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Octal To Binary

Have you ever wondered how computers talk? They use a special language of numbers! In this lesson, we'll explore two of their favorite number systems, octal and binary, and learn the super-simple trick to translate between them. It's like learning a secret code!

What Are Octal and Binary Numbers?

Octal to binary conversion is the process of changing a number from base-8 to base-2. To understand what that means, let's first think about the numbers we use every day. We use the decimal system, or base-10, which has ten different digits to work with: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The octal number system is a little different. The prefix "octo-" means eight, like in "octagon." So, the octal system is a base-8 system. It only uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. You will never see the digit 8 or 9 in an octal number! For example, $(47)_{8}$ is a valid octal number, but $(48)_{8}$ is not.

The binary number system is even simpler. The prefix "bi-" means two, like in "bicycle." The binary system is a base-2 system, and it only uses two digits: 0 and 1. These are often called bits. Computers use binary for everything they do because the 0s and 1s can represent "off" and "on" states in an electrical circuit. A binary number looks like this: $(10110)_{2}$ .

So, when we convert from octal to binary, we are translating a number written with digits 0-7 into a number written with only digits 0-1.

Why Do Computers Use Binary and Octal?

As we mentioned, computers think in binary. Every action, from displaying a picture to playing a sound, is broken down into millions of 1s and 0s. This is the most basic language of electronics. However, binary numbers can get very, very long. For example, the number $(200)_{10}$ in our decimal system is $(11001000)_{2}$ in binary. That's a lot of digits for a human to read!

This is where octal comes in. Octal is like a convenient shorthand for binary. Because $2^{3} = 8$ , there is a special, direct relationship between binary and octal. Each octal digit can represent a group of exactly three binary digits. This makes it much easier for programmers and engineers to read and write long strings of binary code without making mistakes. Instead of writing $11001000$ , they could use a different system called hexadecimal (base-16), or in some older systems, octal. Learning how they relate helps us understand how computers are organized.

What's the Secret Trick to Convert Octal to Binary?

The secret to converting from octal to binary is incredibly simple and powerful. You don't need to do any complicated multiplication or division. Here is the one rule you need to know:

Each octal digit corresponds to a unique group of three binary digits.

That's it! All you have to do is memorize or look up the 3-bit binary equivalent for each of the eight octal digits. Let's look at the conversion table. Understanding how this table is made is also easy. The three binary digits have place values of 4, 2, and 1 (from left to right). To make an octal digit, you just add the place values where there is a '1'. For example, for the octal digit $(6)_{8}$ , you need a 4 and a 2 ( $4 + 2 = 6$ ), so its binary code is $110$ .

Octal to Binary Conversion Table

Octal Digit	Binary Equivalent (3 bits)
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Once you have this table, any conversion becomes a simple lookup task.

How Do You Convert an Octal Number to Binary?

The conversion process involves just two steps. We'll use the conversion table from the previous section.

Separate the Digits: Take your octal number and look at each digit individually.
Convert and Combine: Replace each octal digit with its 3-bit binary equivalent from the table. Then, write these binary groups together in the same order.

Let's try it with our first example. It's as easy as building with blocks!

Example 1

Convert the octal number $(35)_{8}$ to binary.

Step 1: Separate the digits.
The octal number is 35. The digits are 3 and 5.

Step 2: Convert each digit using the table.
Look up the octal digit 3. Its binary equivalent is $011$ .
Look up the octal digit 5. Its binary equivalent is $101$ .

Step 3: Combine the binary groups.
Now, just put the binary groups together in the same order.
The group for 3 comes first, then the group for 5.
$\underset{3}{\underset{⏟}{011}} \underset{5}{\underset{⏟}{101}}$
So, $(35)_{8} = (011101)_{2}$ . Since the leading zero doesn't change the value, we can also write it as $(11101)_{2}$ .

Can We Practice With More Examples?

Of course! The more you practice, the easier it gets. Let's try a couple more, including one with a zero in it.

Example 2

Convert the octal number $(702)_{8}$ to binary.

Step 1: Separate the digits.
The digits are 7, 0, and 2.

Step 2: Convert each digit.
From the table:
The octal digit 7 is $111$ in binary.
The octal digit 0 is $000$ in binary.
The octal digit 2 is $010$ in binary.

Step 3: Combine the binary groups.
Place them together in order: 7, then 0, then 2.
$\underset{7}{\underset{⏟}{111}} \underset{0}{\underset{⏟}{000}} \underset{2}{\underset{⏟}{010}}$
So, $(702)_{8} = (111000010)_{2}$ . Notice how important it is to keep the three zeros for the octal 0. They are placeholders, just like in the decimal number 702!

Example 3

Convert the octal number $(146)_{8}$ to binary.

Step 1: Separate the digits.
The digits are 1, 4, and 6.

Step 2: Convert each digit.
From the table:
The octal digit 1 is $001$ in binary.
The octal digit 4 is $100$ in binary.
The octal digit 6 is $110$ in binary.

Step 3: Combine the binary groups.
Place them together in order: 1, then 4, then 6.
$\underset{1}{\underset{⏟}{001}} \underset{4}{\underset{⏟}{100}} \underset{6}{\underset{⏟}{110}}$
So, $(146)_{8} = (001100110)_{2}$ . We can write the final answer by removing the leading zeros at the very front of the entire number, which gives us $(1100110)_{2}$ .

What Are Some Common Mistakes When Converting?

This conversion is straightforward, but there are a few common traps to watch out for. Being aware of them will help you get the right answer every time!

Forgetting the 3-Bit Rule: A very common mistake is not using three bits for every octal digit. For example, converting $(2)_{8}$ to $(10)_{2}$ instead of the correct $(010)_{2}$ . Every octal digit must create a group of three binary bits.
Dropping Middle Zeros: The binary group for $(0)_{8}$ is $(000)_{2}$ . If you are converting $(401)_{8}$ , you must include the $000$ in the middle to get $(100000001)_{2}$ . Dropping it would give you $(100001)_{2}$ , which is the conversion for $(41)_{8}$ and is a completely different number.
Mixing Up Binary Codes: It's easy to accidentally swap the binary codes for two numbers, like using $101$ (for 5) when you meant $110$ (for 6). Always double-check your conversions with the table until you have them memorized.
Incorrectly Removing Leading Zeros: You can only remove zeros that are at the very beginning of the final, combined binary number. You cannot remove a leading zero from a middle group. For $(16)_{8}$ , the conversion is $\underset{1}{\underset{⏟}{001}} \underset{6}{\underset{⏟}{110}}$ . You cannot remove the '00' from the '001' group. The final answer is $(001110)_{2}$ , which simplifies to $(1110)_{2}$ .

Can I Get a Quick Summary?

Absolutely! Here is a quick summary of everything you need to know to convert octal numbers to binary.

The Main Rule:

Convert each octal digit to its 3-bit binary equivalent and combine the results in order.

The Steps:

Take the octal number, for example $(527)_{8}$ .
Look at each digit separately: 5, 2, 7.
Convert each digit to its 3-bit binary form using the reference table.
- $5 \to 101$
- $2 \to 010$
- $7 \to 111$
Combine the binary groups: $101010111$ .
Write the final answer: $(527)_{8} = (101010111)_{2}$ .

Reference Table

Octal Digit	Binary Equivalent
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Frequently Asked Questions

Why is it called the octal system?

It's called octal because 'octo' is a prefix that means eight. The octal system is a base-8 number system, which means it uses eight unique digits (0, 1, 2, 3, 4, 5, 6, and 7).

Why does one octal digit equal exactly three binary digits?

This is because of the powers of 2. With three binary digits (bits), you can create $2^{3} = 8$ unique combinations. This is the exact number of digits we need to represent the eight digits (0-7) of the octal system.

Can you convert binary back to octal?

Yes! It's the reverse process. You start from the right of the binary number and group the digits into sets of three. Then you convert each 3-bit group back into its single octal digit.

What is the largest single digit in the octal system?

The largest single digit in the octal system is 7. Since it's a base-8 system, it only uses the eight digits from 0 up to 7. There is no '8' or '9' digit.

Do I have to memorize the octal-to-binary conversion table?

Memorizing it makes conversions faster, but you don't have to! You can always figure it out by remembering the binary place values for three digits are 4, 2, and 1. To get any octal digit, just ask yourself: 'Which of 4, 2, or 1 do I need to add up to make this number?'

Why can't an octal number have the digit 8?

A number system's base tells you how many digits it uses, starting from 0. A base-10 system uses 10 digits (0-9). Similarly, a base-8 (octal) system uses only 8 digits (0-7), so the digit '8' does not exist in this system.

Is octal used more than hexadecimal in computers today?

In modern computing, hexadecimal (base-16) is much more common than octal. This is because modern computers organize bits into groups of eight (called a byte), and two hexadecimal digits can perfectly represent one byte. However, octal is still important for learning number systems and is used in some specific areas like file permissions on some computer systems.