Binary Representation of Integers

Binary representation is fundamental in computer science and digital electronics. Understanding how integers are represented in binary is crucial for working with computers, as they operate using binary numbers. This article will provide a detailed explanation of binary representation, including how to convert between binary and decimal, as well as some common applications.

Number Systems Overview

1. Decimal Number System (Base 10):

   • Uses digits from 0 to 9.
   • Each digit in a number represents a power of 10.
   • Example: $345_{10}$ represents $3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 = 300 + 40 + 5 = 345$.

2. Binary Number System (Base 2):

   • Uses only two digits: 0 and 1.
   • Each digit (called a bit) represents a power of 2.
   • Example: $1011_2$ represents $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$ in decimal.

Binary Representation of Integers

An integer can be represented in binary by expressing it as a sum of powers of 2. Each binary digit (bit) indicates whether a particular power of 2 is included in the sum. The rightmost bit (least significant bit) represents $2^0$, the next bit to the left represents $2^1$, and so on.

Step-by-Step Conversion: Decimal to Binary

To convert a decimal integer to binary, follow these steps:

1. Divide the integer by 2.
2. Record the remainder (it will be 0 or 1).
3. Divide the quotient by 2.
4. Repeat the process until the quotient is 0.
5. The binary representation is the sequence of remainders read from bottom to top.

Example 1: Convert $25_{10}$ to binary.

1. $25 \div 2 = 12$ with a remainder of 1.
2. $12 \div 2 = 6$ with a remainder of 0.
3. $6 \div 2 = 3$ with a remainder of 0.
4. $3 \div 2 = 1$ with a remainder of 1.
5. $1 \div 2 = 0$ with a remainder of 1.

Reading the remainders from bottom to top, $25_{10} = 11001_2$.

Example 2: Convert $45_{10}$ to binary.

1. $45 \div 2 = 22$ with a remainder of 1.
2. $22 \div 2 = 11$ with a remainder of 0.
3. $11 \div 2 = 5$ with a remainder of 1.
4. $5 \div 2 = 2$ with a remainder of 1.
5. $2 \div 2 = 1$ with a remainder of 0.
6. $1 \div 2 = 0$ with a remainder of 1.

Reading the remainders from bottom to top, $45_{10} = 101101_2$.

Conversion: Binary to Decimal

To convert a binary number to decimal, simply multiply each bit by its corresponding power of 2 and sum the results.

Example: Convert $101101_2$ to decimal.

• $101101_2 = 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$
• $= 32 + 0 + 8 + 4 + 0 + 1$
• $= 45_{10}$

Binary Representation of Negative Integers

In binary, negative integers are often represented using a method called two’s complement.

Two’s Complement Representation

1. Find the binary representation of the absolute value of the integer.
2. Invert the digits (change 0s to 1s and 1s to 0s).
3. Add 1 to the least significant bit (rightmost bit).

Example: Represent $-13$ in binary (8-bit representation).

1. The binary representation of $13_{10}$ is $00001101_2$ (8-bit).
2. Inverting the digits gives $11110010_2$.
3. Adding 1 gives $11110011_2$.

Thus, $-13$ is represented as $11110011_2$ in an 8-bit system.

Verification: Converting Two’s Complement Back to Decimal

To verify that $11110011_2$ represents $-13$, you can reverse the two’s complement process.

1. Subtract 1 from the binary number: $11110011_2 – 1 = 11110010_2$.
2. Invert the digits: $00001101_2$ which is $13_{10}$.
3. The negative sign is added back, confirming that $11110011_2$ represents $-13$.

Applications of Binary Representation

• Computers and Digital Systems: All data and instructions in a computer are represented in binary, making it the fundamental language of computing.
• Networking: IP addresses and subnet masks are often represented in binary to perform network calculations.
• Digital Electronics: Logic circuits and digital systems operate using binary, with 0 and 1 representing off and on states.

Conclusion

Binary representation of integers is a cornerstone of computer science and digital systems. By understanding how to convert between decimal and binary, and how negative numbers are represented using two’s complement, one can gain a deeper insight into how computers process and store data. This knowledge is essential for fields like programming, networking, and electronics.

References

1. Introduction to Computer Organization and Architecture by Linda Null and Julia Lobur: This textbook provides a detailed explanation of binary representation and other number systems, as well as their application in computer architecture.
2. Digital Logic and Computer Design by M. Morris Mano: A foundational book that covers binary numbers, two’s complement, and other key concepts in digital logic and design.
3. The Art of Computer Programming, Volume 2: Seminumerical Algorithms by Donald E. Knuth: This book delves into number representations, including binary, and their importance in algorithms and computation.
4. Binary Numbers – Khan Academy: An online resource that explains binary numbers, conversion techniques, and applications. Available at: Khan Academy.

These references provide comprehensive information on binary representation, its mathematical foundation, and its applications in various fields.