What is Binary number system full explanation.

 

Binary numbers are a fundamental part of digital systems and computer technology. Understanding binary numbers involves grasping the base-2 numeral system, which is the foundation of binary arithmetic and digital circuit design. Here's a comprehensive explanation:

1. Definition and Basics

• Binary Number System: The binary system is a base-2 numeral system. It uses only two digits: 0 and 1. Each digit in a binary number is called a bit, which stands for binary digit.

2. Binary vs. Decimal Systems

• Decimal System: The standard numeral system used in everyday life is the decimal system, which is base-10 and uses digits from 0 to 9.
• Binary Representation: Each binary digit (bit) represents an increasing power of 2, starting from the rightmost bit, which is the least significant bit (LSB).

3. Binary to Decimal Conversion

To convert a binary number to a decimal number, sum the products of each bit and its corresponding power of 2. For example, the binary number 1011 can be converted to decimal as follows: $1\times 2^3+0\times 2^2+1\times 2^1+1\times 2^0$ $=1\times 8+0\times 4+1\times 2+1\times 1$ $=8+0+2+1$ $=11$

4. Decimal to Binary Conversion

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. Read the remainders from bottom to top. For example, converting the decimal number 13 to binary:

1. 13 ÷ 2 = 6 remainder 1
2. 6 ÷ 2 = 3 remainder 0
3. 3 ÷ 2 = 1 remainder 1
4. 1 ÷ 2 = 0 remainder 1 So, 13 in binary is 1101.

5. Binary Arithmetic

• Addition: Similar to decimal addition, but it carries over when the sum exceeds 1. For example:

  e
    1011
  + 1101
  ------
  11000
  

  Explanation:
  • 1 + 1 = 10 (write 0, carry 1)
  • 1 + 1 + 1 (carry) = 11 (write 1, carry 1)
  • 0 + 1 + 1 (carry) = 10 (write 0, carry 1)
  • 1 (carry) = 1
• Subtraction, Multiplication, and Division: Follows similar rules to decimal arithmetic but with binary digits.

6. Binary Fractions

Binary fractions are represented similarly to decimal fractions but use negative powers of 2. For example, the binary number 110.101 can be converted to decimal as: $1\times 2^2+1\times 2^1+0\times 2^0+1\times 2^{-1}+0\times 2^{-2}+1\times 2^{-3}$ $=4+2+0+0.5+0+0.125$ $=6.625$

7. Use in Computers

• Data Representation: Computers use binary to represent all types of data, including numbers, characters, and instructions. Each type of data is encoded in binary format.
• Logic Gates: Digital circuits in computers use logic gates that operate on binary signals (0 and 1) to perform computations and control functions.
• Memory and Storage: Data in computers is stored in binary format. Memory units like bytes (8 bits) and larger units (kilobytes, megabytes) are all based on binary multiples.

8. Advantages of Binary System

• Simplicity in Digital Circuits: Binary systems simplify the design of digital circuits, which only need to distinguish between two states (on/off, high/low, 1/0).
• Error Detection and Correction: Binary data allows for straightforward methods to detect and correct errors using parity bits and checksums.

Summary

Binary numbers are a base-2 numeral system using digits 0 and 1. They are fundamental to computer systems, representing data and performing arithmetic. Binary is efficient for digital electronics and essential for understanding how computers process information.