Octal

Octal is a special way to write numbers, known as a numeral system. Instead of using ten symbols like we do in everyday math (0 through 9), octal uses only eight symbols, from 0 to 7.

Each position in an octal number represents a power of 8. For example, the octal number 112 means you calculate it like this: 1 × 8² + 1 × 8¹ + 2 × 8⁰, which equals 64 + 8 + 2 = 74 in our usual decimal numbers.

Octal helps us work with numbers in a way that connects closely to binary numbers, which computers use. Octal is useful because each octal digit matches exactly with three binary digits. This makes it easier to change between binary and other number systems. For instance, the binary number for 74, which is 1001010, can be grouped as 001 001 010, turning into the octal number 112. This connection makes octal a helpful tool in computer science and engineering.

Multiplication table

The multiplication table for octal numbers shows how the digits from 0 to 7 multiply together. Each row and column stands for one of these digits. Where they meet, you see the result in octal. This helps us learn how calculations work in the octal system. The octal system uses base 8, which is different from the base 10 we use most often.

Usage

The octal system is a way to write numbers using only the digits 0 to 7, instead of the usual 0 to 9. Each position in an octal number stands for a power of 8, like how each position in our normal numbers stands for a power of 10. For example, the octal number 112 means:

1 × 8² + 1 × 8¹ + 2 × 8⁰

People have used octal in many ways over time. In China, the old trigrams from the I Ching match up with octal digits. Some Native American languages, like the Yuki language in California, also use octal because people count the spaces between their fingers instead of the fingers themselves.

In computing, octal was used a lot in early computers because it fits well with systems that handle data in groups of three bits. Each octal digit stands for exactly three binary digits. Even today, octal is still sometimes used in computer systems, like for setting file permissions in Unix systems.

Conversion between bases

To change a number from decimal (the normal way we count) to octal (base 8), divide the number by 8 again and again and write down the remainders. For example, to turn 125 into octal, break it down like this: 125 = 8 × 15 + 5, then 15 = 8 × 1 + 7, and finally 1 = 8 × 0 + 1. Reading the remainders from last to first gives you 175 in octal.

You can also change octal back to decimal by multiplying each digit by a power of 8 and adding the results. For example, the octal number 764 equals 7 × 8² + 6 × 8¹ + 4 × 8⁰, which is 500 in decimal.

Octal numbers are closely related to binary numbers. Each octal digit matches exactly three binary digits. To turn octal into binary, change each octal digit to its three-digit binary form. To go from binary to octal, group the binary digits into threes (adding zeroes if needed) and change each group into an octal digit.

Real numbers

Octal can show parts of numbers, just like decimal. Since octal only uses the digits 0 to 7, some fractions in octal repeat, but the patterns are usually easy to see.

There are special numbers that cannot be written exactly with any number system, whether decimal or octal. The table below shows how some of these numbers look in both decimal and octal.

Main article: irrational numbers