OCTAL SYSTEM: HISTORY, NUMBERING SYSTEM, CONVERSIONS - MATHS - 2025

The octal system is a base-eight (8) positional numbering system; that is, it consists of eight digits, which are: 0, 1, 2, 3, 4, 5, 6 and 7. Therefore, each digit of an octal number can have any value from 0 to 7. The octal numbers they are formed from binary numbers.

This is so because its base is an exact power of two (2). That is, the numbers that belong to the octal system are formed when they are grouped into three consecutive digits, ordered from right to left, thus obtaining their decimal value.

History

The octal system has its origin in ancient times, when people used their hands to count animals from eight to eight.

For example, to count the number of cows in a stable, one began to count with the right hand, joining the thumb with the little finger; Then, to count the second animal, the thumb was joined with the index finger, and so on with the remaining fingers of each hand, until completing 8.

There is a possibility that in ancient times the octal numbering system was used before the decimal to be able to count interdigital spaces; that is, count all fingers except thumbs.

Later the octal numbering system was established, which originated from the binary system, because it needs many digits to represent only one number; from then on, octal and hexagonal systems were created, which do not require so many digits and can easily be converted to the binary system.

Octal numbering system

The octal system consists of eight digits ranging from 0 to 7. These have the same value as in the case of the decimal system, but their relative value changes depending on the position they occupy. The value of each position is given by the powers of base 8.

The positions of the digits in an octal number have the following weights:

8 ⁴, 8 ³, 8 ², 8 ¹, 8 ⁰, octal point, 8 ^-1, 8 ^-2, 8 ^-3, 8 ^-4, 8 ^-5.

The largest octal digit is 7; thus, when counting in this system, a position of a digit is increased from 0 to 7. When 7 is reached, it is recycled to 0 for the next count; in this way the next digit position is increased. For example, to count sequences, in the octal system it will be:

• 0, 1, 2, 3, 4, 5, 6, 7, 10.
• 53, 54, 55, 56, 57, 60.
• 375, 376, 377, 400.

There is a fundamental theorem that is applied to the octal system, and it is expressed in the following way:

In this expression di represents the digit multiplied by the power of base 8, which indicates the place value of each digit, in the same way that it is ordered in the decimal system.

For example, you have the number 543.2. To bring it to the octal system, it breaks down as follows:

N = ∑ = (5 * 64) + (4 * 8) + (2 * 1) + (2 * 0.125)

N = 320 +32 + 2 + 0.25 = 354 + 0.25 _d

Thus, we have 543.2 _q = 354.25 _d. The subscript q indicates that it is an octal number that can also be represented by the number 8; and the subscript d refers to the decimal number, which can also be represented by the number 10.

Conversion from octal to decimal system

To convert a number from the octal system to its equivalent in the decimal system, simply multiply each octal digit by its place value, starting from the right.

Example 1

732 ₈ = (7 _* 8 ²) + (3 _* 8 ¹) + (2 _* 8 ⁰) = (7 _* 64) + (3 _* 8) + (2 _* 1)

732 ₈ = 448 +24 +2

732 ₈ = 474 ₁₀

Example 2

26.9 ₈ = (2 _* 8 ¹) + (6 _* 8 ⁰) + (9 _* 8 ^-1) = (2 _* 8) + (6 _* 1) + (9 _* 0.125)

26.9 ₈ = 16 + 6 + 1.125

26.9 ₈ = 23.125 ₁₀

Conversion from decimal to octal system

A decimal integer can be converted to an octal number using the repeated division method, where the decimal integer is divided by 8 until the quotient is equal to 0, and the remainders of each division will represent the octal number.

The residuals are ordered from last to first; that is, the first remainder will be the least significant digit of the octal number. That way, the most significant digit will be the last remainder.

Example

Decimal Number Octal 266 ₁₀

- Divide the decimal number 266 by 8 = 266/8 = 33 + remainder of 2.

- Then divide 33 by 8 = 33/8 = 4 + remainder of 1.

- Divide 4 by 8 = 4/8 = 0 + remainder of 4.

As with the last division a quotient less than 1 is obtained, it means that the result has been found; You only have to order the remainders inversely, so that the octal number of decimal 266 is 412, as can be seen in the following image:

Conversion from octal to binary system

Converting from octal to binary is done by converting the octal digit to its equivalent binary digit, consisting of three digits. There is a table that shows how the eight possible digits are converted:

From these conversions, any number from the octal system to binary can be changed, for example, to convert the number 572 _8, we look for its equivalents in the table. Thus, you have to:

5 ₈ = 101

7 ₈ = 111

2 ₈ = 10

Therefore, 572 _{8 is} equivalent in the binary system to 10111110.

Converting from binary to octal

The process of converting binary integers to octal integers is the reverse of the previous process.

That is, the bits of the binary number are grouped into two groups of three bits, starting from right to left. Then, the conversion from binary to octal is done with the table above.

In some cases the binary number will not have groups of 3 bits; to complete it, one or two zeros are added to the left of the first group.

For example, to change the binary number 11010110 to octal, do the following:

- Groups of 3 bits are formed starting from the right (last bit):

11010110

- Since the first group is incomplete, a leading zero is added:

011010110

- The conversion is made from the table:

011 = 3

010 = 2

110 = 6

Thus, the binary number 011010110 equals 326 ₈.

Converting from octal to hexadecimal and vice versa

To change from an octal number to hexadecimal or from hexadecimal to octal, it is necessary to convert the number to binary first, and then to the desired system.

For this, there is a table where each hexadecimal digit is represented with its equivalent in the binary system, made up of four digits.

In some cases, the binary number will not have groups of 4 bits; to complete it, one or two zeros are added to the left of the first group

Example

Convert octal number 1646 to hexadecimal number:

- Convert the number from octal to binary

1 ₈ = 1

6 ₈ = 110

4 ₈ = 100

6 ₈ = 110

- So, 1646 ₈ = 1110100110.

- To convert from binary to hexadecimal, they are first ordered in a group of 4 bits, starting from right to left:

11 1010 0110

- The first group is completed with zeros, so that it can have 4 bits:

0011 1010 0110

- The conversion from binary to hexadecimal is done. The equivalences are substituted by means of the table:

0011 = 3

1010 = A

0110 = 6

Thus, the octal number 1646 equals 3A6 in the hexadecimal system.

References

1. Bressan, AE (1995). Introduction to numbering systems. Argentine University of the Company.
2. Harris, JN (1957). Introduction to the Binary and Octal Numbering Systems: Lexington, Mass. Armed Services Technical Information Agency.
3. Kumar, AA (2016). Fundamentals of Digital Circuits. Learning Pvt.
4. Peris, XC (2009). Single Operative Systems.
5. Ronald J. Tocci, NS (2003). Digital systems: principles and applications. Pearson Education.