Octal (Base₈ )

Why learn the Octal (Base₈ ) number system?

Hexadecimal is most often used as a shorthand for binary numbers, but octal is sometimes used as a shorthand for binary as well. Converting from octal-to-binary and from binary-to-octal is sometimes necessary when working with computer and network systems. One octal digit can be used to represent three binary digits or bits.

What is the Octal (Base₈ ) number system?

The octal base₈ number system consists of two elements:

1. Eight digits are used in the number system (i.e., 0 through 7).
2. Each column represents a value based on a factor of 8.

How does the Octal (Base₈ ) number system work?

Counting in Octal (Base₈)

The standard unit of storage in computing is a byte. A byte consists of 8 bits. A byte can contain the decimal values 0-255 for a total of 256 different numbers or bit combinations. Below is a table that counts through all the possible bit combinations in a byte (decimal values 0-255, octal values 000-377).

Column Values in Octal (Base₈)
For the base₈ octal number system, the 1st four columns look like the following (note that these four columns also represent twelve bits):

For the value of the 1st column, the base is raised to the power of zero (i.e., 8⁰). To get the next column value, one is added to the power of the current column (i.e., 8¹ for the 2nd column, 8² for the 3rd column, 8³ for the 4th column, etc.). Powers or exponents simply determine how many times the number should be multiplied by itself (note that the powers of zero and one are an exception).

8⁰ = 1, any number raised to the power of zero is 1
8¹ = 8, any number raised to the power of one is the number itself
8² = 8 x 8 = 64
8³ = 8 x 8 x 8 = 512

The decimal equivalent of the base₈ octal number 1376₈ is:
(1x512)+(3x64)+(7x8)+(6x1) = 766

In any base system, the next column value to the left can be found by multiplying the current column value by the base number. This works because one is being added to the exponent for the next column. Adding one to the exponent means that the base should be multiplied by itself one more time. In the base₈ octal number system, to get the column value of the next column, just multiply the current column value by 8.

It's easy to extend the column values of a octal number by either adding one to the power of the current column or multiplying the current column by eight. Here's what the column values are for a 24-bit or 3-byte octal number.

Because the column values begin to get quite large, it makes sense to begin using a shorthand to indicate these larger values. The number 1024 can be represented by Kilo (K). The number 1024K can be represented by Meg (M). Using this shorthand makes the multiples of 1024 much easier to remember. Below is the same table using K to represent 1024 and M to represent 1024K.

Here's what 4K and above mean:
4K = 4 x 1024 = 4,096
32K = 32 x 1024 = 32,768
256K = 256 x 1024 = 262,144
2M = 2 x 1024K = 2,097,152

Often in computing, K is rounded to 1,000. Although this isn't quite accurate, it does make it easier to compare numbers when rough values are all that's needed. Note that many numeric shorthand letters in computing are based on the same concept as K (i.e., 1024K = 1Meg or 1M or roughly 1 million, 1024M = 1Gig or 1G or roughly 1 billion).

The decimal equivalent of the 24-bit or 3-byte base₈ octal number 14567320₈ is:
(1x2M)+(4x256K)+(5x32K)+(6x4K)+(7x512)+(3x64)+(2x8)+(0x1) = 3,337,936

Fortunately, octal is most often used as a shorthand for binary. It is usually not necessary to convert from octal-to-decimal or vice versa. However, it is often necessary to convert between octal and binary.

Converting from Octal (Base₈) to Binary (Base₂)

Determining the binary value of an octal number simply requires converting each octal digit to three binary digits or bits. For example, the binary equivalent of the base₈ octal number 1475₈ is 001100111101₂.

Method: Convert each octal digit to a 3-bit binary number and combine all the binary numbers.

For example, determining the binary equivalent of the octal number 2360₈ requires that the four octal digits 2, 3, 6 and 0 be converted to four 3-bit binary numbers.

The leftmost digit of the octal number 2360₈ is 2. Converting 2 in octal to a 3-bit binary number produces 010₂.

The next digit of the octal number 2360₈ is 3. Converting 3 in octal to a 3-bit binary number produces 011₂.

The next digit of the octal number 2360₈ is 6. Converting 6 in octal to a 3-bit binary number produces 110₂.

The last digit of the octal number 2360₈ is 0. Converting 0 in octal to a 3-bit binary number produces 000₂.

The four 3-bit binary numbers together are 010 011 110 000.
Octal number 2360₈ = 010011110000₂ in binary.

Converting from Binary (Base₂) to Octal (Base₈)

Determining the octal value of a binary number simply requires converting every three binary digits or bits to a single octal digit. For example, the octal equivalent of the base₂ binary number 111010101001₂ is 7251₈.

Method: Convert every three bits  to an octal digit and combine all the octal digits.

For example, determining the octal equivalent of the binary number 11010111110₂ requires that every three bits 011, 010, 111 and 110 be converted to an octal digit. Notice that the bits must be counted from the right, then if needed, the leftmost set of bits are padded with leading zeros to create three bits for that set.

The leftmost three bits of the binary number 11 010 111 110₂ are 011 (note that the leading zero was added to 11 to pad out to three bits). Binary 011₂ converts to octal 3₈.

The next three bits of the binary number 11 010 111 110₂ are 010. Binary 010₂ converts to octal 2₈.

The next three bits of the binary number 11 010 111 110₂ are 111. Binary 111₂ converts to octal 7₈.

The last three bits of the binary number 11 010 111 110₂ are 110. Binary 110₂ converts to octal 6₈.

Combining the four octal digits gives the result 3276.
Binary 11010111110₂ = Octal 3276₈

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