Question

How do you convert hexadecimal to octal and vice versa?

Answer 1

To convert between hexadecimal and octal, it's often easiest to use binary as an intermediary. Here’s how you can do it:

### Hexadecimal to Octal

1. **Convert Hexadecimal to Binary:**
   Each hex digit converts to a 4-bit binary equivalent.
   - Example: \(A\) in hex is \(1010\) in binary.

2. **Group Binary into Sets of 3:**
   Starting from the right, group the binary digits into sets of three. If necessary, add leading zeros to the leftmost group to make it complete.
   - Example: \(1010\) becomes \(001 010\).

3. **Convert Binary to Octal:**
   Each 3-bit binary group converts to an octal digit.
   - Example: \(001\) is \(1\) and \(010\) is \(2\), so \(1010_{16} = 12_{8}\).

### Octal to Hexadecimal

1. **Convert Octal to Binary:**
   Each octal digit converts to a 3-bit binary equivalent.
   - Example: \(5\) in octal is \(101\) in binary.

2. **Group Binary into Sets of 4:**
   Starting from the right, group the binary digits into sets of four. If necessary, add leading zeros.
   - Example: \(101\) becomes \(0101\).

3. **Convert Binary to Hexadecimal:**
   Each 4-bit binary group converts to a hex digit.
   - Example: \(0101\) is \(5\), so \(5_{8} = 5_{16}\).

### Summary

- **Hex to Octal:** Hex → Binary → Group into 3 → Octal
- **Octal to Hex:** Octal → Binary → Group into 4 → Hex

Let me know if you’d like examples or further clarification!

Answer 2

Converting between hexadecimal and octal involves a few steps. Both are positional numeral systems but with different bases: hexadecimal is base-16, and octal is base-8. Here’s a detailed process for converting between these systems:

### Converting Hexadecimal to Octal

1. **Convert Hexadecimal to Binary:**
   - Each hexadecimal digit corresponds to a 4-bit binary sequence. Use a lookup table to convert each hexadecimal digit to its 4-bit binary equivalent.
   - Example: To convert `2F` from hexadecimal to binary:
     - `2` in hex is `0010` in binary.
     - `F` in hex is `1111` in binary.
     - So, `2F` in binary is `0010 1111`.

2. **Group Binary Digits into Sets of Three:**
   - Octal digits correspond to 3-bit binary sequences. Starting from the right (least significant bits), group the binary digits into sets of three.
   - Example: For `0010 1111`, group as `00 101 111 1`. Add leading zeros to make complete groups if necessary: `001 011 111`.

3. **Convert Each 3-Bit Group to Octal:**
   - Convert each 3-bit group into its octal equivalent using a lookup table.
   - Example: `001` in binary is `1` in octal, `011` is `3`, `111` is `7`.
   - So, `0010 1111` in octal is `137`.

### Converting Octal to Hexadecimal

1. **Convert Octal to Binary:**
   - Each octal digit corresponds to a 3-bit binary sequence. Use a lookup table to convert each octal digit to its 3-bit binary equivalent.
   - Example: To convert `137` from octal to binary:
     - `1` in octal is `001` in binary.
     - `3` in octal is `011` in binary.
     - `7` in octal is `111` in binary.
     - So, `137` in binary is `001 011 111`.

2. **Group Binary Digits into Sets of Four:**
   - Hexadecimal digits correspond to 4-bit binary sequences. Starting from the right, group the binary digits into sets of four.
   - Example: For `001 011 111`, group as `0001 0111 1110`. Add leading zeros to make complete groups if necessary: `0001 0111 1111`.

3. **Convert Each 4-Bit Group to Hexadecimal:**
   - Convert each 4-bit group into its hexadecimal equivalent using a lookup table.
   - Example: `0001` in binary is `1` in hex, `0111` is `7`, `1111` is `F`.
   - So, `001 011 111` in hexadecimal is `17F`.

### Summary

- **Hexadecimal to Octal:** Convert hex to binary, group binary digits into sets of three, and convert each group to octal.
- **Octal to Hexadecimal:** Convert octal to binary, group binary digits into sets of four, and convert each group to hex.

This approach ensures accurate conversion between the two numeral systems.