Which of the following is a non-positional number system?

Question

Prepp Team · Accepted Answer

Understanding Number Systems

Number systems are ways to represent numbers using symbols. Different number systems have different rules for assigning value to symbols and combining them to represent quantities.

Number systems can be broadly classified into two types: positional and non-positional number systems.

Positional vs. Non-Positional Number Systems

Let's look at the key differences between these two types of number systems.

Feature	Positional Number System	Non-Positional Number System
Value of a Digit	Depends on its position within the number as well as its intrinsic value. Each position has a weight (usually a power of the base).	Does not depend on its position. Each symbol has a fixed intrinsic value.
Complexity	More complex for manual calculations for large numbers, but efficient for calculations using machines (computers).	Simpler for basic representation, but complex for performing arithmetic operations.
Number of Symbols	Uses a limited set of symbols (digits) defined by the base of the system.	May use a limited or unlimited set of symbols.
Example	Decimal (Base 10), Binary (Base 2), Octal (Base 8), Hexadecimal (Base 16)	Roman Numerals, Ancient Egyptian Numerals

Analyzing the Options

Let's examine each option provided in the question:

Decimal System: This is the number system we use daily. It uses digits 0-9. The value of a digit depends on its position. For example, in 123, the '1' is in the hundreds place, '2' in the tens place, and '3' in the units place. This is a positional number system.
Octal System: This system uses digits 0-7 and has a base of 8. The value of each digit depends on its position, where positions represent powers of 8. This is a positional number system.
Binary System: This system uses only two digits, 0 and 1, and has a base of 2. It is the fundamental number system used by computers. The value of each digit depends on its position, representing powers of 2. This is a positional number system.
Roman System: This system uses symbols like I, V, X, L, C, D, and M. The value of each symbol is fixed (I=1, V=5, X=10, etc.). While there are rules for combining symbols (like subtracting when a smaller value precedes a larger one, e.g., IV = 5-1 = 4), the intrinsic value of the symbol itself does not change based on its position. It is considered a non-positional number system.

Conclusion

Based on the analysis, the Decimal, Octal, and Binary systems are all positional number systems because the value of a digit depends on its position. The Roman numeral system is one where the value of a symbol is fixed regardless of its position (though its contribution can be affected by adjacent symbols), classifying it as a non-positional or additive-subtractive system. Therefore, the Roman system is the non-positional number system among the given choices.

Revision Table: Number Systems

Number System	Type	Base	Symbols Used
Decimal	Positional	10	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Binary	Positional	2	0, 1
Octal	Positional	8	0, 1, 2, 3, 4, 5, 6, 7
Roman	Non-Positional	N/A	I, V, X, L, C, D, M

Additional Information: Non-Positional Systems

Non-positional number systems are historically significant and were used in various ancient civilizations. They are often additive, where the total value is the sum of the values of individual symbols. While simple for basic representation and counting, they become cumbersome for performing complex arithmetic operations like multiplication or division compared to positional systems. The lack of a zero symbol in many ancient non-positional systems also posed challenges for placeholding, which is crucial in positional systems.