# Octal

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The octal, or oct for short, is the base-8 positional numeral system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 number system, hence a true octal system might use different vocabulary.

Numeral systems, bits and Gray code
hexdecoct3210step
0hex0dec0oct00000
1hex1dec1oct00011
2hex2dec2oct00103
3hex3dec3oct00112
4hex4dec4oct01007
5hex5dec5oct01016
6hex6dec6oct01104
7hex7dec7oct01115
8hex8dec10oct1000F
9hex9dec11oct1001E
Ahex10dec12oct1010C
Bhex11dec13oct1011D
Chex12dec14oct11008
Dhex13dec15oct11019
Ehex14dec16oct1110B
Fhex15dec17oct1111A

In the decimal system, each 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.

Octal numerals can be easily converted from binary representations (similar to a quaternary numeral system) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: (00)1 001 010, corresponding to the octal digits 1 1 2, yielding the octal representation 112.

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 × 1 2 3 4 5 6 7 10 1 1 2 3 4 5 6 7 10 2 2 4 6 10 12 14 16 20 3 3 6 11 14 17 22 25 30 4 4 10 14 20 24 30 34 40 5 5 12 17 24 31 36 43 50 6 6 14 22 30 36 44 52 60 7 7 16 25 34 43 52 61 70 10 10 20 30 40 50 60 70 100
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