Binary Numbering System Examples

The binary numbering system, also known as base-2, represents numeric values using two symbols: 0 and 1. Each digit in a binary number is referred to as a bit. Binary numbers are the foundation of digital systems and computers, where binary logic is used to perform calculations and store data. Understanding binary is crucial for computer science and electronics.

Below are examples demonstrating the binary system and conversion between binary and decimal systems.

Examples

Binary to Decimal

Binary: 1011

• Calculation: (1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0)
• Decimal: (8 + 0 + 2 + 1 = 11)

Binary: 1101

• Calculation: (1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0)
• Decimal: (8 + 4 + 0 + 1 = 13)

Decimal to Binary:

Decimal: 9

• Calculation: (9 \div 2 = 4 \text{ remainder } 1), (4 \div 2 = 2 \text{ remainder } 0), (2 \div 2 = 1 \text{ remainder } 0), (1 \div 2 = 0 \text{ remainder } 1)
• Binary: 1001

Decimal: 14

• Calculation: (14 \div 2 = 7 \text{ remainder } 0), (7 \div 2 = 3 \text{ remainder } 1), (3 \div 2 = 1 \text{ remainder } 1), (1 \div 2 = 0 \text{ remainder } 1)
• Binary: 1110

References

1. Tanenbaum, A. S., & Austin, T. (2012). Structured Computer Organization (6th ed.). Pearson.
2. Patterson, D. A., & Hennessy, J. L. (2013). Computer Organization and Design: The Hardware/Software Interface (5th ed.). Morgan Kaufmann.
3. Stallings, W. (2015). Computer Organization and Architecture (10th ed.). Pearson.
4. Knuth, D. E. (1997). The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley.
5. Tokheim, R. (2003). Digital Electronics: Principles and Applications (7th ed.). McGraw-Hill.