=\"29\">Basics of Digital Logic<\/strong><\/h3>\n<p data-start=\"33\" data-end=\"298\">Digital logic forms the foundation of <strong data-start=\"71\" data-end=\"114\">computer systems and electronic devices<\/strong>. It is based on binary operations and logical computations using <strong data-start=\"180\" data-end=\"262\">number systems, codes, arithmetic operations, and signed number representation<\/strong>. Let&#8217;s break down these concepts:<\/p>\n<h3 data-start=\"305\" data-end=\"351\"><strong data-start=\"308\" data-end=\"349\">\u00a01. Number Systems in Digital Logic<\/strong><\/h3>\n<p data-start=\"353\" data-end=\"474\">A <strong data-start=\"355\" data-end=\"372\">number system<\/strong> defines how numbers are represented and used in digital circuits. The four main number systems are:<\/p>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"476\" data-end=\"795\">\n<thead data-start=\"476\" data-end=\"548\">\n<tr data-start=\"476\" data-end=\"548\">\n<th data-start=\"476\" data-end=\"496\"><strong data-start=\"478\" data-end=\"495\">Number System<\/strong><\/th>\n<th data-start=\"496\" data-end=\"515\"><strong data-start=\"498\" data-end=\"514\">Base (Radix)<\/strong><\/th>\n<th data-start=\"515\" data-end=\"533\"><strong data-start=\"517\" data-end=\"532\">Digits Used<\/strong><\/th>\n<th data-start=\"533\" data-end=\"548\"><strong data-start=\"535\" data-end=\"546\">Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"616\" data-end=\"795\">\n<tr data-start=\"616\" data-end=\"658\">\n<td><strong data-start=\"618\" data-end=\"637\">Binary (Base 2)<\/strong><\/td>\n<td>2<\/td>\n<td>0, 1<\/td>\n<td>1011\u2082<\/td>\n<\/tr>\n<tr data-start=\"659\" data-end=\"697\">\n<td><strong data-start=\"661\" data-end=\"679\">Octal (Base 8)<\/strong><\/td>\n<td>8<\/td>\n<td>0-7<\/td>\n<td>57\u2088<\/td>\n<\/tr>\n<tr data-start=\"698\" data-end=\"741\">\n<td><strong data-start=\"700\" data-end=\"721\">Decimal (Base 10)<\/strong><\/td>\n<td>10<\/td>\n<td>0-9<\/td>\n<td>45\u2081\u2080<\/td>\n<\/tr>\n<tr data-start=\"742\" data-end=\"795\">\n<td><strong data-start=\"744\" data-end=\"769\">Hexadecimal (Base 16)<\/strong><\/td>\n<td>16<\/td>\n<td>0-9, A-F<\/td>\n<td>1A3\u2081\u2086<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3 data-start=\"797\" data-end=\"842\"><strong data-start=\"801\" data-end=\"840\">Conversions Between Number Systems:<\/strong><\/h3>\n<p data-start=\"843\" data-end=\"1202\"><strong data-start=\"846\" data-end=\"868\">Binary to Decimal:<\/strong> Multiply each binary digit by <strong data-start=\"899\" data-end=\"913\">2^position<\/strong> and sum them.<br data-start=\"927\" data-end=\"930\" \/><strong data-start=\"933\" data-end=\"955\">Decimal to Binary:<\/strong> Repeatedly <strong data-start=\"967\" data-end=\"982\">divide by 2<\/strong> and record the remainders.<br data-start=\"1009\" data-end=\"1012\" \/><strong data-start=\"1015\" data-end=\"1041\">Binary to Hexadecimal:<\/strong> Group <strong data-start=\"1048\" data-end=\"1058\">4 bits<\/strong> together and convert them into a hexadecimal digit.<br data-start=\"1110\" data-end=\"1113\" \/><strong data-start=\"1116\" data-end=\"1136\">Binary to Octal:<\/strong> Group <strong data-start=\"1143\" data-end=\"1153\">3 bits<\/strong> together and convert them into an octal digit.<\/p>\n<p data-start=\"1204\" data-end=\"1250\"><strong data-start=\"1207\" data-end=\"1219\">Example:<\/strong> Convert <strong data-start=\"1228\" data-end=\"1237\">1011\u2082<\/strong> to Decimal<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">1\u00d723+0\u00d722+1\u00d721+1\u00d720=8+0+2+1=11101 \u00d7 2^3 + 0 \u00d7 2^2 + 1 \u00d7 2^1 + 1 \u00d7 2^0 = 8 + 0 + 2 + 1 = 11\u2081\u2080<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">2<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">2<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">2<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">2<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mord\">1<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"1324\" data-end=\"1370\"><strong data-start=\"1327\" data-end=\"1368\">\u00a02. Digital Codes in Logic Circuits<\/strong><\/h3>\n<p data-start=\"1372\" data-end=\"1472\">Binary codes are used to represent data in digital systems. Some important coding schemes include<strong data-start=\"1478\" data-end=\"1514\">a) BCD (Binary-Coded Decimal)<\/strong><\/p>\n<ul data-start=\"1515\" data-end=\"1612\">\n<li data-start=\"1515\" data-end=\"1577\">Each decimal digit (0-9) is represented by <strong data-start=\"1560\" data-end=\"1576\">4-bit binary<\/strong>.<\/li>\n<li data-start=\"1578\" data-end=\"1612\">Example: <strong data-start=\"1589\" data-end=\"1604\">9\u2081\u2080 = 1001\u2082<\/strong> in BCD.<\/li>\n<\/ul>\n<h3 data-start=\"1614\" data-end=\"1637\"><strong data-start=\"1618\" data-end=\"1637\">\u00a0b) Gray Code<\/strong><\/h3>\n<ul data-start=\"1638\" data-end=\"1824\">\n<li data-start=\"1638\" data-end=\"1713\">A <strong data-start=\"1642\" data-end=\"1657\">binary code<\/strong> where two successive values differ by only <strong data-start=\"1701\" data-end=\"1712\">one bit<\/strong>.<\/li>\n<li data-start=\"1714\" data-end=\"1786\">Used to <strong data-start=\"1724\" data-end=\"1761\">reduce errors in digital circuits<\/strong> (e.g., rotary encoders).<\/li>\n<li data-start=\"1787\" data-end=\"1824\">Example: <strong data-start=\"1798\" data-end=\"1823\">Binary 110 \u2192 Gray 101<\/strong>.<\/li>\n<\/ul>\n<h3 data-start=\"1826\" data-end=\"1898\"><strong data-start=\"1830\" data-end=\"1898\">\u00a0c) ASCII (American Standard Code for Information Interchange)<\/strong><\/h3>\n<ul data-start=\"1899\" data-end=\"1995\">\n<li data-start=\"1899\" data-end=\"1959\"><strong data-start=\"1901\" data-end=\"1919\">7-bit or 8-bit<\/strong> codes used to represent <strong data-start=\"1944\" data-end=\"1958\">characters<\/strong>.<\/li>\n<li data-start=\"1960\" data-end=\"1995\">Example: <strong data-start=\"1971\" data-end=\"1994\">A = 65\u2081\u2080 = 1000001\u2082<\/strong>.<\/li>\n<\/ul>\n<h3 data-start=\"1997\" data-end=\"2024\"><strong data-start=\"2001\" data-end=\"2024\">\u00a0d) Excess-3 Code<\/strong><\/h3>\n<ul data-start=\"2025\" data-end=\"2142\">\n<li data-start=\"2025\" data-end=\"2081\">A self-complementary code used in <strong data-start=\"2061\" data-end=\"2080\">error detection<\/strong>.<\/li>\n<li data-start=\"2082\" data-end=\"2142\">Example: <strong data-start=\"2093\" data-end=\"2141\">Decimal 5 \u2192 Binary 0101 \u2192 Excess-3 Code 1000<\/strong>.<\/li>\n<\/ul>\n<h3 data-start=\"2149\" data-end=\"2202\"><strong data-start=\"2152\" data-end=\"2200\">\u00a03. Arithmetic Operations in Digital Logic<\/strong><\/h3>\n<h3 data-start=\"2204\" data-end=\"2235\"><strong data-start=\"2208\" data-end=\"2233\">\u00a0a) Binary Addition<\/strong><\/h3>\n<p data-start=\"2236\" data-end=\"2292\">Similar to decimal addition but uses <strong data-start=\"2273\" data-end=\"2289\">only 0 and 1<\/strong>.<\/p>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"2294\" data-end=\"2423\">\n<thead data-start=\"2294\" data-end=\"2323\">\n<tr data-start=\"2294\" data-end=\"2323\">\n<th data-start=\"2294\" data-end=\"2298\">A<\/th>\n<th data-start=\"2298\" data-end=\"2302\">B<\/th>\n<th data-start=\"2302\" data-end=\"2314\">Sum (A+B)<\/th>\n<th data-start=\"2314\" data-end=\"2323\">Carry<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2352\" data-end=\"2423\">\n<tr data-start=\"2352\" data-end=\"2369\">\n<td>0<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr data-start=\"2370\" data-end=\"2387\">\n<td>0<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr data-start=\"2388\" data-end=\"2405\">\n<td>1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr data-start=\"2406\" data-end=\"2423\">\n<td>1<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p data-start=\"2425\" data-end=\"2442\"><strong data-start=\"2428\" data-end=\"2440\">Example:<\/strong><\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">   1011<br \/>\n<span class=\"hljs-section\">+  0110<br \/>\n------------<\/span><br \/>\n10001 (Binary Sum)<br \/>\n<\/code><\/div>\n<\/div>\n<h3 data-start=\"2502\" data-end=\"2559\"><strong data-start=\"2506\" data-end=\"2557\">\u00a0b) Binary Subtraction (Using 2\u2019s Complement)<\/strong><\/h3>\n<ul data-start=\"2560\" data-end=\"2679\">\n<li data-start=\"2560\" data-end=\"2630\">To subtract <strong data-start=\"2574\" data-end=\"2583\">A &#8211; B<\/strong>, take <strong data-start=\"2590\" data-end=\"2608\">2\u2019s complement<\/strong> of B and add it to A.<\/li>\n<li data-start=\"2631\" data-end=\"2679\"><strong data-start=\"2633\" data-end=\"2651\">2\u2019s Complement<\/strong>: Invert the bits and add 1.<\/li>\n<\/ul>\n<p data-start=\"2681\" data-end=\"2726\"><strong data-start=\"2684\" data-end=\"2696\">Example:<\/strong><br data-start=\"2696\" data-end=\"2699\" \/>Find <strong data-start=\"2704\" data-end=\"2713\">5 &#8211; 3<\/strong> in Binary:<\/p>\n<ul data-start=\"2727\" data-end=\"2808\">\n<li data-start=\"2727\" data-end=\"2743\">5 \u2192 <strong data-start=\"2733\" data-end=\"2741\">0101<\/strong><\/li>\n<li data-start=\"2744\" data-end=\"2760\">3 \u2192 <strong data-start=\"2750\" data-end=\"2758\">0011<\/strong><\/li>\n<li data-start=\"2761\" data-end=\"2799\">2\u2019s Complement of <strong data-start=\"2781\" data-end=\"2786\">3<\/strong> \u2192 <strong data-start=\"2789\" data-end=\"2797\">1101<\/strong><\/li>\n<li data-start=\"2800\" data-end=\"2808\">Add:<\/li>\n<\/ul>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">  0101<br \/>\n<span class=\"hljs-section\">+ 1101<br \/>\n-----------<\/span><br \/>\n0010  (2 in Decimal)<br \/>\n<\/code><\/div>\n<\/div>\n<h3 data-start=\"2867\" data-end=\"2904\"><strong data-start=\"2871\" data-end=\"2902\">\u00a0c) Binary Multiplication<\/strong><\/h3>\n<p data-start=\"2905\" data-end=\"2988\">Follows the same rules as decimal multiplication but only involves <strong data-start=\"2972\" data-end=\"2985\">0s and 1s<\/strong>.<\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">  101 (5)<br \/>\n<span class=\"hljs-section\">\u00d7  11 (3)<br \/>\n------------<\/span><br \/>\n101<br \/>\n<span class=\"hljs-section\">+1010  (Shifted Left)<br \/>\n------------<\/span><br \/>\n1111  (15 in Decimal)<br \/>\n<\/code><\/div>\n<\/div>\n<hr data-start=\"3098\" data-end=\"3101\" \/>\n<h3 data-start=\"3103\" data-end=\"3146\"><strong data-start=\"3106\" data-end=\"3144\">\u00a04. Signed Number Representation<\/strong><\/h3>\n<p data-start=\"3148\" data-end=\"3249\">In digital systems, <strong data-start=\"3168\" data-end=\"3186\">signed numbers<\/strong> are used to represent <strong data-start=\"3209\" data-end=\"3246\">both positive and negative values<\/strong>.<\/p>\n<h3 data-start=\"3251\" data-end=\"3296\"><strong data-start=\"3255\" data-end=\"3294\">\u00a0a) Sign-Magnitude Representation<\/strong><\/h3>\n<ul data-start=\"3297\" data-end=\"3462\">\n<li data-start=\"3297\" data-end=\"3393\">The <strong data-start=\"3303\" data-end=\"3333\">MSB (Most Significant Bit)<\/strong> is used as the <strong data-start=\"3349\" data-end=\"3361\">sign bit<\/strong> (0 = Positive, 1 = Negative).<\/li>\n<li data-start=\"3394\" data-end=\"3462\">Example: <strong data-start=\"3405\" data-end=\"3418\">+5 = 0101<\/strong>, <strong data-start=\"3420\" data-end=\"3433\">-5 = 1101<\/strong> (in 4-bit representation).<\/li>\n<\/ul>\n<h3 data-start=\"3464\" data-end=\"3494\"><strong data-start=\"3468\" data-end=\"3492\">\u00a0b) 1\u2019s Complement<\/strong><\/h3>\n<ul data-start=\"3495\" data-end=\"3583\">\n<li data-start=\"3495\" data-end=\"3545\"><strong data-start=\"3497\" data-end=\"3516\">Invert all bits<\/strong> to get the negative value.<\/li>\n<li data-start=\"3546\" data-end=\"3583\">Example: <strong data-start=\"3557\" data-end=\"3582\">+5 (0101) \u2192 -5 (1010)<\/strong>.<\/li>\n<\/ul>\n<h3 data-start=\"3585\" data-end=\"3629\"><strong data-start=\"3589\" data-end=\"3627\">\u00a0c) 2\u2019s Complement (Most Common)<\/strong><\/h3>\n<ul data-start=\"3630\" data-end=\"3771\">\n<li data-start=\"3630\" data-end=\"3668\"><strong data-start=\"3632\" data-end=\"3651\">Invert the bits<\/strong> and <strong data-start=\"3656\" data-end=\"3665\">add 1<\/strong>.<\/li>\n<li data-start=\"3669\" data-end=\"3708\">Example: <strong data-start=\"3680\" data-end=\"3705\">+5 (0101) \u2192 -5 (1011)<\/strong>.<\/li>\n<li data-start=\"3709\" data-end=\"3771\">Used in most computers for <strong data-start=\"3738\" data-end=\"3768\">easy arithmetic operations<\/strong>.<\/li>\n<\/ul>\n<p data-start=\"3773\" data-end=\"3968\"><strong data-start=\"3776\" data-end=\"3803\">Why Use 2\u2019s Complement?<\/strong><br data-start=\"3803\" data-end=\"3806\" \/><strong data-start=\"3808\" data-end=\"3850\">No separate subtraction circuit needed<\/strong>\u2014addition handles both positive and negative numbers.<br data-start=\"3903\" data-end=\"3906\" \/><strong data-start=\"3908\" data-end=\"3941\">Only one representation for 0<\/strong> (unlike 1\u2019s complement).<\/p>\n<h3 data-start=\"3975\" data-end=\"4016\"><strong data-start=\"3978\" data-end=\"4014\">\u00a0Applications of Digital Logic<\/strong><\/h3>\n<p data-start=\"4017\" data-end=\"4351\"><strong data-start=\"4019\" data-end=\"4050\">Microprocessors &amp; Computers<\/strong> \u2013 Use binary logic for computations.<br data-start=\"4087\" data-end=\"4090\" \/><strong data-start=\"4092\" data-end=\"4112\">Embedded Systems<\/strong> \u2013 Digital circuits in <strong data-start=\"4135\" data-end=\"4177\">automobiles, IoT devices, and robotics<\/strong>.<br data-start=\"4178\" data-end=\"4181\" \/><strong data-start=\"4183\" data-end=\"4208\">Digital Communication<\/strong> \u2013 Used in <strong data-start=\"4219\" data-end=\"4266\">networking, encoding, and data transmission<\/strong>.<br data-start=\"4267\" data-end=\"4270\" \/><strong data-start=\"4272\" data-end=\"4299\">Cryptography &amp; Security<\/strong> \u2013 Binary operations in <strong data-start=\"4323\" data-end=\"4348\">encryption algorithms<\/strong>.<\/p>\n<h3 data-start=\"4358\" data-end=\"4380\"><strong data-start=\"4361\" data-end=\"4378\">\u00a0Conclusion<\/strong><\/h3>\n<p data-start=\"4381\" data-end=\"4597\">Digital logic is the <strong data-start=\"4402\" data-end=\"4436\">foundation of modern computing<\/strong>. Understanding <strong data-start=\"4452\" data-end=\"4546\">number systems, coding techniques, arithmetic operations, and signed number representation<\/strong> is crucial for <strong data-start=\"4562\" data-end=\"4594\">designing efficient circuits<\/strong>.<\/p>\n<p data-start=\"4599\" data-end=\"4693\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"4602\" data-end=\"4690\">Would you like additional practice problems or real-world applications on any topic?<\/strong><\/p>\n<p data-start=\"0\" data-end=\"241\">Here is a clear and concise explanation of the <strong data-start=\"47\" data-end=\"74\">Basics of Digital Logic<\/strong>, covering the key topics: <strong data-start=\"101\" data-end=\"183\">Number Systems, Codes, Arithmetic Operations, and Signed Number Representation<\/strong> \u2014 suitable for students, beginners, or revision material.<\/p>\n<hr data-start=\"243\" data-end=\"246\" \/>\n<h1 data-start=\"248\" data-end=\"280\">\ud83d\udca1 <strong data-start=\"253\" data-end=\"280\">Basics of Digital Logic<\/strong><\/h1>\n<p data-start=\"282\" data-end=\"476\">Digital logic is the foundation of computer systems, electronics, and digital devices. It uses <strong data-start=\"377\" data-end=\"405\">binary signals (0 and 1)<\/strong> to represent and process data. Let&#8217;s explore the core building blocks:<\/p>\n<hr data-start=\"478\" data-end=\"481\" \/>\n<h2 data-start=\"483\" data-end=\"510\">\ud83d\udd22 <strong data-start=\"489\" data-end=\"510\">1. Number Systems<\/strong><\/h2>\n<h3 data-start=\"512\" data-end=\"538\">a. <strong data-start=\"519\" data-end=\"538\">Binary (Base 2)<\/strong><\/h3>\n<ul data-start=\"539\" data-end=\"652\">\n<li data-start=\"539\" data-end=\"555\">\n<p data-start=\"541\" data-end=\"555\">Digits: 0, 1<\/p>\n<\/li>\n<li data-start=\"556\" data-end=\"652\">\n<p data-start=\"558\" data-end=\"652\">Used in computers because hardware circuits understand two states: <strong data-start=\"625\" data-end=\"635\">ON (1)<\/strong> and <strong data-start=\"640\" data-end=\"651\">OFF (0)<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"654\" data-end=\"682\">b. <strong data-start=\"661\" data-end=\"682\">Decimal (Base 10)<\/strong><\/h3>\n<ul data-start=\"683\" data-end=\"731\">\n<li data-start=\"683\" data-end=\"701\">\n<p data-start=\"685\" data-end=\"701\">Digits: 0 to 9<\/p>\n<\/li>\n<li data-start=\"702\" data-end=\"731\">\n<p data-start=\"704\" data-end=\"731\">Common human number system.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"733\" data-end=\"788\">c. <strong data-start=\"740\" data-end=\"758\">Octal (Base 8)<\/strong> and <strong data-start=\"763\" data-end=\"788\">Hexadecimal (Base 16)<\/strong><\/h3>\n<ul data-start=\"789\" data-end=\"893\">\n<li data-start=\"789\" data-end=\"813\">\n<p data-start=\"791\" data-end=\"813\">Octal digits: 0 to 7<\/p>\n<\/li>\n<li data-start=\"814\" data-end=\"847\">\n<p data-start=\"816\" data-end=\"847\">Hex digits: 0 to 9 and A to F<\/p>\n<\/li>\n<li data-start=\"848\" data-end=\"893\">\n<p data-start=\"850\" data-end=\"893\">Used to <strong data-start=\"858\" data-end=\"877\">simplify binary<\/strong> representation.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"895\" data-end=\"917\">\u2705 <strong data-start=\"901\" data-end=\"917\">Conversions:<\/strong><\/h3>\n<ul data-start=\"918\" data-end=\"1045\">\n<li data-start=\"918\" data-end=\"974\">\n<p data-start=\"920\" data-end=\"974\"><strong data-start=\"920\" data-end=\"940\">Binary \u2192 Decimal<\/strong>: Multiply each bit by 2\u207f and sum.<\/p>\n<\/li>\n<li data-start=\"975\" data-end=\"1045\">\n<p data-start=\"977\" data-end=\"1045\"><strong data-start=\"977\" data-end=\"997\">Decimal \u2192 Binary<\/strong>: Divide by 2 repeatedly and collect remainders.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1047\" data-end=\"1050\" \/>\n<h2 data-start=\"1052\" data-end=\"1087\">\ud83e\uddfe <strong data-start=\"1058\" data-end=\"1087\">2. Codes in Digital Logic<\/strong><\/h2>\n<h3 data-start=\"1089\" data-end=\"1126\">a. <strong data-start=\"1096\" data-end=\"1126\">BCD (Binary Coded Decimal)<\/strong><\/h3>\n<ul data-start=\"1127\" data-end=\"1246\">\n<li data-start=\"1127\" data-end=\"1200\">\n<p data-start=\"1129\" data-end=\"1200\">Each decimal digit is represented by <strong data-start=\"1166\" data-end=\"1182\">4-bit binary<\/strong> (e.g., 9 = 1001).<\/p>\n<\/li>\n<li data-start=\"1201\" data-end=\"1246\">\n<p data-start=\"1203\" data-end=\"1246\">Used in <strong data-start=\"1211\" data-end=\"1245\">calculators and digital clocks<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1248\" data-end=\"1268\">b. <strong data-start=\"1255\" data-end=\"1268\">Gray Code<\/strong><\/h3>\n<ul data-start=\"1269\" data-end=\"1353\">\n<li data-start=\"1269\" data-end=\"1306\">\n<p data-start=\"1271\" data-end=\"1306\">Only <strong data-start=\"1276\" data-end=\"1305\">one bit changes at a time<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1307\" data-end=\"1353\">\n<p data-start=\"1309\" data-end=\"1353\">Used in <strong data-start=\"1317\" data-end=\"1336\">rotary encoders<\/strong> to avoid errors.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1355\" data-end=\"1424\">c. <strong data-start=\"1362\" data-end=\"1424\">ASCII (American Standard Code for Information Interchange)<\/strong><\/h3>\n<ul data-start=\"1425\" data-end=\"1496\">\n<li data-start=\"1425\" data-end=\"1496\">\n<p data-start=\"1427\" data-end=\"1496\">Represents <strong data-start=\"1438\" data-end=\"1452\">characters<\/strong> using 7 or 8 bits (e.g., A = 65 = 1000001).<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1498\" data-end=\"1534\">d. <strong data-start=\"1505\" data-end=\"1534\">Excess-3 and Parity Codes<\/strong><\/h3>\n<ul data-start=\"1535\" data-end=\"1602\">\n<li data-start=\"1535\" data-end=\"1602\">\n<p data-start=\"1537\" data-end=\"1602\"><strong data-start=\"1537\" data-end=\"1556\">Error detection<\/strong>, <strong data-start=\"1558\" data-end=\"1576\">data integrity<\/strong> in communication systems.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1604\" data-end=\"1607\" \/>\n<h2 data-start=\"1609\" data-end=\"1642\">\u2795 <strong data-start=\"1614\" data-end=\"1642\">3. Arithmetic Operations<\/strong><\/h2>\n<h3 data-start=\"1644\" data-end=\"1676\">a. <strong data-start=\"1651\" data-end=\"1676\">Binary Addition Rules<\/strong><\/h3>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\"><\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\">0 + 0 = 0<br \/>\n0 + 1 = 1<br \/>\n1 + 0 = 1<br \/>\n1 + 1 = 10 (carry 1)<br \/>\n<\/code><\/div>\n<\/div>\n<h3 data-start=\"1743\" data-end=\"1778\">b. <strong data-start=\"1750\" data-end=\"1778\">Binary Subtraction Rules<\/strong><\/h3>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\"><\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\">0 - 0 = 0<br \/>\n1 - 0 = 1<br \/>\n1 - 1 = 0<br \/>\n0 - 1 = 1 (borrow 1)<br \/>\n<\/code><\/div>\n<\/div>\n<h3 data-start=\"1845\" data-end=\"1890\">c. <strong data-start=\"1852\" data-end=\"1890\">Binary Multiplication and Division<\/strong><\/h3>\n<ul data-start=\"1891\" data-end=\"1942\">\n<li data-start=\"1891\" data-end=\"1942\">\n<p data-start=\"1893\" data-end=\"1942\">Follows same logic as decimal but with 0s and 1s.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1944\" data-end=\"1947\" \/>\n<h2 data-start=\"1949\" data-end=\"1989\">\u2796 <strong data-start=\"1954\" data-end=\"1989\">4. Signed Number Representation<\/strong><\/h2>\n<p data-start=\"1991\" data-end=\"2053\">Digital systems must handle <strong data-start=\"2019\" data-end=\"2052\">positive and negative numbers<\/strong>.<\/p>\n<h3 data-start=\"2055\" data-end=\"2080\">a. <strong data-start=\"2062\" data-end=\"2080\">Sign-Magnitude<\/strong><\/h3>\n<ul data-start=\"2081\" data-end=\"2192\">\n<li data-start=\"2081\" data-end=\"2158\">\n<p data-start=\"2083\" data-end=\"2120\">MSB (Most Significant Bit) is sign:<\/p>\n<ul data-start=\"2123\" data-end=\"2158\">\n<li data-start=\"2123\" data-end=\"2139\">\n<p data-start=\"2125\" data-end=\"2139\">0 = positive<\/p>\n<\/li>\n<li data-start=\"2142\" data-end=\"2158\">\n<p data-start=\"2144\" data-end=\"2158\">1 = negative<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2159\" data-end=\"2192\">\n<p data-start=\"2161\" data-end=\"2192\">Value stored in remaining bits.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"2194\" data-end=\"2219\">b. <strong data-start=\"2201\" data-end=\"2219\">1\u2019s Complement<\/strong><\/h3>\n<ul data-start=\"2220\" data-end=\"2284\">\n<li data-start=\"2220\" data-end=\"2284\">\n<p data-start=\"2222\" data-end=\"2284\">Negative number = <strong data-start=\"2240\" data-end=\"2259\">Invert all bits<\/strong> of the positive version.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"2286\" data-end=\"2325\">c. <strong data-start=\"2293\" data-end=\"2311\">2\u2019s Complement<\/strong> (most common)<\/h3>\n<ul data-start=\"2326\" data-end=\"2464\">\n<li data-start=\"2326\" data-end=\"2361\">\n<p data-start=\"2328\" data-end=\"2361\">Negative = <strong data-start=\"2339\" data-end=\"2361\">1\u2019s complement + 1<\/strong><\/p>\n<\/li>\n<li data-start=\"2362\" data-end=\"2399\">\n<p data-start=\"2364\" data-end=\"2399\">Makes subtraction easier in binary.<\/p>\n<\/li>\n<li data-start=\"2400\" data-end=\"2464\">\n<p data-start=\"2402\" data-end=\"2410\">Range:<\/p>\n<ul data-start=\"2413\" data-end=\"2464\">\n<li data-start=\"2413\" data-end=\"2436\">\n<p data-start=\"2415\" data-end=\"2436\">For 4-bit: \u20138 to +7<\/p>\n<\/li>\n<li data-start=\"2439\" data-end=\"2464\">\n<p data-start=\"2441\" data-end=\"2464\">For 8-bit: \u2013128 to +127<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr data-start=\"2466\" data-end=\"2469\" \/>\n<h2 data-start=\"2471\" data-end=\"2508\">\ud83d\udccc Example: 2\u2019s Complement (4-bit)<\/h2>\n<ul data-start=\"2509\" data-end=\"2579\">\n<li data-start=\"2509\" data-end=\"2524\">\n<p data-start=\"2511\" data-end=\"2524\">+5 = <code data-start=\"2516\" data-end=\"2522\">0101<\/code><\/p>\n<\/li>\n<li data-start=\"2525\" data-end=\"2579\">\n<p data-start=\"2527\" data-end=\"2579\">\u20135 = <code data-start=\"2532\" data-end=\"2538\">1011<\/code> (invert <code data-start=\"2547\" data-end=\"2553\">0101<\/code> \u2192 <code data-start=\"2556\" data-end=\"2562\">1010<\/code>, add 1 \u2192 <code data-start=\"2572\" data-end=\"2578\">1011<\/code>)<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2581\" data-end=\"2584\" \/>\n<h2 data-start=\"2586\" data-end=\"2615\">\ud83e\udde0 <strong data-start=\"2592\" data-end=\"2615\">Quick Summary Table<\/strong><\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"2617\" data-end=\"2866\">\n<thead data-start=\"2617\" data-end=\"2639\">\n<tr data-start=\"2617\" data-end=\"2639\">\n<th data-start=\"2617\" data-end=\"2627\" data-col-size=\"sm\">Concept<\/th>\n<th data-start=\"2627\" data-end=\"2639\" data-col-size=\"md\">Key Idea<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2662\" data-end=\"2866\">\n<tr data-start=\"2662\" data-end=\"2709\">\n<td data-start=\"2662\" data-end=\"2678\" data-col-size=\"sm\">Number System<\/td>\n<td data-col-size=\"md\" data-start=\"2678\" data-end=\"2709\">Binary, Octal, Decimal, Hex<\/td>\n<\/tr>\n<tr data-start=\"2710\" data-end=\"2738\">\n<td data-start=\"2710\" data-end=\"2718\" data-col-size=\"sm\">Codes<\/td>\n<td data-start=\"2718\" data-end=\"2738\" data-col-size=\"md\">BCD, Gray, ASCII<\/td>\n<\/tr>\n<tr data-start=\"2739\" data-end=\"2798\">\n<td data-start=\"2739\" data-end=\"2752\" data-col-size=\"sm\">Arithmetic<\/td>\n<td data-col-size=\"md\" data-start=\"2752\" data-end=\"2798\">Binary addition\/subtraction\/multiplication<\/td>\n<\/tr>\n<tr data-start=\"2799\" data-end=\"2866\">\n<td data-start=\"2799\" data-end=\"2823\" data-col-size=\"sm\">Signed Representation<\/td>\n<td data-col-size=\"md\" data-start=\"2823\" data-end=\"2866\">Sign-magnitude, 1&#8217;s, and 2\u2019s complement<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2868\" data-end=\"2871\" \/>\n<h2 data-start=\"2873\" data-end=\"2893\">\ud83c\udfaf <strong data-start=\"2879\" data-end=\"2893\">Conclusion<\/strong><\/h2>\n<p data-start=\"2895\" data-end=\"2958\">Understanding the <strong data-start=\"2913\" data-end=\"2940\">basics of digital logic<\/strong> is essential for:<\/p>\n<ul data-start=\"2959\" data-end=\"3043\">\n<li data-start=\"2959\" data-end=\"2980\">\n<p data-start=\"2961\" data-end=\"2980\">Digital electronics<\/p>\n<\/li>\n<li data-start=\"2981\" data-end=\"3004\">\n<p data-start=\"2983\" data-end=\"3004\">Computer architecture<\/p>\n<\/li>\n<li data-start=\"3005\" data-end=\"3023\">\n<p data-start=\"3007\" data-end=\"3023\">Embedded systems<\/p>\n<\/li>\n<li data-start=\"3024\" data-end=\"3043\">\n<p data-start=\"3026\" data-end=\"3043\">Programming logic<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"3045\" data-end=\"3048\" \/>\n<p data-start=\"3050\" data-end=\"3073\">Would you like this as:<\/p>\n<ul data-start=\"3074\" data-end=\"3157\">\n<li data-start=\"3074\" data-end=\"3095\">\n<p data-start=\"3076\" data-end=\"3095\">\ud83d\udcdd A printable PDF?<\/p>\n<\/li>\n<li data-start=\"3096\" data-end=\"3124\">\n<p data-start=\"3098\" data-end=\"3124\">\ud83d\udcca A PowerPoint for class?<\/p>\n<\/li>\n<li data-start=\"3125\" data-end=\"3157\">\n<p data-start=\"3127\" data-end=\"3157\">\ud83d\udcf9 A video explanation script?<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3159\" data-end=\"3176\" data-is-last-node=\"\" data-is-only-node=\"\">Just let me know!<\/p>\n<h3 data-start=\"3159\" data-end=\"3176\"><a href=\"https:\/\/www.sctevtservices.nic.in\/docs\/website\/pdf\/140294.pdf\" target=\"_blank\" rel=\"noopener\">Basic of digital logic- Number system,Codes,Arithmetic Operations,Signed Number Representation.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.jsscacs.edu.in\/sites\/default\/files\/Department%20Files\/Number%20System%20.pdf\" target=\"_blank\" rel=\"noopener\">DIGITAL ELECTRONICS Number System<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.sathyabama.ac.in\/sites\/default\/files\/course-material\/2020-10\/unit1_7.pdf\" target=\"_blank\" rel=\"noopener\">1. Number System<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\" style=\"text-align: left\"><a href=\"https:\/\/www.pvpsiddhartha.ac.in\/dep_it\/lecture%20notes\/FDLD\/unit1.pdf\" target=\"_blank\" rel=\"noopener\">Chapter 1: Digital Systems and Binary Numbers<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Basic of digital logic- Number system,Codes,Arithmetic Operations,Signed Number Representation. [fvplayer id=&#8221;294&#8243;] Basics of Digital Logic Digital logic forms the foundation of computer systems and electronic devices. It is based on binary operations and logical computations using number systems, codes, arithmetic operations, and signed number representation. Let&#8217;s break down these concepts: \u00a01. Number Systems in Digital [&hellip;]<\/p>\n","protected":false},"author":66,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111],"tags":[],"class_list":["post-3196","post","type-post","status-publish","format-standard","hentry","category-digital-electronics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3196","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=3196"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3196\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}