Octal Number System

What Is the Octal Number System?

The octal number system is a way of counting that uses a base of 8. This means it only uses eight unique digits to represent all possible numbers: 0, 1, 2, 3, 4, 5, 6, and 7. You might notice that the digits 8 and 9 are missing! That's the most important rule of the octal system.

Think about the number system you use every day. We call it the decimal or 'base-10' system because it uses ten digits (0 through 9). The word 'octal' comes from the Latin root 'octo,' which means eight. So, just as 'decimal' relates to ten, 'octal' relates to eight.

Imagine you are an octopus counting your treasures. You might find it natural to group things in eights. This is the core idea behind the octal system. Instead of place values being powers of ten (like ones, tens, hundreds), in octal, they are powers of eight (ones, eights, sixty-fours, and so on).

How Do We Count in Octal?

Counting in octal starts off just like counting in decimal. You go 0, 1, 2, 3, 4, 5, 6, 7. But what happens next? In the decimal system, after 9, we run out of single digits, so we roll over to 10, which means 'one group of ten and zero ones'.

The octal system follows the exact same logic, but it rolls over after 7. When we count past 7 in octal, we've run out of single digits. So, the next number is written as 10. In octal, $(10{)}_8$ doesn't mean 'ten'; it means 'one group of eight and zero ones'. The number after $(10{)}_8$ is $(11{)}_8$ (one eight and one one), then $(12{)}_8$ (one eight and two ones), all the way up to $(17{)}_8$. After $(17{)}_8$, we roll over again to $(20{)}_8$, which means 'two groups of eight and zero ones'.

Here is a table to help you see the pattern:

How Does Place Value Work in Octal?

Understanding place value is the key to mastering any number system. In our familiar decimal system, each position in a number represents a power of 10. For the number $(425{)}_{10}$:

• The '5' is in the ones place (${10}^0$)
• The '2' is in the tens place (${10}^1$)
• The '4' is in the hundreds place (${10}^2$)

The octal system works the same way, but the place values are powers of 8.

• The rightmost digit is the ones place ($8^0=1$)
• The next digit to the left is the eights place ($8^1=8$)
• The next is the sixty-fours place ($8^2=64$)
• And the next is the five hundred twelves place ($8^3=512$), and so on.

Let's look at the octal number $(345{)}_8$. It does NOT mean three hundred forty-five! Here’s what it really represents:

• The '5' is in the $8^0$ (ones) place. Its value is $5\times 1=5$.
• The '4' is in the $8^1$ (eights) place. Its value is $4\times 8=32$.
• The '3' is in the $8^2$ (sixty-fours) place. Its value is $3\times 64=192$.

To find the total value in decimal, you just add these up: $192+32+5=229$. So, the octal number $(345{)}_8$ is equal to the decimal number $(229{)}_{10}$.

How Do You Convert an Octal Number to a Decimal Number?

Converting from octal to decimal is a straightforward process once you understand place value. You just need to multiply each octal digit by its corresponding power of 8 and then add all the results together.

Here are the steps:

1. Write down the octal number and identify the place value for each digit, starting from the right ($8^0,8^1,8^2,$ etc.).
2. Multiply each digit by its place value.
3. Add up all the products from step 2. The sum is your equivalent decimal number.

How Do You Convert a Decimal Number to an Octal Number?

Converting from decimal to octal can feel a bit like a puzzle. The most common method is called repeated division by 8. You repeatedly divide your decimal number by 8 and keep track of the remainders. The remainders, read in reverse order, form your octal number.

Let's break it down into steps:

1. Take your starting decimal number.
2. Divide the number by 8.
3. Write down the remainder (it will always be a number from 0 to 7).
4. Take the whole number part of the quotient from the division and repeat step 2.
5. Continue this process until your quotient is 0.
6. Your octal number is the list of remainders you wrote down, read from the bottom up.

Why Is the Octal System Used?

You might be wondering why anyone would use a base-8 system when we have the perfectly good base-10 system. The primary reason is its special relationship with the binary (base-2) number system, which is the fundamental language of computers.

Computers think in binary, using only 0s and 1s. A long string of binary code like $110101111001$ is very hard for humans to read and remember. This is where octal comes in. Because $8=2^3$, every single octal digit corresponds perfectly to a group of exactly three binary digits (bits).

This allows us to use octal as a convenient shorthand for binary. Instead of writing a long binary number, a programmer can represent it with a much shorter and more readable octal number.

Here's the conversion chart:

• $(0{)}_8=(000{)}_2$
• $(1{)}_8=(001{)}_2$
• $(2{)}_8=(010{)}_2$
• $(3{)}_8=(011{)}_2$
• $(4{)}_8=(100{)}_2$
• $(5{)}_8=(101{)}_2$
• $(6{)}_8=(110{)}_2$
• $(7{)}_8=(111{)}_2$

What Are Some Common Mistakes with Octal Numbers?

When first learning about the octal system, it's easy to make a few common mistakes. Being aware of them can help you avoid them!

• Using the Digits 8 or 9: This is the most frequent error. Remember, the octal system only uses digits from 0 to 7. A number like $481$ can never be an octal number.
• Forgetting the Subscript: When working with different number systems, it's crucial to label them. Is the number 10 supposed to be ten (in decimal) or eight (in octal)? Always write $(10{)}_{10}$ or $(10{)}_8$ to avoid confusion. This is called the 'base' or 'radix'.
• Reading Remainders in the Wrong Order: When converting from decimal to octal using the division method, remember to read the remainders from the bottom up. Reading them from top to bottom will give you the wrong answer.
• Confusing Place Values: When converting from octal to decimal, don't accidentally use powers of 10. The place values are always powers of 8 (1, 8, 64, 512, etc.).

Quick Summary: Key Octal Concepts

Feeling a little overwhelmed? Don't worry! Here are the most important ideas to remember about the octal number system.

• Base-8: The octal system is a base-8 system.
• Eight Digits: It uses only the digits 0, 1, 2, 3, 4, 5, 6, and 7.
• Place Values: Place values are powers of 8 ($8^0,8^1,8^2,$ and so on).
• Counting Rollover: After the digit 7, the count 'rolls over' to 10, which means one group of eight.
• Computer Shorthand: It's often used in computer science as a more human-friendly way to represent binary numbers, since one octal digit equals three binary digits.