Number System Conversions and Arithmetic

Hexadecimal Conversions

• Hex to Binary: Break the hexadecimal number into four-bit groups, starting from the right, and replace each group with its hexadecimal equivalent.
  • Example: Convert $1084_{16}$ to binary.
    • Break $4$ into its four-bit sequence: $0100$.
    • $8$ becomes $1000$.
    • $0$ becomes $0000$.
    • $1$ becomes $0001$.
    • The binary equivalent is $0001 0000 1000 0100$.
• Negative Hexadecimal Numbers: Use two's complement; a method will be covered but is not on the test.

Hexadecimal to Decimal Conversion

• Method 1: Convert hexadecimal to binary (four-bit groups), then convert binary to decimal.
• Method 2: Use the sum of weights method.
  • The first character is the ones place ($16^0$).
  • The second character is the sixteens place ($16^1$).
  • The third character is the 256s place ($16^2$).
  • The fourth character is the 4,096s place ($16^3$).

Visual Aid for Number System Conversions

• A visual aid is available on Teams to help organize conversions between hexadecimal, binary, and decimal.
• Four bits per hex character when converting to binary.
• Three bits represent an octal character.
• Blue indicates the sum of weights method.
• Red indicates repeated division.
• BCD (Binary Coded Decimal) will be covered in a later lecture.

Applying Sum of Weights in Decimal and Hexadecimal

• Decimal Example: The number $142$ in base 10 can be broken down as:
  • $(1 * 10^2) + (4 * 10^1) + (2 * 10^0) = 100 + 40 + 2 = 142$.
• Hexadecimal: The sequence for hex numbers is based on powers of 16.
  • Ones place ($16^0$).
  • Sixteens place ($16^1$).
  • 256s place ($16^2$).
  • 4,096s place ($16^3$).

Examples

• Example 1: Convert $1C_{16}$ to decimal.
  • $(1 * 16^1) + (12 * 16^0) = 16 + 12 = 28_{10}$.
• Example 2: Convert $85_{16}$ to decimal.
  • $(8 * 16^1) + (5 * 16^0) = (8 * 16) + 5 = 128 + 5 = 133_{10}$.
• An alternate method is to convert the hexadecimal to its four-bit binary sequences (nibbles) and then use the sum of weights method.

Converting Decimal to Hexadecimal

• Repeated division by 16.

Hexadecimal Addition

• Think in decimal but subtract 16 for each carry-over.
• Example: $23<em>{16} + 16</em>{16} = 39_{16}$. (Not 39 in decimal terms).
  • If the result exceeds 15, subtract 16 to carry over.
• Example: $58<em>{16} + 22</em>{16} = 7A_{16}$.
• Example: $2B<em>{16} + 84</em>{16} = AF_{16}$.

Complex Hexadecimal Addition

• Example: $DF<em>{16} + AC</em>{16}$.
  • $F$ (15) + $C$ (12) = 27. Since 27 > 15, subtract 16 resulting in 11 ($B$) with a carry-over of 1.
  • $D$ (13) + $A$ (10) + 1 (carry-over) = 24. Since 24 > 15, subtract 16 resulting in 8 with a carry-over of 1.
  • Final Answer: $18B_{16}$.
• The process involves carrying over values, similar to decimal addition, but subtracting by 16 instead of 10.

Hexadecimal Subtraction Using Two's Complement

• Convert the hexadecimal number to binary, perform the two's complement on the binary, and then convert back to hex.

Methods for Computing Two's Complement of Hexadecimal Numbers

• Method 1: Convert to binary, perform two's complement, convert back to hex.
  • Example: Find the two's complement of $2A<em>{16}$. Convert $2A</em>{16}$ to binary, perform the two's complement operation, and the two’s complement is $D6_{16}$.
• Method 2: Subtract from the maximum value and add one.
• Method 3: A translation table where each hex digit is converted to its complement and then one is added.

Octal Number System

• Base 8, using digits 0 through 7.
• Octal values:
  • 10 (in decimal) is represented as 12 in octal (1 eight and 2 ones).
  • 11: 13
• Weighted values are powers of 8.
  • $8^0$: Ones place.
  • $8^1$: Eights place.
  • $8^2$: 64s place.

Octal to Decimal Conversion

• Use the sum of weights method.
  • Multiply the character by the weighted column, then add the next weighted portion.
  • Example: Convert $2374_8$ to decimal.
    • $(2 * 8^3) + (3 * 8^2) + (7 * 8^1) + (4 * 8^0)$
    • $(2 * 512) + (3 * 64) + (7 * 8) + 4$
    • $1024 + 192 + 56 + 4 = 1276_{10}$.
• Repeated division by 8 to convert a decimal to octal.

Octal to Binary Conversion

• Each octal character is represented by three bits.
• To convert, replace each octal digit with its three-bit binary equivalent and concatenate.
  • Example: Convert $13<em>8$ to binary. 3 is 011, 1 is 001, thus $13</em>8$ is 001011.

Binary to Octal Conversion

• Start at the least significant bit (LSB).
• Group the bits into groups of three.
• Convert each group to an octal character.

Key Takeaways

• Convert everything to binary as an intermediary step.