Number Systems

Overview

Digital circuits operate on discrete voltage levels representing binary digits (bits). Use this quick reference after a programming language course when you want to understand what integers, characters, addresses, and overflow look like under the code. Understanding number systems is essential for interpreting data, addresses, and encodings at the hardware level.

Positional Number Systems

Every positional system has a radix (base). A number's value is the sum of each digit multiplied by the base raised to its position power.

General formula: (dₙdₙ₋₁…d₁d₀)ᵣ = Σ dᵢ × rⁱ

Example — binary 1011₂: 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11₁₀

Binary ↔ Decimal Conversion

Decimal → Binary (successive division)

Divide by 2 repeatedly; remainders (bottom-up) form the binary number.

45 ÷ 2 = 22 R 1  (LSB)
22 ÷ 2 = 11 R 0
11 ÷ 2 =  5 R 1
 5 ÷ 2 =  2 R 1
 2 ÷ 2 =  1 R 0
 1 ÷ 2 =  0 R 1  (MSB)
45₁₀ = 101101₂

Decimal → Binary (fractions, successive multiplication)

Multiply fraction by 2; integer parts (top-down) form the binary fraction.

0.625 × 2 = 1.25  → 1
0.25  × 2 = 0.5   → 0
0.5   × 2 = 1.0   → 1
0.625₁₀ = 0.101₂

Hexadecimal ↔ Binary

Each hex digit maps to exactly 4 bits.

Example: 0xAF = 1010 1111₂ = 175₁₀

Signed Number Representations

Sign-Magnitude

MSB is the sign bit (0 = positive, 1 = negative). Two representations of zero (+0 and −0).

One's Complement

Negate by flipping all bits. Still has two zeros.

Two's Complement (standard)

Negate by flipping all bits and adding 1. One representation of zero; used in virtually all modern hardware.

Range for n bits: −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1

Quick negate: flip bits, add 1. Example: +5 = 0101 → flip → 1010 → +1 → 1011 = −5 ✓

Binary Arithmetic

Addition

Carry rules identical to decimal but base-2:

  0111  (+7)
+ 0001  (+1)
------
  1000  (+8)

Carry chain:  1 1 1 0

Overflow Detection (Two's Complement)

Overflow occurs when adding two numbers of the same sign and getting the opposite sign.

• Two positive numbers → negative result: overflow
• Two negative numbers → positive result: overflow
• Mixed signs: never overflow

Subtraction via Two's Complement

A − B = A + (−B) = A + (~B + 1)

7 − 3 = 7 + (−3)
  0111
+ 1101  (−3 in two's complement)
------
 10100  → ignore carry → 0100 = 4 ✓

Binary Coded Decimal (BCD)

Each decimal digit encoded separately in 4 bits (0000–1001 valid; 1010–1111 invalid).

BCD is used in financial and display hardware where human-readable digits matter more than compact storage.

Character Encodings

ASCII layout landmarks:

• 0x30–0x39 → digits '0'–'9'
• 0x41–0x5A → uppercase 'A'–'Z'
• 0x61–0x7A → lowercase 'a'–'z'
• Lowercase = uppercase + 0x20 (set bit 5)

Powers of 2 Reference