How to Convert Between Binary, Decimal, Hexadecimal, and Octal?

Converting between binary, decimal, hexadecimal, and octal number systems involves understanding their bases and using specific methods for each conversion. Here's a comprehensive guide:

Understanding Number Bases

• Binary (Base-2): Uses only 0 and 1.
• Decimal (Base-10): The standard number system we use daily (0-9).
• Hexadecimal (Base-16): Uses 0-9 and A-F (A=10, B=11, C=12, D=13, E=14, F=15).
• Octal (Base-8): Uses 0-7.

Conversion Methods

1. Binary to Decimal:

• Multiply each bit by 2 raised to the power of its position (starting from 0 on the rightmost bit).

• Sum the results.

  Example: Binary 101101 becomes:

  (1 * 2^5) + (0 * 2^4) + (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 32 + 0 + 8 + 4 + 0 + 1 = 45 (Decimal)

2. Decimal to Binary:

• Repeatedly divide the decimal number by 2.

• Note the remainders (0 or 1) at each step.

• Write the remainders in reverse order to get the binary equivalent.

  Example: Decimal 45 becomes:

  • 45 / 2 = 22, Remainder = 1
  • 22 / 2 = 11, Remainder = 0
  • 11 / 2 = 5, Remainder = 1
  • 5 / 2 = 2, Remainder = 1
  • 2 / 2 = 1, Remainder = 0
  • 1 / 2 = 0, Remainder = 1

  Result: 101101 (Binary)

3. Binary to Hexadecimal:

• Group the binary digits into sets of 4, starting from the right. If necessary, add leading zeros to complete the leftmost group.

• Convert each group of 4 binary digits into its corresponding hexadecimal digit.

  Example: Binary 11010110 becomes:

  • 1101 0110
  • D 6

  Result: D6 (Hexadecimal)

  Binary	Hexadecimal
  0000	0
  0001	1
  0010	2
  0011	3
  0100	4
  0101	5
  0110	6
  0111	7
  1000	8
  1001	9
  1010	A
  1011	B
  1100	C
  1101	D
  1110	E
  1111	F

4. Hexadecimal to Binary:

• Convert each hexadecimal digit into its corresponding 4-bit binary representation.

  Example: Hexadecimal D6 becomes:

  • D 6
  • 1101 0110

  Result: 11010110 (Binary)

5. Binary to Octal:

• Group the binary digits into sets of 3, starting from the right. Add leading zeros if necessary.

• Convert each group of 3 binary digits into its corresponding octal digit.

  Example: Binary 1011010 becomes:

  • 010 110 10 (Add a leading zero)
  • 2 6 2

  Result: 262 (Octal)

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

6. Octal to Binary:

• Convert each octal digit into its corresponding 3-bit binary representation.

  Example: Octal 262 becomes:

  • 2 6 2
  • 010 110 010

  Result: 010110010 (Binary) Leading zeros can be dropped, resulting in 10110010.

7. Decimal to Hexadecimal:

• Repeatedly divide the decimal number by 16.

• Note the remainders at each step (0-15). If the remainder is 10-15, represent it as A-F.

• Write the remainders in reverse order.

  Example: Decimal 214 becomes:

  • 214 / 16 = 13, Remainder = 6
  • 13 / 16 = 0, Remainder = 13 (D)

  Result: D6 (Hexadecimal)

8. Hexadecimal to Decimal:

• Multiply each hexadecimal digit by 16 raised to the power of its position (starting from 0 on the rightmost digit). Remember that A=10, B=11, C=12, D=13, E=14, F=15.

• Sum the results.

  Example: Hexadecimal D6 becomes:

  • (13 16^1) + (6 16^0) = (13 16) + (6 1) = 208 + 6 = 214 (Decimal)

9. Decimal to Octal:

• Repeatedly divide the decimal number by 8.

• Note the remainders at each step (0-7).

• Write the remainders in reverse order.

  Example: Decimal 170 becomes:

  • 170 / 8 = 21, Remainder = 2
  • 21 / 8 = 2, Remainder = 5
  • 2 / 8 = 0, Remainder = 2

  Result: 252 (Octal)

10. Octal to Decimal:

• Multiply each octal digit by 8 raised to the power of its position (starting from 0 on the rightmost digit).

• Sum the results.

  Example: Octal 252 becomes:

  • (2 8^2) + (5 8^1) + (2 8^0) = (2 64) + (5 8) + (2 1) = 128 + 40 + 2 = 170 (Decimal)

By following these methods, you can accurately convert between binary, decimal, hexadecimal, and octal number systems.