Binary Number System

Overview

Binary (base 2) is the fundamental number system of computing. Every piece of data in a computer — numbers, text, images, instructions — is ultimately represented in binary.

Binary Basics

Each digit (bit) represents a power of 2:

Position:  7    6    5    4    3    2    1    0
Power:     2^7  2^6  2^5  2^4  2^3  2^2  2^1  2^0
Value:     128  64   32   16   8    4    2    1

Binary to Decimal

10110101₂ = 1×128 + 0×64 + 1×32 + 1×16 + 0×8 + 1×4 + 0×2 + 1×1
          = 128 + 32 + 16 + 4 + 1
          = 181₁₀

Decimal to Binary

Method 1: Division by 2

181 ÷ 2 = 90  remainder 1
 90 ÷ 2 = 45  remainder 0
 45 ÷ 2 = 22  remainder 1
 22 ÷ 2 = 11  remainder 0
 11 ÷ 2 = 5   remainder 1
  5 ÷ 2 = 2   remainder 1
  2 ÷ 2 = 1   remainder 0
  1 ÷ 2 = 0   remainder 1

Read remainders bottom to top: 10110101₂

Method 2: Subtraction

181 - 128 = 53  → bit 7 = 1
 53 - 64  = -11 → bit 6 = 0
 53 - 32  = 21  → bit 5 = 1
 21 - 16  = 5   → bit 4 = 1
  5 - 8   = -3  → bit 3 = 0
  5 - 4   = 1   → bit 2 = 1
  1 - 2   = -1  → bit 1 = 0
  1 - 1   = 0   → bit 0 = 1

Result: 10110101₂

Binary Arithmetic

Addition

  1011  (11)
+ 1101  (13)
------
11000  (24)

Rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0 carry 1)
1 + 1 + 1 = 11 (1 carry 1)

Subtraction

  1101  (13)
- 0101  (5)
------
  1000  (8)

Rules:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1)

Multiplication

  101  (5)
× 110  (6)
------
  000  (101 × 0)
 101   (101 × 1, shift left)
101    (101 × 1, shift left)
------
11110  (30)

Binary Representations

Unsigned Binary

Range for n bits: 0 to 2^n - 1

Signed Binary (Sign-Magnitude)

MSB is sign bit (0=positive, 1=negative):

+5 = 00000101
-5 = 10000101

Problem: Two representations of zero (+0 and -0).

Two’s Complement

The standard for signed integers. See Two’s Complement.

Binary Coded Decimal (BCD)

Each decimal digit is encoded in 4 bits:

92₁₀ = 1001 0010 (BCD)

Use: Financial calculations (exact decimal representation).

Binary in Computing

Interview Questions

1. Q: Convert 42 to binary. A: 42 ÷ 2 = 21 R0, 21÷2 = 10 R1, 10÷2 = 5 R0, 5÷2 = 2 R1, 2÷2 = 1 R0, 1÷2 = 0 R1. Reading bottom-up: 101010₂. Verification: 32+8+2 = 42.

2. Q: How many bits are needed to represent 1000? A: 2^9 = 512, 2^10 = 1024. Need 10 bits (range 0-1023). log₂(1000) ≈ 9.97, round up to 10.

3. Q: What is overflow in binary addition? A: When the result exceeds the representable range. For 8-bit unsigned: 200+100=300 > 255, overflow. For signed: 127+1=128 > 127 (8-bit signed max), overflow.

4. Q: Why is hexadecimal used in computing? A: Each hex digit maps to exactly 4 binary digits. Hex is more compact (0xFF vs 11111111) and easier to read. Used for memory addresses, MAC addresses, color codes.

Common Mistakes

• Confusing bit positions (MSB is leftmost, not rightmost)
• Forgetting that 2^n has n+1 bits (100…0)
• Not understanding overflow in fixed-width arithmetic
• Confusing sign-magnitude with two’s complement

Summary

Binary is the foundation of all computer data. Understanding conversions, arithmetic, and representations (unsigned, sign-magnitude, two’s complement) is essential. Hex and octal are convenient shorthand for binary.

Cross-References

• Number Systems Overview
• Hexadecimal — Compact binary representation
• Two’s Complement — Signed integers
• Floating Point — Real numbers

Cross References

• Hexadecimal
• Two’s Complement
• ALU