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Binary system

The binary system is a base-2 numeral system that represents numeric values using two symbols: 0 and 1. Unlike the decimal system, which is base-10 and uses ten digits (0 to 9), the binary system has only two digits. This system forms the core of computing because computers operate using binary values, processing data through a series of electrical signals that represent binary states (0 and 1).

Why Binary System in Computers?

Computers use the binary system because it aligns with their physical electronic components, which operate with two states: on and off. This binary representation can be efficiently managed by electrical circuits, where:

  • 0 represents a low voltage (off state).
  • 1 represents a high voltage (on state).

Using binary is reliable, as it’s easier to distinguish between two states (on/off) than multiple voltage levels, which could lead to more errors.

Binary Representation and Counting

  1. Binary Digits (Bits):
    • Each digit in a binary number is called a bit (binary digit), the smallest unit of data in computing.
    • In binary, each position represents a power of 2, just as each position in decimal represents a power of 10.
  2. Counting in Binary:
    • In the binary system, counting proceeds as follows:
      • 0 = 0 in decimal
      • 1 = 1 in decimal
      • 10 = 2 in decimal
      • 11 = 3 in decimal
      • 100 = 4 in decimal
      • 101 = 5 in decimal, and so on.
  3. Positional Values:
    • Just as in decimal, where the rightmost position represents 10010^0100, the binary system’s rightmost position represents 202^020.
    • For example, in the binary number 1011:
      • Rightmost 1 = 20=12^0 = 120=1
      • Second position 1 = 21=22^1 = 221=2
      • Third position 0 = 22=02^2 = 022=0
      • Leftmost 1 = 23=82^3 = 823=8
    • Adding these values gives: 8+0+2+1=118 + 0 + 2 + 1 = 118+0+2+1=11 in decimal.

Converting Between Decimal and Binary

  1. Decimal to Binary:
    • To convert a decimal number to binary, you repeatedly divide the number by 2 and record the remainder. The binary result is the sequence of remainders read in reverse order.
    • For example, to convert 13 to binary:
      • 13 ÷ 2 = 6, remainder 1
      • 6 ÷ 2 = 3, remainder 0
      • 3 ÷ 2 = 1, remainder 1
      • 1 ÷ 2 = 0, remainder 1
      • Binary result: 1101
  2. Binary to Decimal:
    • To convert a binary number to decimal, you multiply each bit by 2 raised to the power of its position (starting from 0 on the right) and then add the results.
    • For example, to convert 1101 to decimal:
      • 1×23+1×22+0×21+1×20=8+4+0+1=131 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 131×23+1×22+0×21+1×20=8+4+0+1=13

Binary Arithmetic

Binary arithmetic follows the same principles as decimal arithmetic, with operations on bits instead of decimal digits.

  1. Addition:
    • Binary addition rules:
      • 0 + 0 = 0
      • 1 + 0 = 1
      • 1 + 1 = 10 (0 with a carry of 1 to the next higher bit position)
    • Example: 101 + 11 in binary:markdownCopy code 101 + 011 ----- 1000 Result: 1000 (8 in decimal)
  2. Subtraction, Multiplication, and Division:
    • Similar rules apply as in decimal, using binary digits.

Applications of the Binary System

  1. Data Storage and Processing:
    • All data in computers (text, images, sound) is stored as binary values. Each character or color pixel is represented by a unique binary code.
  2. Digital Logic Circuits:
    • Binary forms the basis of logic gates in processors, such as AND, OR, and NOT gates, which perform basic logical operations on binary inputs.
  3. Network Data Transmission:
    • Data is transmitted in binary format over digital networks, encoded as electrical signals, light pulses, or radio waves representing 0s and 1s.
  4. Programming and Machine Code:
    • High-level programming languages are ultimately translated into binary machine code that the CPU executes.

Binary System and Hexadecimal

Due to the complexity of binary numbers, computer scientists use hexadecimal (base-16) as a shorthand. Each hexadecimal digit represents four binary digits (bits). This makes it easier to read and write binary values for programming and debugging purposes.

For example:

  • Binary 11010100 = Hexadecimal D4

Conclusion

The binary system is fundamental to computers, providing a simple, reliable way to represent and process data using on-off electrical states. This base-2 system allows computers to handle complex tasks and is integral to everything from basic data storage to intricate processing operations in modern computing.