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{{Table Numeral Systems}}The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal 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. Two zeroes can be added at the left: {{nowrap|(00)1 001 010}}, corresponding the octal digits {{nowrap|1 1 2}}, yielding the octal representation 112.In the decimal system each decimal place is a power of ten. For example:
mathbf{74}_{10} = mathbf{7} times 10^1 + mathbf{4} times 10^0
In the octal system each place is a power of eight. For example:
mathbf{112}_8 = mathbf{1} times 8^2 + mathbf{1} times 8^1 + mathbf{2} times 8^0
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.{| class="wikitable" style="float:right; text-align:center"|+ The octal multiplication table
1 >2 >3 >4 >5 >6 >7 >| 10
1 >| 10
2 >| 20
3 >| 30
4 >| 40
5 >| 50
6 >| 60
7 >| 70
10 >| 100

## Usage

### By Native Americans

The Yuki language in California and the Pamean languagesJOURNAL
, Avelino
, Heriberto
, The typology of Pame number systems and the limits of Mesoamerica as a linguistic area
, Linguistic Typology
, 2006
, 10
, 1
, 41â€“60
, 10.1515/LINGTY.2006.002
,
, in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.JOURNAL, 2686959, Ethnomathematics: A Multicultural View of Mathematical Ideas, Marcia Ascher, The College Mathematics Journal,

### By Europeans

• It has been suggested that the reconstructed Proto-Indo-European word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal number system, though the evidence supporting this is slim.BOOK

, Winter
, Werner
, Some thoughts about Indo-European numerals
, Indo-European numerals
, Trends in Linguistics
, 57
, GvozdanoviÄ‡
, 1991
, Mouton de Gruyter
, Berlin
, 3-11-011322-8
, 13â€“14
, 2013-06-09
,

, Wilkins
, John
, An Essay Towards a Real Character and a Philosophical Language
, 1668
,
, London
, 190
, 2015-02-08
,
• 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 the base. In 1718 Swedenborg wrote (but did not publish) a manuscript: "En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 1-7 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.Donald Knuth, The Art of Computer Programming
• Writing under the pseudonym "Hirossa Ap-Iccim" in The Gentleman's Magazine, (London) July 1745, Hugh Jones proposed an octal system for British coins, weights and measures. "Whereas reason and convenience indicate to us an uniform standard for all quantities; which I shall call the Georgian standard; and that is only to divide every integer in each species into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho' all nations count universally by tens (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ten is uncapable...." In a later treatise on Octave computation (1753) Jones concluded: "Arithmetic by Octaves seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic."See H.R. Phalen, "Hugh Jones and Octave Computation," The American Mathematical Monthly 56 (Augustâ€“September 1949): 461-65.
• In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic. He suggested base 8, for which he coined the term octal. His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of English units was already, to a remarkable extent, an octal system.James Anderson, On Octal Arithmetic [title appears only in page headers], Recreations in Agriculture, Natural-History, Arts, and Miscellaneous Literature, Vol. IV, No. 6 (Feb. 1801), T. Bensley, London; pages 437-448.
• In the mid 19th century, Alfred B. Taylor concluded that "Our octonary [base 8] radix is, therefore, beyond all comparison the "best possible one" for an arithmetical system." The proposal included a graphical notation for the digits and new names for the numbers, suggesting that we should count "un, du, the, fo, pa, se, ki, unty, unty-un, unty-du" and so on, with successive multiples of eight named "unty, duty, thety, foty, paty, sety, kity and under." So, for example, the number 65 (101 in octal) would be spoken in octonary as under-un.A.B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, Sept. 15, 1859. See pages and 48 and 53.Alfred B. Taylor, Octonary numeration and its application to a system of weights and measures, Proc. Amer. Phil. Soc. Vol XXIV, Philadelphia, 1887; pages 296-366. See pages 327 and 330. Taylor also republished some of Swedenborg's work on octonary as an appendix to the above-cited publications.

### In aviation

Transponders in aircraft transmit a code, expressed as a four-octal-digit number, when interrogated by ground radar. This code is used to distinguish different aircraft on the radar screen.

## Conversion between bases

### Decimal to octal conversion

#### Method of successive Euclidean division by 8

To convert integer decimals to octal, divide the original number by the largest possible power of 8 and 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 12510 to octal:
125 = 82 Ã— 1 + 61 61 = 81 Ã— 7 + 5 5 = 80 Ã— 5 + 0
Therefore, 12510 = 1758.Another example:
900 = 83 Ã— 1 + 388 388 = 82 Ã— 6 + 4 4 = 81 Ã— 0 + 4 4 = 80 Ã— 4 + 0
Therefore, 90010 = 16048.

#### Method of successive multiplication by 8

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:
0.1640625â€‰Ã—â€‰8 = 1.3125 = 1 + 0.3125 0.3125â€‰Ã—â€‰8 = 2.5 = 2 + 0.5 0.5â€‰Ã—â€‰8 = 4.0 = 4 + 0
Therefore, 0.164062510 = 0.1248.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.

#### Method of successive duplication

To convert integer decimals to octal, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix:Double the value to the left side of the radix, using octal rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. If the moved radix point crosses over a digit that is 8 or 9, convert it to 0 or 1 and add the carry to the next leftward digit of the current value. Add octally those digits to the left of the radix and simply drop down those digits to the right, without modification.Example:
0.4 9 1 8 decimal value
+0
-----
4.9 1 8
+1 0
----
6 1.1 8
+1 4 2
----
7 5 3.8
+1 7 2 6
----
1 1 4 6 6. octal value

### Octal to decimal conversion

To convert a number {{mvar|k}} to decimal, use the formula that defines its base-8 representation:
k = sum_{i=0}^n left( a_itimes 8^i right)
In this formula, {{math|ai}} is an individual octal digit being converted, where {{mvar|i}} is the position of the digit (counting from 0 for the right-most digit).Example: Convert 7648 to decimal:

7648 = 7â€‰Ã—â€‰82 + 6â€‰Ã—â€‰81 + 4â€‰Ã—â€‰80 = 448 + 48 + 4 = 50010
For double-digit octal numbers this method amounts to multiplying the lead digit by 8 and adding the second digit to get the total.Example: 658 = 6â€‰Ã—â€‰8 + 5 = 5310

#### Method of successive duplication

To convert octals to decimals, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix: Double the value to the left side of the radix, using decimal rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. Subtract decimally those digits to the left of the radix and simply drop down those digits to the right, without modification.Example:
0.1 1 4 6 6 octal value
-0
-------
1.1 4 6 6
- 2
------
9.4 6 6
- 1 8
------
7 6.6 6
- 1 5 2
------
6 1 4.6
- 1 2 2 8
------
4 9 1 8. decimal value

### Octal to binary conversion

To convert octal to binary, replace each octal digit by its binary representation. Example: Convert 518 to binary:
58 = 1012 18 = 0012
Therefore, 518 = 101 0012.

### Binary to octal conversion

The process is the reverse of the previous algorithm. The binary digits are grouped by threes, starting from the least significant bit and proceeding to the left and to the right. Add leading zeroes (or trailing zeroes 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:
{| border="1" cellspacing="0" cellpadding="4" align="center"
| 100
align="center"
| 4
Therefore, 10101111002 = 12748.Convert binary 11100.01001 to octal:
{| border="1" cellspacing="0" cellpadding="4" align="center"
| 010
align="center"
| 2
Therefore, 11100.010012 = 34.228.

### Octal to hexadecimal conversion

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:
To binary: {| border="1" cellspacing="0" cellpadding="4" align="center"
| 7
align="center"
| 111
then to hexadecimal: {| border="1" cellspacing="0" cellpadding="4" align="center"
| 1111
align="center"
| F
Therefore, 10578 = 22F16.

### Hexadecimal to octal conversion

Hexadecimal to octal conversion proceeds by first converting the hexadecimal digits to 4-bit binary values, then regrouping the binary bits into 3-bit octal digits.For example, to convert 3FA516:
To binary: {| border="1" cellspacing="0" cellpadding="4" align="center"
| 5
align="center"
| 0101
then to octal: {| border="1" cellspacing="0" cellpadding="4" align="center"
| 101
align="center"
| 5
Therefore, 3FA516 = 376458.

## Real numbers

### Fractions

Due to having only factors of two, many octal fractions have repeating digits, although these tend to be fairly simple:{|class="wikitable" Decimal basePrime factors of the base: 2, 5Prime factors of one below the base: 3Prime factors of one above the base: 11Other Prime factors: 7 13 17 19 23 29 31 Octal basePrime factors of the base: 2Prime factors of one below the base: 7Prime factors of one above the base: 3Other Prime factors: 5 13 15 21 23 27 35 37
Fraction Prime factorsof the denominator Positional representation Positional representation Prime factorsof the denominator Fraction
1/2 2| 0.5| 0.4 2 1/2
1/3 3 0.3333... = 0.{{overline|3}} 0.2525... = 0.{{overline|25}} 3 1/3
1/4 2| 0.25| 0.2 2 1/4
1/5 5| 0.2 0.{{overline|1463}} 5 1/5
1/6 2, 3 0.1{{overline|6}} 0.1{{overline|25}} 2, 3 1/6
1/7 7 0.{{overline|142857}} 0.{{overline|1}} 7 1/7
1/8 2| 0.125| 0.1 2 1/10
1/9 3 0.{{overline|1}} 0.{{overline|07}} 3 1/11
1/10 2, 5| 0.1 0.0{{overline|6314}} 2, 5 1/12
1/11 11 0.{{overline|09}} 0.{{overline|0564272135}} 13 1/13
1/12 2, 3 0.08{{overline|3}} 0.0{{overline|52}} 2, 3 1/14
1/13 13 0.{{overline|076923}} 0.{{overline|0473}} 15 1/15
1/14 2, 7 0.0{{overline|714285}} 0.0{{overline|4}} 2, 7 1/16
1/15 3, 5 0.0{{overline|6}} 0.{{overline|0421}} 3, 5 1/17
1/16 2| 0.0625| 0.04 2 1/20
1/17 17 0.{{overline|0588235294117647}} 0.{{overline|03607417}} 21 1/21
1/18 2, 3 0.0{{overline|5}} 0.0{{overline|34}} 2, 3 1/22
1/19 19 0.{{overline|052631578947368421}} 0.{{overline|032745}} 23 1/23
1/20 2, 5| 0.05 0.0{{overline|3146}} 2, 5 1/24
1/21 3, 7 0.{{overline|047619}} 0.{{overline|03}} 3, 7 1/25
1/22 2, 11 0.0{{overline|45}} 0.0{{overline|2721350564}} 2, 13 1/26
1/23 23 0.{{overline|0434782608695652173913}} 0.{{overline|02620544131}} 27 1/27
1/24 2, 3 0.041{{overline|6}} 0.0{{overline|25}} 2, 3 1/30
1/25 5| 0.04 0.{{overline|02436560507534121727}} 5 1/31
1/26 2, 13 0.0{{overline|384615}} 0.0{{overline|2354}} 2, 15 1/32
1/27 3 0.{{overline|037}} 0.{{overline|022755}} 3 1/33
1/28 2, 7 0.03{{overline|571428}} 0.0{{overline|2}} 2, 7 1/34
1/29 29 0.{{overline|0344827586206896551724137931}} 0.{{overline|0215173454106475626043236713}} 35 1/35
1/30 2, 3, 5 0.0{{overline|3}} 0.0{{overline|2104}} 2, 3, 5 1/36
1/31 31 0.{{overline|032258064516129}} 0.{{overline|02041}} 37 1/37
1/32 2| 0.03125| 0.02 2 1/40

### Irrational numbers

The table below gives the expansions of some common irrational numbers in decimal and octal.{| class="wikitable"! rowspan=2 | Number! colspan=2 | Positional representation! Decimal! Octal
Square root of 2>{{sqrt {{small>(the length of the diagonal of a unit square)}}1.414213562373095048}}...| 1.3240 4746 3177 1674...
Square root of 3>{{sqrt {{small>(the length of the diagonal of a unit cube)}}1.732050807568877293}}...| 1.5666 3656 4130 2312...
Square root of 5>{{sqrt {{small>(the length of the diagonal of a 1Ã—2 rectangle)}}2.236067977499789696}}...| 2.1706 7363 3457 7224...
Golden ratio>{{mvar {{small>1=(phi, the golden ratio = {{math5}})/2}})}}1.618033988749894848}}...| 1.4743 3571 5627 7512...
Pi>{{mvar {{small>(pi, the ratio of circumference to diameter of a circle)}}3.141592653589793238462643}}{{val|383279502884197169399375105}}...| 3.1103 7552 4210 2643...
E (mathematical constant)>{{mvar {{small>(the base of the natural logarithm)}}2.718281828459045235}}...| 2.5576 0521 3050 5355...

## References

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