What is the classification of number system definition?
Number systems are broadly categorized as positional or non-positional. Positional systems, like binary and decimal, assign a value to each digit based on its position. Non-positional systems do not.
Classification of Number Systems: Positional vs. Non-Positional
Number systems serve as the foundation for representing numerical quantities. They provide a structured way to express and perform arithmetic operations. Based on their fundamental properties, number systems are broadly classified into two main categories: positional and non-positional.
Positional Number Systems
Positional number systems, such as the binary system (base 2) and the decimal system (base 10), are characterized by the principle that the value of each digit in a number is determined by its position within the number. This means that the same digit can represent different values depending on its position.
For example, in the decimal system, the digit ‘3’ represents the value 3 when it appears as the rightmost digit, but it represents the value 30 when it appears as the second digit from the right.
Non-Positional Number Systems
In contrast to positional number systems, non-positional number systems do not rely on the position of digits to determine their value. Instead, each symbol or character represents a specific numerical value that remains constant regardless of its placement within the number.
An example of a non-positional number system is Roman numerals. In this system, each symbol represents a specific value, such as I for one, V for five, and X for ten. The value of a number is simply the sum of the values of its constituent symbols.
Comparison of Positional and Non-Positional Systems
| Feature | Positional Systems | Non-Positional Systems |
|---|---|---|
| Value of digits | Determined by position | Fixed, regardless of position |
| Calculation of value | Add values based on positions | Add values directly |
| Efficiency | Efficient for large numbers | Less efficient for large numbers |
| Base | Can have any base (e.g., binary, decimal) | Do not have a base |
| Examples | Binary, decimal, hexadecimal | Roman numerals, Egyptian numerals |
Applications
Positional number systems, such as binary and decimal, are widely used in modern computing and communication systems due to their efficiency and ease of manipulation. Non-positional number systems, such as Roman numerals, are primarily of historical interest and are occasionally used in specific contexts, such as clock faces and numbering of chapters in books.