Question

How to convert octal to hexadecimal and vice versa?

Answer 1

Converting between octal (base-8) and hexadecimal (base-16) number systems involves a few steps, but it's straightforward once you get the hang of it. Here’s a step-by-step guide to help you with both conversions:

### Converting Octal to Hexadecimal

1. **Convert Octal to Decimal:**
   - Each digit in an octal number represents a power of 8. Convert the octal number to decimal by expanding it in terms of powers of 8.
   - For example, to convert the octal number \( 345_8 \) to decimal:
     \[
     345_8 = 3 \cdot 8^2 + 4 \cdot 8^1 + 5 \cdot 8^0 = 192 + 32 + 5 = 229_{10}
     \]

2. **Convert Decimal to Hexadecimal:**
   - Once you have the decimal equivalent, convert it to hexadecimal by repeatedly dividing the decimal number by 16 and noting the remainders.
   - For 229 in decimal:
     \[
     229 \div 16 = 14 \text{ remainder } 5
     \]
     \[
     14 \div 16 = 0 \text{ remainder } 14 \text{ (which is } E \text{ in hexadecimal)}
     \]
     - Read the remainders from bottom to top: \( E5_{16} \)

   Thus, \( 345_8 \) converts to \( E5_{16} \).

### Converting Hexadecimal to Octal

1. **Convert Hexadecimal to Decimal:**
   - Each digit in a hexadecimal number represents a power of 16. Convert the hexadecimal number to decimal by expanding it in terms of powers of 16.
   - For example, to convert the hexadecimal number \( A7_{16} \) to decimal:
     \[
     A7_{16} = 10 \cdot 16^1 + 7 \cdot 16^0 = 160 + 7 = 167_{10}
     \]

2. **Convert Decimal to Octal:**
   - Convert the decimal number to octal by repeatedly dividing the decimal number by 8 and noting the remainders.
   - For 167 in decimal:
     \[
     167 \div 8 = 20 \text{ remainder } 7
     \]
     \[
     20 \div 8 = 2 \text{ remainder } 4
     \]
     \[
     2 \div 8 = 0 \text{ remainder } 2
     \]
     - Read the remainders from bottom to top: \( 247_8 \)

   Thus, \( A7_{16} \) converts to \( 247_8 \).

### Summary

1. **Octal to Hexadecimal:**
   - Convert octal to decimal.
   - Convert decimal to hexadecimal.

2. **Hexadecimal to Octal:**
   - Convert hexadecimal to decimal.
   - Convert decimal to octal.

Understanding these conversions can be helpful in various computing and programming scenarios, such as when dealing with different numeral systems or low-level programming.

Answer 2

Converting between octal and hexadecimal involves a few steps, primarily converting through binary since both octal and hexadecimal systems align well with binary representation. Here’s how you can do it:

### Octal to Hexadecimal

1. **Convert Octal to Binary:**
   - Each octal digit corresponds to a 3-bit binary sequence. For example:
     - Octal `0` is Binary `000`
     - Octal `1` is Binary `001`
     - Octal `2` is Binary `010`
     - Octal `3` is Binary `011`
     - Octal `4` is Binary `100`
     - Octal `5` is Binary `101`
     - Octal `6` is Binary `110`
     - Octal `7` is Binary `111`
   - Convert each octal digit to its 3-bit binary equivalent and combine them.

2. **Convert Binary to Hexadecimal:**
   - Group the binary digits into sets of 4, starting from the right (add leading zeros if necessary).
   - Convert each 4-bit binary group to its hexadecimal equivalent. For example:
     - Binary `0000` is Hexadecimal `0`
     - Binary `0001` is Hexadecimal `1`
     - Binary `0010` is Hexadecimal `2`
     - Binary `0011` is Hexadecimal `3`
     - Binary `0100` is Hexadecimal `4`
     - Binary `0101` is Hexadecimal `5`
     - Binary `0110` is Hexadecimal `6`
     - Binary `0111` is Hexadecimal `7`
     - Binary `1000` is Hexadecimal `8`
     - Binary `1001` is Hexadecimal `9`
     - Binary `1010` is Hexadecimal `A`
     - Binary `1011` is Hexadecimal `B`
     - Binary `1100` is Hexadecimal `C`
     - Binary `1101` is Hexadecimal `D`
     - Binary `1110` is Hexadecimal `E`
     - Binary `1111` is Hexadecimal `F`

### Example

- **Octal `345` to Hexadecimal:**

  1. **Convert to Binary:**
     - Octal `3` is Binary `011`
     - Octal `4` is Binary `100`
     - Octal `5` is Binary `101`
     - Combined binary is `011100101`

  2. **Pad and Group Binary:**
     - Binary `011100101` (pad to 12 bits) becomes `0011 1001 0101`.

  3. **Convert to Hexadecimal:**
     - Binary `0011` is Hexadecimal `3`
     - Binary `1001` is Hexadecimal `9`
     - Binary `0101` is Hexadecimal `5`
     - Thus, Hexadecimal `395`.

### Hexadecimal to Octal

1. **Convert Hexadecimal to Binary:**
   - Each hexadecimal digit corresponds to a 4-bit binary sequence. For example:
     - Hexadecimal `0` is Binary `0000`
     - Hexadecimal `1` is Binary `0001`
     - Hexadecimal `2` is Binary `0010`
     - Hexadecimal `3` is Binary `0011`
     - Hexadecimal `4` is Binary `0100`
     - Hexadecimal `5` is Binary `0101`
     - Hexadecimal `6` is Binary `0110`
     - Hexadecimal `7` is Binary `0111`
     - Hexadecimal `8` is Binary `1000`
     - Hexadecimal `9` is Binary `1001`
     - Hexadecimal `A` is Binary `1010`
     - Hexadecimal `B` is Binary `1011`
     - Hexadecimal `C` is Binary `1100`
     - Hexadecimal `D` is Binary `1101`
     - Hexadecimal `E` is Binary `1110`
     - Hexadecimal `F` is Binary `1111`
   - Convert each hexadecimal digit to its 4-bit binary equivalent and combine them.

2. **Convert Binary to Octal:**
   - Group the binary digits into sets of 3, starting from the right (add leading zeros if necessary).
   - Convert each 3-bit binary group to its octal equivalent. For example:
     - Binary `000` is Octal `0`
     - Binary `001` is Octal `1`
     - Binary `010` is Octal `2`
     - Binary `011` is Octal `3`
     - Binary `100` is Octal `4`
     - Binary `101` is Octal `5`
     - Binary `110` is Octal `6`
     - Binary `111` is Octal `7`

### Example

- **Hexadecimal `2F` to Octal:**

  1. **Convert to Binary:**
     - Hexadecimal `2` is Binary `0010`
     - Hexadecimal `F` is Binary `1111`
     - Combined binary is `00101111`

  2. **Pad and Group Binary:**
     - Binary `00101111` (pad to 9 bits) becomes `001 011 111`.

  3. **Convert to Octal:**
     - Binary `001` is Octal `1`
     - Binary `011` is Octal `3`
     - Binary `111` is Octal `7`
     - Thus, Octal `137`.

These steps help in converting octal numbers to hexadecimal and vice versa by leveraging the binary system as an intermediary.