{"id":41415,"date":"2026-09-16T11:13:00","date_gmt":"2026-09-16T11:13:00","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=41415"},"modified":"2026-09-17T09:33:00","modified_gmt":"2026-09-17T09:33:00","slug":"octal-number-system","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/octal-number-system\/","title":{"rendered":"Octal Number System Explained With Examples"},"content":{"rendered":"\t\t
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\r\n Imagine counting normally, but after 7<\/strong>, instead of writing 8, you write 10<\/strong>. Sounds strange at first, right? That is exactly how counting works in the octal number system<\/strong>.\r\n <\/p>\r\n\r\n

\r\n

\r\n The octal number system is a base-8 positional number system<\/strong> that uses only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7<\/strong>. Each position represents a power of 8 rather than a power of 10.\r\n <\/p>\r\n <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n How Does the Octal Number System Work?\r\n <\/h2>\r\n\r\n

\r\n We normally use the decimal system, or base 10, which has ten digits from 0 to 9. Octal uses only eight digits, so after 7, the next number is 10.\r\n <\/p>\r\n\r\n

\r\n

Here is a quick comparison:<\/p>\r\n <\/div>\r\n\r\n

\r\n \r\n \r\n \r\n \r\n
Decimal<\/th>\r\n Octal<\/th>\r\n <\/tr>\r\n <\/thead>\r\n
0<\/td>0<\/td><\/tr>\r\n
1<\/td>1<\/td><\/tr>\r\n
2<\/td>2<\/td><\/tr>\r\n
3<\/td>3<\/td><\/tr>\r\n
4<\/td>4<\/td><\/tr>\r\n
5<\/td>5<\/td><\/tr>\r\n
6<\/td>6<\/td><\/tr>\r\n
7<\/td>7<\/td><\/tr>\r\n
8<\/td>10<\/td><\/tr>\r\n
9<\/td>11<\/td><\/tr>\r\n
10<\/td>12<\/td><\/tr>\r\n <\/tbody>\r\n <\/table>\r\n <\/div>\r\n\r\n
\r\n

\r\n So, 10 in octal does not mean ten<\/strong>. It means eight in decimal<\/strong>.\r\n <\/p>\r\n

\r\n That small difference is the key to understanding the whole octal number system.\r\n <\/p>\r\n <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n What Are the Place Values in Octal?\r\n <\/h2>\r\n\r\n

\r\n Just as decimal numbers use powers of 10, octal numbers use powers of 8.\r\n <\/p>\r\n\r\n

\r\n

\r\n For example, consider 345\u2088<\/span>:\r\n <\/p>\r\n <\/div>\r\n\r\n

\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
Digit<\/th>\r\n Place value<\/th>\r\n Value<\/th>\r\n <\/tr>\r\n <\/thead>\r\n
3<\/td>\r\n 8\u00b2 = 64<\/td>\r\n 3 \u00d7 64 = 192<\/td>\r\n <\/tr>\r\n

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<\/td>\r\n
<\/td>\r\n <\/tr>\r\n
4<\/td>\r\n 8\u00b9 = 8<\/td>\r\n 4 \u00d7 8 = 32<\/td>\r\n <\/tr>\r\n
5<\/td>\r\n 8\u2070 = 1<\/td>\r\n 5 \u00d7 1 = 5<\/td>\r\n <\/tr>\r\n <\/tbody>\r\n <\/table>\r\n <\/div>\r\n\r\n
\r\n

\r\n Add them:
\r\n 192 + 32 + 5 = 229<\/strong>\r\n <\/p>\r\n

\r\n Therefore,
\r\n 345\u2088 = 229\u2081\u2080<\/strong>\r\n <\/p>\r\n <\/div>\r\n\r\n

\r\n The small subscript tells us which number system we are using. This matters because the same written digits can represent different values in different bases. NCERT explains this positional relationship as a fundamental part of number systems.\r\n <\/p>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n Why Does Octal Connect So Easily to Binary?\r\n <\/h2>\r\n\r\n
\r\n

\r\n Here is the clever part.\r\n <\/p>\r\n

\r\n 8 = 2\u00b3\r\n <\/p>\r\n <\/div>\r\n\r\n

\r\n That means every octal digit can be represented using exactly 3 binary bits<\/strong>.\r\n <\/p>\r\n\r\n

\r\n \r\n \r\n \r\n \r\n
Octal<\/th>\r\n Binary<\/th>\r\n <\/tr>\r\n <\/thead>\r\n
0<\/td>000<\/td><\/tr>\r\n
1<\/td>001<\/td><\/tr>\r\n
2<\/td>010<\/td><\/tr>\r\n
3<\/td>011<\/td><\/tr>\r\n
4<\/td>100<\/td><\/tr>\r\n
5<\/td>101<\/td><\/tr>\r\n
6<\/td>110<\/td><\/tr>\r\n
7<\/td>111<\/td><\/tr>\r\n <\/tbody>\r\n <\/table>\r\n <\/div>\r\n\r\n
\r\n

For example:<\/strong><\/p>\r\n

57\u2088<\/p>\r\n

5 \u2192 101<\/p>\r\n

7 \u2192 111<\/p>\r\n

57\u2088 = 101111\u2082<\/p>\r\n <\/div>\r\n\r\n

\r\n This is why octal was useful for representing binary information more compactly. NCERT specifically describes octal as a compact representation of binary numbers and explains the three-bit relationship.\r\n <\/p>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n How to Solve Octal Number System Conversions\r\n <\/h2>\r\n\r\n

\r\n There are several conversions you may encounter in school or computer-science problems.\r\n <\/p>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Octal to Decimal<\/h3>\r\n\r\n

\r\n Multiply each digit by its corresponding power of 8 and add.\r\n <\/p>\r\n\r\n

\r\n

Example:<\/p>\r\n

247\u2088<\/p>\r\n

\r\n = 2 \u00d7 8\u00b2 + 4 \u00d7 8\u00b9 + 7 \u00d7 8\u2070
\r\n = 2 \u00d7 64 + 4 \u00d7 8 + 7 \u00d7 1
\r\n = 128 + 32 + 7
\r\n = 167\u2081\u2080<\/strong>\r\n <\/p>\r\n <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Decimal to Octal<\/h3>\r\n\r\n

\r\n Use repeated division by 8.\r\n <\/p>\r\n\r\n

\r\n For example, convert 83\u2081\u2080<\/strong> to octal:\r\n <\/p>\r\n\r\n

\r\n 83 \u00f7 8 = 10 remainder 3
\r\n 10 \u00f7 8 = 1 remainder 2
\r\n 1 \u00f7 8 = 0 remainder 1\r\n <\/div>\r\n\r\n
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\r\n Read the remainders from bottom to top:\r\n <\/p>\r\n

\r\n 83\u2081\u2080 = 123\u2088\r\n <\/p>\r\n <\/div>\r\n\r\n

\r\n This repeated-division method is also the standard approach described in NCERT's number-system conversion material.\r\n <\/p>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Recommended reading:<\/strong><\/p>