Question

How to convert an octal number to binary?

Answer 1

Converting an octal number to binary is straightforward because each octal digit corresponds to three binary digits (bits). Here’s how to do it step by step:

1. **Write down the octal number.** For example, let's use the octal number 257.

2. **Convert each octal digit to its binary equivalent.** Use the following conversions:
   - 0 → 000
   - 1 → 001
   - 2 → 010
   - 3 → 011
   - 4 → 100
   - 5 → 101
   - 6 → 110
   - 7 → 111

   For 257:
   - 2 → 010
   - 5 → 101
   - 7 → 111

3. **Combine the binary equivalents.** Write them together without spaces:
   - 2 → 010
   - 5 → 101
   - 7 → 111

   So, 257 in octal converts to:
   \[
   010 \, 101 \, 111
   \]

4. **Remove any leading zeros if necessary.** The final binary representation is:
   \[
   10101111
   \]

So, the octal number \(257\) converts to the binary number \(10101111\).

Answer 2

Converting an octal number to binary is straightforward because each octal digit maps directly to a 3-bit binary number. Here’s a step-by-step guide:

### 1. **Understand the Octal and Binary Systems:**

- **Octal (Base-8):** Uses digits 0 through 7.
- **Binary (Base-2):** Uses digits 0 and 1.

Each octal digit can be directly converted to a 3-bit binary equivalent.

### 2. **Create a Conversion Table:**

Here's a table of octal digits and their binary equivalents:

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

### 3. **Convert Each Octal Digit to Binary:**

1. **Write Down the Octal Number:**
   - For example, let's convert the octal number `572`.

2. **Convert Each Digit Individually:**
   - Break down the octal number into its individual digits: `5`, `7`, and `2`.
   - Using the table:
     - Octal `5` → Binary `101`
     - Octal `7` → Binary `111`
     - Octal `2` → Binary `010`

3. **Combine the Binary Results:**
   - Combine the binary representations of each digit to get the final binary number.
   - For `572` in octal, this becomes `101 111 010` in binary.

4. **Remove Leading Zeros (if necessary):**
   - The final binary number `101111010` doesn’t need any leading zeros.

### Example Conversion:

Let’s do an example with the octal number `341`.

1. **Break It Down:**
   - `3` → `011`
   - `4` → `100`
   - `1` → `001`

2. **Combine:**
   - Combine them: `011 100 001`

3. **Result:**
   - The binary representation of octal `341` is `11100001`.

### Summary

To convert an octal number to binary:

1. Look up each octal digit’s binary equivalent.
2. Write down the binary equivalents.
3. Concatenate them to get the final binary number.

This method is efficient because it leverages the direct relationship between octal and binary systems, simplifying the conversion process.