## What Is the Octal to Binary Conversion?

**Octal to binary conversion refers to converting an octal number into an equivalent binary number. **

The base of the octal number system is 8. The base of the binary number system is 2. Each octal digit represents three binary digits. A single octal digit can represent three binary bits.

There are two different methods that we can use to convert a number from the octal system to a binary system: direct method and indirect method. We will discuss them in detail.

We can convert octal to binary using this online tool to carry out the conversion instantly. However, the knowledge of the various steps involved in the conversion method helps gain a better understanding.

**Binary Number System**

The number system that uses only two digits, 0 and 1, is called binary. It is also referred to as the base-2 system. It is popularly used in computer systems. In a binary system, the digits are called bits.

Example: $(01001)_{2},\; (01110010)_{2}$

**Octal Number System**

The octal number system comprises digits from 0 to 7. The digits 8 and 9 are not included in the octal system. It is also referred to as the base-8 system. Like a binary system, the octal number system finds usage in minicomputers.

Example: $(63)_{8},\; (45)_{8}$

## How to Convert Octal to Binary

We can convert a number from octal to binary using two ways:

- Indirect Method: Octal to decimal followed by decimal to binary
- Direct Method: Converting an octal number directly into the binary number system

## Indirect Method: Octal to Decimal to Binary

Let’s discuss the octal to binary conversion steps for the indirect method.

**Step 1: Convert octal to decimal.**

To convert octal to decimal, we multiply each digit by the power of 8 based on the position starting from the right. We will multiply the first digit from the right by $8^{0}$. Next, we will multiply the second digit by $8^{1}$ and so on.

**Step 2: Convert decimal to binary.**

For converting decimal to binary, we will divide the given number by 2 and record the quotient and reminder. We will repeat the process until we obtain 0 as the quotient.

**Example: Convert from octal to binary: **$54_{8}$**. **

**Step 1: Octal to decimal**

$54_{8} = 4 \times 8^{0} + 5 \times 8^{1}$

$54_{8} = 4 \times 1 + 5 \times 8$

$54_{8} = 4 + 40$

$54_{8} = 44_{10}$

**Step 2: Decimal to binary **

Division | Quotient | Remainder |
---|---|---|

$44 \div 2$ | 22 | 0 |

$22 \div 2$ | 11 | 0 |

$11 \div 2$ | 5 | 1 |

$5 \div 2$ | 2 | 1 |

$2 \div 2$ | 1 | 0 |

$1 \div 2$ | 0 | 1 |

On arranging all the remainders in the reverse order, we will obtain the following binary number:

Thus, $44_{10} = 101100_{2}$

## Direct Method: Octal to Binary Conversion Using the Chart

There is no specific octal to binary formula for conversion. However, if you are looking for an easier and less complicated octal to binary conversion method, the direct method involves the following steps:

In octal to binary conversion, each digit in the octal number has a three-digit binary representation. Using it, we can easily convert a number from octal to binary. We can convert octal to binary by choosing the binary equivalent of every digit of the octal number from the below-mentioned chart.

**Octal to Binary Conversion Table**

The following table includes the octal numbers 0 to 7 and their equivalent three-digit binary representation to aid octal to binary conversion.

Octal Number | Equivalent Three-digit Binary Representation |
---|---|

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

Let’s convert $54_{8}$ to binary.

The number 54 has two digits: 5 and 4.

From the above chart, we will note down their binary equivalents.

$5 → 101$

$4 → 100$

Now, combining the two, we get the following binary number: $101100_{2}$.

## Octal to Binary Conversion without Using the Conversion Table

Let’s understand the steps with the help of an example. We will also understand how the three-digit binary representation is obtained for each octal digit.

**Example: Convert **$765_{8}$**to binary.**

**Step 1:** Write the octal number by separating the digits.

7 | 6 | 5 |

**Step 2:** Each octal digit represents 3 binary bits. Starting from right to left, the value of these three digits is $2^{0} = 1,\; 2^{1} = 2$, and $2^{2} = 4$ respectively. Thus, write (4, 2, 1) below each octal digit.

7 | 6 | 5 | ||||||

4 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 1 |

**Step 3:** Identify the numbers among 4, 2, and 1 (powers of 2), which add up to the octal number written on the top. Write 1 below if the number is used. Write 0 below the number that is not used in the sum. For example, $7 = 4 + 2 + 1$, so we write 1 under all the three numbers.

7 | 6 | 5 | ||||||

4 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 1 |

1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 |

**Step 4:** Write the 1s and 0s from left to write to find the binary equivalent of the given octal number.

Thus, $765_{8} = 111110101_{2}$

## Octal to Binary Conversion: Examples

Let’s look at some examples based on all the methods we learned.

**Example 1: Find the binary equivalent of the octal number 34**_{8}**.**

**Octal to decimal:**

$34_{8} = (3 \times 8^{1}) + (4 \times 8^{0})$

$34_{8} = 24 + 4$

$34_{8} = 28_{10}$

**Decimal to binary:**

Division by 2 | Quotient | Remainder |
---|---|---|

$28 \div 2$ | 14 | 0 |

$14 \div 2$ | 7 | 0 |

$7 \div 2$ | 3 | 1 |

$3 \div 2$ | 1 | 1 |

$1 \div 2$ | 0 | 1 |

Thus, $34_{8} = 28_{10} = 11100_{2}$

**Example 2: Convert 4567**_{8 }**from octal to binary.**

4 | 5 | 6 | 7 | ||||||||

4 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 1 |

1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |

Thus, $4567_{8} = 100101110111_{2}$

## Facts about Octal and Binary Conversion

- Octal to binary conversion is all about converting numbers from base 8 to base 2.
- Each octal digit has a three-digit binary representation. So, the octal to binary conversion is simple.
- The binary number system forms the base of all operations and computing systems.
- Octal number system has great applications in computer programming.

## Conclusion

In this article, we learned about octal to binary conversion, different methods with steps, and the conversion chart. Let’s solve a few examples and practice problems to understand it better.

## Solved Examples of Octal to Binary Conversion

**1. Convert the given octal number system to binary: **$64_{8}$**.**

**Solution:**

From the table, we get the following binary equivalents:

$6 = 110_{2}$

$4 = 100_{2}$

Thus, the octal number $64_{8}$ can be written as $110100_{2}$.

**2. Convert the octal number **$52_{8}$** to binary using the indirect decimal conversion method.**

**Solution: **

Given octal number $52_{8}$. Using the decimal conversion method, we will convert $52_{8}$ as follows:

$⇒ 52_{8} = 2 \times 8^{0} + 5 \times 8^{1}$

$⇒ 52_{8} = 2 \times 1 + 5 \times 8$

$= 52_{8} = 2 + 40$

$= 52_{8} = 42_{10}$

Now, we will convert 42_{10} to a binary number as follows.

Division | Quotient | Remainder |
---|---|---|

$42 \div 2$ | 21 | 0 |

$21 \div 2$ | 10 | 1 |

$10 \div 2$ | 5 | 0 |

$5 \div 2$ | 2 | 1 |

$2 \div 2$ | 1 | 0 |

$1 \div 2$ | 0 | 1 |

On arranging all the remainders in the reverse order, we get

$42_{10} = 101010$.

Thus, the octal to binary conversion for the number $52_{8}$_{ }is 101010.

**3. Convert the octal number **$25_{8}$** to its binary equivalent using the direct method. **

**Solution:**

Using the binary conversion table, we have

$2_{8} = 010_{2}$

$5_{8} = 101_{2}$

On combining the two bits, we get $25_{8} = 010101_{2}$

**4. Convert 213**_{8}** from octal to binary.**

**Solution: **

2 | 1 | 3 | ||||||

4 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 1 |

0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |

Thus, $213_{8} = 10001011_{2}$

## Practice Problems on Octal to Binary Conversion

## Octal to Binary Conversion - Definition, Table, Examples, Facts

### Which of the following is the right representation of the octal number 23?

On substituting the values from the octal to the binary conversion table, we get $2 = 010$ and $3 = 011$. Thus, by combining the two, we get 10011.

### What is the binary equivalent of the octal number $73_{8}$?

On substituting the values from the octal to the binary conversion table, we get

$7_{8} = 111_{2}$ and $3_{8} = 011_{2}$

$73_{8} = 111011_{2}$

### What is the equivalent of the octal number 4 in binary?

The binary equivalent of 4 is $100_{2}$.

### Convert $777_{8}$ into binary.

$7_{8} \rightarrow 111_{2}$

$777_{8} = 111111111_{2}$

## Frequently Asked Questions on Octal to Binary Conversion

**What is a base in the number system?**

A base is also called a radix in a number system. It refers to the total number of digits used in a particular number system to form other numbers. For instance, the binary number system has base 2, i.e., it has two digits, 1 and 2.

**Does the octal system have numbers 8 and 9?**

No. The base of the octal system is 8. It uses the digits 0 to 7. The number 8 is written in the octal system as 10. The number 9 is written as 11.

**How can we convert a number from binary to octal?**

We know that each octal digit is a 3-bit binary number. The binary to octal conversion can be done by grouping 3 binary bits and identifying the octal number from the conversion chart.

**Example:** $10110100_{2}$^{ }can be grouped using 3 bits as

0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |

$010_{2} = 2_{8}$ | $110_{2} = 6_{8}$ | $100_{2} = 4_{8}$ |

The octal equivalent of 10110100 is $264_{8}$.