Number System Notes & Formulas

1. Definition

A system for representing numeric values. Digital systems use Binary (base-2). Other common bases: Octal (8), Hexadecimal (16), Decimal (10).

2. Conversion Methods (Very Important)

• Decimal → Binary: Repeated division by 2, remainders read bottom-up
• Binary → Decimal: Σ (bit × 2^position)
• Binary → Octal: Group bits in 3s from right
• Binary → Hex: Group bits in 4s from right
• Fractional part: Multiply by base repeatedly, take integer part

3. Complements (Very Important)

r's complement (2's complement for binary):

• 2's complement = 1's complement + 1
• 1's complement = flip all bits
• Used for signed number representation and subtraction

Subtraction using 2's complement:
A − B = A + (2's complement of B). If carry out = 1 → result positive; no carry → negative result (take 2's complement of result)

4. Signed Number Representation

| Format | Range (n bits) | Negation |
|---|---|---|
| Sign-Magnitude | −(2ⁿ⁻¹−1) to +(2ⁿ⁻¹−1) | Flip MSB |
| 1's Complement | −(2ⁿ⁻¹−1) to +(2ⁿ⁻¹−1) | Flip all bits |
| 2's Complement | −2ⁿ⁻¹ to +(2ⁿ⁻¹−1) | Flip all + 1 |

5. Codes

• BCD (Binary Coded Decimal): Each decimal digit encoded in 4 bits. 0000-1001 valid, 1010-1111 invalid
• Gray Code: Successive values differ by 1 bit only. Used in K-maps and shaft encoders
• Excess-3: BCD + 3. Self-complementing code
• ASCII: 7-bit character encoding

6. IEEE 754 Floating Point

• Single (32-bit): 1 sign + 8 exponent (bias 127) + 23 mantissa
• Double (64-bit): 1 sign + 11 exponent (bias 1023) + 52 mantissa
• Stored as: (−1)^s × 1.mantissa × 2^(exponent−bias)

7. GATE Trick

2's complement range is asymmetric - one extra negative number. BCD addition: if sum > 9 or carry generated, add 6 (0110) to correct. Gray to binary: MSB same, each subsequent bit = Gray bit XOR previous binary bit.