Question

How to convert octal hexadecimal to binary and vice versa?

Answer 1

Converting between octal, hexadecimal, and binary systems involves understanding how numbers are represented in each system and using simple conversion rules. Here’s a detailed guide on how to convert between these bases:

### Converting Octal to Binary

1. **Write down the octal number.** Each digit in an octal number is in the range 0-7.
2. **Convert each octal digit to a 3-bit binary equivalent.** Each octal digit directly maps to a 3-bit binary number.
   - Octal 0 = Binary 000
   - Octal 1 = Binary 001
   - Octal 2 = Binary 010
   - Octal 3 = Binary 011
   - Octal 4 = Binary 100
   - Octal 5 = Binary 101
   - Octal 6 = Binary 110
   - Octal 7 = Binary 111
3. **Combine the 3-bit binary groups.** Concatenate the binary values of each octal digit to get the final binary number.

**Example: Convert octal 345 to binary.**
- 3 in octal = 011 in binary
- 4 in octal = 100 in binary
- 5 in octal = 101 in binary
- Combined binary = 011 100 101

### Converting Binary to Octal

1. **Write down the binary number.**
2. **Group the binary digits into sets of three, starting from the right.** If the number of binary digits is not a multiple of three, pad the leftmost group with zeros.
3. **Convert each 3-bit binary group to its octal equivalent.**
   - Binary 000 = Octal 0
   - Binary 001 = Octal 1
   - Binary 010 = Octal 2
   - Binary 011 = Octal 3
   - Binary 100 = Octal 4
   - Binary 101 = Octal 5
   - Binary 110 = Octal 6
   - Binary 111 = Octal 7
4. **Combine the octal digits.**

**Example: Convert binary 110101 to octal.**
- Grouped binary = 110 101
- 110 in binary = 6 in octal
- 101 in binary = 5 in octal
- Combined octal = 65

### Converting Hexadecimal to Binary

1. **Write down the hexadecimal number.** Each digit in hexadecimal is in the range 0-15 (represented as 0-9 and A-F).
2. **Convert each hexadecimal digit to a 4-bit binary equivalent.**
   - Hex 0 = Binary 0000
   - Hex 1 = Binary 0001
   - Hex 2 = Binary 0010
   - Hex 3 = Binary 0011
   - Hex 4 = Binary 0100
   - Hex 5 = Binary 0101
   - Hex 6 = Binary 0110
   - Hex 7 = Binary 0111
   - Hex 8 = Binary 1000
   - Hex 9 = Binary 1001
   - Hex A = Binary 1010
   - Hex B = Binary 1011
   - Hex C = Binary 1100
   - Hex D = Binary 1101
   - Hex E = Binary 1110
   - Hex F = Binary 1111
3. **Combine the 4-bit binary groups.**

**Example: Convert hexadecimal 2F to binary.**
- 2 in hexadecimal = 0010 in binary
- F in hexadecimal = 1111 in binary
- Combined binary = 0010 1111

### Converting Binary to Hexadecimal

1. **Write down the binary number.**
2. **Group the binary digits into sets of four, starting from the right.** If the number of binary digits is not a multiple of four, pad the leftmost group with zeros.
3. **Convert each 4-bit binary group to its hexadecimal equivalent.**
   - Binary 0000 = Hex 0
   - Binary 0001 = Hex 1
   - Binary 0010 = Hex 2
   - Binary 0011 = Hex 3
   - Binary 0100 = Hex 4
   - Binary 0101 = Hex 5
   - Binary 0110 = Hex 6
   - Binary 0111 = Hex 7
   - Binary 1000 = Hex 8
   - Binary 1001 = Hex 9
   - Binary 1010 = Hex A
   - Binary 1011 = Hex B
   - Binary 1100 = Hex C
   - Binary 1101 = Hex D
   - Binary 1110 = Hex E
   - Binary 1111 = Hex F
4. **Combine the hexadecimal digits.**

**Example: Convert binary 11010101 to hexadecimal.**
- Grouped binary = 1101 0101
- 1101 in binary = D in hexadecimal
- 0101 in binary = 5 in hexadecimal
- Combined hexadecimal = D5

By following these steps, you can efficiently convert between octal, hexadecimal, and binary systems.

Answer 2

Converting between octal, hexadecimal, and binary systems is quite straightforward once you understand the base systems involved. Each of these numbering systems is related to each other in a systematic way, and conversion between them is often done through binary as an intermediate step. Here's a detailed guide on how to convert between these systems:

### Binary to Octal

1. **Group the Binary Digits**: Start by grouping the binary digits into sets of three, starting from the right (least significant bit). Add leading zeros if necessary to make complete groups of three.

   For example, for the binary number `101110`:
   - Group into threes: `10 111 10` (add leading zeros): `010 111 110`

2. **Convert Each Group**: Convert each group of three binary digits to its octal equivalent.
   - `010` (binary) = `2` (octal)
   - `111` (binary) = `7` (octal)
   - `110` (binary) = `6` (octal)

   Therefore, `101110` (binary) = `276` (octal).

### Octal to Binary

1. **Convert Each Octal Digit**: Convert each octal digit to its 3-bit binary equivalent.

   For example, for the octal number `276`:
   - `2` (octal) = `010` (binary)
   - `7` (octal) = `111` (binary)
   - `6` (octal) = `110` (binary)

2. **Combine the Binary Digits**: Concatenate the binary equivalents together.

   So, `276` (octal) = `010 111 110` (binary) = `101110` (binary).

### Binary to Hexadecimal

1. **Group the Binary Digits**: Group the binary digits into sets of four, starting from the right. Add leading zeros if necessary to make complete groups of four.

   For example, for the binary number `101110`:
   - Group into fours: `10 1110` (add leading zeros): `0010 1110`

2. **Convert Each Group**: Convert each group of four binary digits to its hexadecimal equivalent.
   - `0010` (binary) = `2` (hexadecimal)
   - `1110` (binary) = `E` (hexadecimal)

   Therefore, `101110` (binary) = `2E` (hexadecimal).

### Hexadecimal to Binary

1. **Convert Each Hexadecimal Digit**: Convert each hexadecimal digit to its 4-bit binary equivalent.

   For example, for the hexadecimal number `2E`:
   - `2` (hexadecimal) = `0010` (binary)
   - `E` (hexadecimal) = `1110` (binary)

2. **Combine the Binary Digits**: Concatenate the binary equivalents together.

   So, `2E` (hexadecimal) = `0010 1110` (binary) = `101110` (binary).

### Octal to Hexadecimal and Vice Versa

To convert directly between octal and hexadecimal, you can use binary as an intermediary:

1. **Octal to Binary**: Convert the octal number to binary.
2. **Binary to Hexadecimal**: Convert the resulting binary number to hexadecimal.

**Example:**
Convert `276` (octal) to hexadecimal:

1. **Octal to Binary**: `276` (octal) = `010 111 110` (binary) = `101110` (binary).
2. **Binary to Hexadecimal**: `101110` (binary) = `2E` (hexadecimal).

**Direct Conversion:**
To convert directly from octal to hexadecimal:
1. Convert each octal digit to its 3-bit binary equivalent.
2. Combine the binary digits.
3. Convert the resulting binary to hexadecimal.

**Example:**
Convert `276` (octal) to hexadecimal:

1. Convert `276` (octal) to binary: `010 111 110`.
2. Combine the binary: `010111110`.
3. Group into four bits for hexadecimal: `0010 1111 10` (add leading zero): `0010 1111 1110`.
4. Convert `0010 1111 1110` (binary) to hexadecimal: `2FE` (hexadecimal).

Similarly, for hexadecimal to octal:
1. Convert hexadecimal to binary.
2. Convert binary to octal.

With these steps, you can easily convert between octal, hexadecimal, and binary systems.