Numeral systems by culture  

HinduArabic numerals  
Eastern Arabic Indian family Khmer 
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East Asian numerals  
Chinese Counting rods Japanese 
Korean Suzhou Vietnamese 
Alphabetic numerals  
Abjad Armenian Āryabhaṭa Cyrillic 
Ge'ez Greek (Ionian) Hebrew 
Other systems  
Attic Babylonian Brahmi Egyptian Etruscan Inuit 
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List of numeral system topics  
Positional systems by base  
Decimal (10)  
1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more…  
The octal numeral system, or oct for short, is the base8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010, which can be grouped into (00)1 001 010 — so the octal representation is 112.
In decimal systems each decimal place is a base of 10. For example:
In octal numerals each place is a power with base 8. For example:
By performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal.
Octal is sometimes used in computing instead of hexadecimal.
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The Yuki language in California and the Pamean languages^{[1]} in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves^{[2]}.
In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number system based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as base. In 1718 Swedenborg wrote a manuscript, which has not been published: "En ny räknekonst som omväxlas vid talet 8 istället för det vanliga vid talet 10" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 17 are there denoted by the consonants l, s, n, m, t, f, u (v) and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule.^{[3]}
Octal is sometimes used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with file permissions under Unix systems (see chmod). It has the advantage of not requiring any extra symbols as digits (the hexadecimal system is base16 and therefore needs six additional symbols beyond 0–9). It is also used for digital displays.
At the time when octal originally became widely used in computing, systems such as the ICL 1900 and IBM mainframes employed 24bit (or 36bit) words. Octal was an ideal abbreviation of binary for these machines because eight (or twelve) digits could concisely display an entire machine word (each octal digit covering three binary digits). It also cut costs by allowing Nixie tubes, sevensegment displays, and calculators to be used for the operator consoles, where binary displays were too complex to use, decimal displays needed complex hardware to convert radixes, and hexadecimal displays needed to display more numerals.
All modern computing platforms, however, use 16, 32, or 64bit words, further divided into eightbit bytes. On such systems three octal digits per byte would be required, with the most significant octal digit representing two binary digits (plus one bit of the next significant byte, if any). Octal representation of a 16bit word requires 6 digits, but the most significant octal digit represents (quite inelegantly) only one bit (0 or 1). This representation offers no way to easily read the most significant byte, because it's smeared over four octal digits. Therefore, hexadecimal is more commonly used in programming languages today, since two hexadecimal digits exactly specify one byte. Some platforms with a poweroftwo word size still have instruction subwords that are more easily understood if displayed in octal; this includes the PDP11 and Motorola 68000 family. The modernday ubiquitous x86 architecture belongs to this category as well, but octal is rarely used on this platform.
In programming languages, octal literals are typically identified with a variety of prefixes, including the digit 0, the letters o or q, or the digit–letter combination 0o. For example, the literal 73 (base 8) might be represented as 073, o73, q73, or 0o73 in various languages. Newer languages have been abandoning the prefix 0, as decimal numbers are often represented with leading zeroes. The prefix q was introduced to avoid the prefix o being mistaken for a zero, while the prefix 0o was introduced to avoid starting a numerical literal with an alphabetic character (like o or q), since these might cause the literal to be confused with a variable name. The prefix 0o also follows the model set by the prefix 0x used for hexadecimal literals in the C language.^{[4]}^{[5]}^{[6]}
Octal numbers that are used in some programming languages (C, Perl, PostScript…) for textual/graphical representations of byte strings when some byte values (unrepresented in a code page, nongraphical, having special meaning in current context or otherwise undesired) have to be to escaped as \nnn. Octal representation of nonASCII bytes may be particularly handy with UTF8, where any start byte has octal value \3nn and any continuation byte has octal value \2nn.
For more information and other bases, see Conversion among bases.
To convert integer decimals to octal, divide the original number by the largest possible power of 8 and successively divide the remainders by successively smaller powers of 8 until the power is 1. The octal representation is formed by the quotients, written in the order generated by the algorithm.
For example, to convert 125_{10} to octal:
Another example:
To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds.
Example: Convert 0.1640625 to octal:
These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part.
To convert a number k to decimal, use the formula that defines its base8 representation:
Example: Convert 764_{8} to decimal:
For doubledigit octal numbers this method amounts to multiplying the lead digit by 8 and adding the second digit to get the total.
Example: 65_{8} = 6x8 + 5 = 53_{10}
To convert octal to binary, replace each octal digit by its binary representation.
Example: Convert 51_{8} to binary:
The process is the reverse of the previous algorithm. The binary digits are grouped by threes, starting from the decimal point and proceeding to the left and to the right. Add leading 0s (or trailing zeros to the right of decimal point) to fill out the last group of three if necessary. Then replace each trio with the equivalent octal digit.
For instance, convert binary 1010111100 to octal:
001  010  111  100 
1  2  7  4 
Thus 1010111100_{2} = 1274_{8}
Convert binary 11100.01001 to octal:
011  100  .  010  010 
3  4  .  2  2 
Thus 11100.01001_{2} = 34.22_{8}
The conversion is made in two steps using binary as an intermediate base. Octal is converted to binary and then binary to hexadecimal, grouping digits by fours, which correspond each to a hexadecimal digit.
For instance, convert octal 1057 to hexadecimal:
1  0  5  7 
001  000  101  111 
0010  0010  1111 
2  2  F 
Thus 1057_{8} = 22F_{16}
Reverse the previous algorithm.
The octal numeral system is a base 8 numeral system. It uses the numerals 0 through 7. The system is similar to binary (base 2) and hexadecimal (base 16). Octal numerals are written using the letter o before the numeral, for example, o04 or o1242.
At one time, the octal system was used mainly for work with computers. It provided an easier way to work with binary numbers. As computers changed from using 24 bit systems to 32 and 64 bit systems, hexadecimal replaced octal for most uses. Certain groups , for example, Native Americans using the Yuki language in California and the Pamean languages^{[1]} in Mexico also use octal numbering system. They do this because when they count, they use the spaces between their fingers instead of counting the actual fingers.
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The octal numeral system uses a three bit binary coding. Each digit in an octal numeral is the same as three digits in a binary numeral. The grouping of the binary digits is done from right to left. The first three binary digits from the right are grouped into the last part of the octal numeral then the next three digits form the next to the last part of the numeral.



In the Decimal system (base 10), each digit in Octal is equal to that digit multiplied by the exponent of 8 that is equal to its location minus one.
Location  

6  5  4  3  2  1  
Value  32768 (8^{5})  4096 (8^{4})  512 (8^{3})  64 (8^{2})  8(8^{1})  1 (8^{0}) 
Example: o3425 to decimal

Octal is similar to hexadecimal because they are both easily converted to binary. Where Octal is equal to three digit binary, hexadecimal is equal to four digit binary. Where octal numerals start with the letter "o", hexadecimal numeral end with the letter "h". The easiest way to convert from one to the other is to convert to binary and then to the other system.
Octal  Binary  Hexadecimal  

three digit  four digit  
o4  100  0100  04h  
o15  001  101  1101  0Dh  
o306  011  000  110  1100  0110  C6h  
o54253  101  100  010  101  011  0101  1000  1010  1011  58ABh 
