🔢 Signed Number Representations in Binary

🧠 What Are Signed Numbers?

Signed numbers allow binary systems to represent both positive and negative values. The most significant bit (MSB) typically acts as the sign bit:

• 0 → positive
• 1 → negative

🔧 Common Encoding Schemes

1️⃣ Sign-Magnitude

+5 → 0101
−5 → 1101

• MSB is the sign; remaining bits are magnitude
• Two representations of zero: 0000 (+0), 1000 (−0)

2️⃣ One’s Complement

+5 → 0101
−5 → 1010  (bitwise NOT of +5)

• Negative numbers are bitwise inverses of positives
• Two zeros: 0000 (+0), 1111 (−0)
• Requires end-around carry in addition

3️⃣ Two’s Complement ✅ (Most Common)

+5 → 0101
−5 → 1011  (bitwise NOT of +5 → 1010, then +1)

• Single representation of zero: 0000
• Clean arithmetic: no special rules for subtraction
• Range: −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1

🧮 Range Comparison (4-bit)