Number System in Computer

A Detailed Note on Number System in Computer

Number System

A number system is a systematic method to represent numbers. It provides a consistent way to denote the quantity and values of numbers. Understanding different number systems is foundational to many branches of mathematics and computing.

1. Base or Radix

The base or radix of a number system refers to the number of unique digits (including zero) used to represent numbers in that system. The most commonly used systems are:

• Decimal System (Base-10): Uses ten digits from 0 to 9. It is the system we use in everyday life.
• Binary System (Base-2): Uses only two digits, 0 and 1. Fundamental in computing.
• Octal System (Base-8): Uses eight digits from 0 to 7.
• Hexadecimal System (Base-16): Uses sixteen digits: 0-9 and A-F (where A represents 10, B for 11, and so on).

2. Conversion between Bases

Converting numbers between different bases is essential in computing and other applications. Methods include:

• Decimal to Binary: Divide by 2 and note remainders.
• Binary to Decimal: Multiply each binary digit by its place value and sum up.
• Similar conversions exist for other bases.

3. Properties of Number Systems

• Addition and Subtraction: Rules are consistent across all systems.
• Multiplication: Multiplying numbers in different bases involves carrying over values based on the base.
• Division: Also follows consistent rules but can be more complex due to remainders.

4. Applications

• Computing: Binary numbers are foundational in digital computing due to their direct mapping to on/off states of computer switches.
• Cryptography: Hexadecimal is commonly used to represent cryptographic keys and values.
• Networking: IP addresses are represented in a dotted-decimal format, which is a mix of decimal and binary representations.

5. Positional Notation

In a positional number system, the value of a digit depends on its position. For instance, in the decimal system, the value of the digit ‘5’ in ’53’ is 5, but in ’50’, its value is 50. This positional nature is consistent across all bases.

6. Non-positional Systems

While most common number systems are positional, there are non-positional systems, like the Roman numeral system. Here, the value of a symbol doesn’t solely depend on its position but also on the symbols around it.

7. Floating-point Representation

Apart from whole numbers, number systems can represent fractions. In binary, this involves representing numbers as a sum of powers of 2. However, due to limitations in representation, some numbers might have approximations.

8. Number Bases in Real-world Context

• Money: Some cultures historically used base-20 for counting due to using fingers and toes, resulting in systems where numbers like 60 (3 x 20) were significant.
• Time: Ancient systems, like the Babylonian, had a base-60 system for time, leading to divisions like 60 seconds in a minute, 60 minutes in an hour.

9. Limitations and Approximations

• Representation Limitations: Not all numbers can be represented exactly in a finite number of digits, leading to approximations in some systems.
• Precision Issues: In computing, representing real numbers in binary can lead to rounding errors, especially when performing repeated operations.

Conclusion:

Number systems are foundational in mathematics, computing, and many real-world applications. While the principles of representing numbers remain consistent, the choice of base and system depends on the specific requirements and contexts of different fields and cultures.

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