In-Depth Notes on Number Systems, Operations, and Codes in Digital Electronics

Decimal Numbers

• Definition: The decimal number system is a weighted number system with a radix (base) of 10, using digits 0 through 9.

• Column Weights: Each digit place represents a power of ten, increasing from right to left:

  • Whole numbers: $10^0, 10^1, 10^2, 10^3, …$
  • Fractional numbers: $10^{-1}, 10^{-2}, 10^{-3}, …$

• Example: Representing 568.23 in weighted sums:

  • $5 \times 10^2 + 6 \times 10^1 + 8 \times 10^0 + 2 \times 10^{-1} + 3 \times 10^{-2}$
  • Calculation: $500 + 60 + 8 + 0.2 + 0.03 = 568.23$

Binary Numbers

• Definition: The binary number system is used in digital systems, having a radix (base) of 2 with digits 0 and 1.

• Column Weights: Powers of two increase from right to left:

  • Whole numbers: $2^0, 2^1, 2^2, …$
  • Fractional numbers: $2^{-1}, 2^{-2}, …$

• Counting Up To: With $n$ bits, the highest number is $2^n - 1$.

• Example with 6 bits: Maximum = $2^6 - 1 = 63$ (0 to 63)

Binary Weighting Structure

• Right-most bit = LSB (Least Significant Bit)
• Left-most bit = MSB (Most Significant Bit)
• For fractional binary, weights are negative powers of 2.

Binary to Decimal Conversion

• Procedure: Add values of bits equal to 1 and ignore bits that are 0.

• Example: Binary number 1101101:

  • $1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 64 + 32 + 0 + 8 + 4 + 0 + 1 = 109$

• Fractional Example: Binary number 0.1011:

  • $1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4} = 0.5 + 0 + 0.125 + 0.0625 = 0.6875$

Decimal to Binary Conversion

1. Sum of Weights Method: Identify powers of 2 that sum to the decimal number.
   • Example: Convert 25: $16 + 8 + 1 \rightarrow 11001$
2. Repeated Division by 2: Continually divide by 2 and track remainders.

• Example: Decimal 19 to binary:
  1. 19 / 2 = 9 remainder 1
  2. 9 / 2 = 4 remainder 1
  3. 4 / 2 = 2 remainder 0
  4. 2 / 2 = 1 remainder 0
  5. 1 / 2 = 0 remainder 1
  • Thus, 19 = 10011

Binary Arithmetic Operations

1. Binary Addition Rules:

   • $0+0=0$, $0+1=1$, $1+0=1$, $1+1=10$ (carry 1 to next bit)
   • Example: $11 + 11 = 110$ (3 + 3 = 6)

2. Binary Subtraction Rules:

   • Requires borrowing like decimal subtraction.
   • Example: $11 - 01 = 10$ (3 - 1 = 2)

3. Binary Multiplication:

   • Similar to decimal multiplication; involves partial products.
   • Example: $11 \times 11 = 1001$ (3*3=9)

4. Binary Division:

   • Follows the same principles as decimal division.

Complements in Binary Numbers

• 1's Complement: Invert all bits.
• 2's Complement: Add 1 to the 1's complement.
  • Example: 1's complement of 10110010 is 01001101; adding 1 makes it 01001110.

Signed Binary Numbers

• Sign Bit: Left-most bit indicates the sign (0 = positive, 1 = negative).
  • Sign-Magnitude Form: Sign bit + magnitude.
  • Example: +81 as 01010001, -25 as 10011001.
• 1's Complement Form: Positive same as sign-magnitude, negative as 1's complement.
  • Example: +25 as 00011001, -25 as 11100110 (1's complement).
• 2's Complement Form: Positive same as elsewhere, negative as 2's complement of positive.
  • Example: +25 as 00110001, -25 as 11001111 (2's complement).

Hexadecimal Number System

• Definition: Base 16 system using digits 0-9 and letters A-F.
• Conversion:
  • Binary to Hexadecimal: Groups of 4 bits.
  • Hexadecimal to Binary: Reverse process.

ASCII and Digital Codes

• ASCII: American Standard Code for Information interchange, using 7-bit binary.
• Example: Character 'A' is 1000001.
• Gray Code: A binary numeral system where two successive values differ in only one bit.