How to change from octal to binary?
2 Answers
To convert a number from octal (base 8) to binary (base 2), you can follow a straightforward method because both number systems are based on powers of 2. Here’s a detailed step-by-step guide:
### Step-by-Step Conversion
1. **Understand the Octal System**:
- The octal system uses digits from 0 to 7. Each digit represents a power of 8.
- For example, the octal number \(345_8\) can be broken down as:
\[
3 \times 8^2 + 4 \times 8^1 + 5 \times 8^0 = 192 + 32 + 5 = 229_{10} \text{ (in decimal)}
\]
2. **Convert Each Octal Digit to Binary**:
- Each octal digit can be represented as a 3-digit binary number (since \(2^3 = 8\)).
- Here’s a quick reference for octal to binary conversions:
- \(0_8 = 000_2\)
- \(1_8 = 001_2\)
- \(2_8 = 010_2\)
- \(3_8 = 011_2\)
- \(4_8 = 100_2\)
- \(5_8 = 101_2\)
- \(6_8 = 110_2\)
- \(7_8 = 111_2\)
3. **Example Conversion**:
- Let’s say we want to convert \(345_8\) to binary.
- Convert each digit:
- \(3_8 = 011_2\)
- \(4_8 = 100_2\)
- \(5_8 = 101_2\)
4. **Combine the Binary Digits**:
- Now, concatenate the binary representations of each digit together:
- \(011 \, 100 \, 101\)
5. **Final Result**:
- So, the octal number \(345_8\) is equivalent to \(011100101_2\) in binary.
- You can omit the leading zeros, resulting in \(11100101_2\).
### Summary of the Process
- **Convert** each octal digit to a 3-bit binary number.
- **Concatenate** the binary strings to get the final binary representation.
### Why This Works
Each octal digit directly corresponds to a set of three binary digits because \(2^3 = 8\). Therefore, you can efficiently convert from octal to binary without having to go through decimal as an intermediary step.
This method is systematic and can be applied to any octal number, making it easy to perform conversions quickly and accurately.
### Step-by-Step Conversion
1. **Understand the Octal System**:
- The octal system uses digits from 0 to 7. Each digit represents a power of 8.
- For example, the octal number \(345_8\) can be broken down as:
\[
3 \times 8^2 + 4 \times 8^1 + 5 \times 8^0 = 192 + 32 + 5 = 229_{10} \text{ (in decimal)}
\]
2. **Convert Each Octal Digit to Binary**:
- Each octal digit can be represented as a 3-digit binary number (since \(2^3 = 8\)).
- Here’s a quick reference for octal to binary conversions:
- \(0_8 = 000_2\)
- \(1_8 = 001_2\)
- \(2_8 = 010_2\)
- \(3_8 = 011_2\)
- \(4_8 = 100_2\)
- \(5_8 = 101_2\)
- \(6_8 = 110_2\)
- \(7_8 = 111_2\)
3. **Example Conversion**:
- Let’s say we want to convert \(345_8\) to binary.
- Convert each digit:
- \(3_8 = 011_2\)
- \(4_8 = 100_2\)
- \(5_8 = 101_2\)
4. **Combine the Binary Digits**:
- Now, concatenate the binary representations of each digit together:
- \(011 \, 100 \, 101\)
5. **Final Result**:
- So, the octal number \(345_8\) is equivalent to \(011100101_2\) in binary.
- You can omit the leading zeros, resulting in \(11100101_2\).
### Summary of the Process
- **Convert** each octal digit to a 3-bit binary number.
- **Concatenate** the binary strings to get the final binary representation.
### Why This Works
Each octal digit directly corresponds to a set of three binary digits because \(2^3 = 8\). Therefore, you can efficiently convert from octal to binary without having to go through decimal as an intermediary step.
This method is systematic and can be applied to any octal number, making it easy to perform conversions quickly and accurately.
To convert an octal number (base 8) to a binary number (base 2), follow these steps:
### Step-by-Step Process:
1. **Understand the Basics:**
- An octal number system uses digits from 0 to 7.
- A binary number system uses digits 0 and 1.
2. **Convert Each Octal Digit to a 3-bit Binary Equivalent:**
- Each digit in an octal number can be represented by a 3-bit binary number because \(2^3 = 8\).
- Below is a conversion table for quick reference:
| Octal Digit | Binary Equivalent |
|-------------|-------------------|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
3. **Write the Binary Equivalent for Each Octal Digit:**
- Replace each octal digit with its corresponding 3-bit binary equivalent from the table.
4. **Combine the Binary Groups:**
- After converting each octal digit, combine all the 3-bit binary groups together to get the final binary number.
### Example:
Let's convert the octal number **245** to binary.
1. **Break down the octal number:**
- 2 → 010
- 4 → 100
- 5 → 101
2. **Combine the binary digits:**
- 245 (octal) = 010 100 101 (binary)
So, the binary equivalent of the octal number **245** is **010100101**.
### Summary:
To convert from octal to binary, simply replace each octal digit with its corresponding 3-bit binary value and combine them. This method is straightforward and avoids the need for any complex calculations.
### Step-by-Step Process:
1. **Understand the Basics:**
- An octal number system uses digits from 0 to 7.
- A binary number system uses digits 0 and 1.
2. **Convert Each Octal Digit to a 3-bit Binary Equivalent:**
- Each digit in an octal number can be represented by a 3-bit binary number because \(2^3 = 8\).
- Below is a conversion table for quick reference:
| Octal Digit | Binary Equivalent |
|-------------|-------------------|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
3. **Write the Binary Equivalent for Each Octal Digit:**
- Replace each octal digit with its corresponding 3-bit binary equivalent from the table.
4. **Combine the Binary Groups:**
- After converting each octal digit, combine all the 3-bit binary groups together to get the final binary number.
### Example:
Let's convert the octal number **245** to binary.
1. **Break down the octal number:**
- 2 → 010
- 4 → 100
- 5 → 101
2. **Combine the binary digits:**
- 245 (octal) = 010 100 101 (binary)
So, the binary equivalent of the octal number **245** is **010100101**.
### Summary:
To convert from octal to binary, simply replace each octal digit with its corresponding 3-bit binary value and combine them. This method is straightforward and avoids the need for any complex calculations.