The binary number system is a base-2 numeral system that uses only two digits: 0 and 1. It is widely used in digital electronics and computer systems because it aligns well with the on/off states of electronic circuits.
Key Features of the Binary Number System:
Base-2 System: Each digit in a binary number represents a power of 2, starting from 2 to the power of 0 on the far right.
Digits (Bits): Only 0 and 1 are used.
Positional Value: The value of a digit depends on its position in the number. For example:
Binary: 1010
Decimal Equivalent: (1 × 2 to the power of 3)+(0 × 2 to the power of 2)+(1 × 2 to the power of 1)+(0 × 2 the power of 0) 1×22=4= 8 + 0 + 2 + 0 =10
Conversion Between Binary and Decimal:
Binary to Decimal: Multiply each binary digit by 2position2^{\text{position}}2position (position starts from 0 on the right) and sum up the results.
Example: 110111011101
1 × 2 to the power of 3 = 8
1 × 2 to the power of 2 = 4
0 × 2 to the power of 1 = 0
1 × 2 to the power of 0 = 1
Total: 8 + 4 + 0 + 1 = 138 + 4 + 0 + 1 = 138 + 4 + 0 + 1 = 13
Decimal to Binary: Repeatedly divide the decimal number by 2, record the remainder (0 or 1), and read the remainders in reverse order.
Example: Convert 19 to binary:
19 ÷ 2 = 9 remainder 1
9 ÷ 2 = 4 remainder 1
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Then read all the remainders from bottom to top
The Binary is 10011
Binary Arithmetic:
Addition:
Follows the rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1 to the next column)
Example: 101 + 110 = 1011
Subtraction:
Uses borrowing, similar to decimal subtraction.
Example: 110 − 101 = 1
Multiplication:
Similar to decimal multiplication, but simpler since it’s only 0 or 1.
Example: 101 × 10 = 1010101 \times 10 = 1010101 × 10 = 1010
Division:
Similar to long division in the decimal system.
Advantages of Binary:
Simplicity in Circuit Design: Only two states (0 and 1) are needed to represent data, making it ideal for digital circuits.
Error Detection: Easier to implement error detection and correction methods.
Reliability: Less prone to noise and signal degradation compared to higher base systems.
Applications of Binary:
Computer Systems: Data storage, processing, and instructions are all encoded in binary.
Digital Electronics: Circuit logic, like AND, OR, and XOR gates, works on binary inputs.
Communication Systems: Encoding and transmitting data in binary simplifies signal processing.
That is the explanation of Binary.
