- Position-Independent Number Systems
- Position-Dependent Number Systems
- Advantages of Position-Dependent Number Systems
The same number can be represented in different ways using different number systems. Broadly speaking, number systems fall into two categories: position-independent and position-dependent systems.
Position-Independent Number Systems
In a position-independent number system, the value of a symbol remains constant regardless of its position in the sequence. A classic example is the Roman numeral system.
In this system, symbols such as I, V and X always represent the same values: 1, 5, and 10 respectively. For example:
- IV
- VI
In both cases, I is worth 1 and V is worth 5, regardless of their position. The interpretation of the entire number may change due to ordering, but the value of each symbol does not.
Position-Dependent Number Systems
Modern number systems, such as decimal and binary, are position-dependent. This means the value of a digit depends on its position in the number.
Take the decimal number 35, for instance. The 3 here represents 3 * 10 = 30 because it is in the tens place. However, in 23, the 3 represents just 3, as it is in the units place.
This is in stark contrast to the Roman system, where the value of V remains 5 in both IV and VI, regardless of its position.
Advantages of Position-Dependent Number Systems
Ancient position-independent systems like the Roman numerals were sufficient for representing small numbers but lacked scalability and made arithmetic operations difficult.
Position-dependent systems are:
- Scalable. They can represent very large or very small numbers compactly by leveraging positional magnitude.
- Efficient for computation. They are better suited for arithmetic operations, especially by machines.
- Expressive. A number like 1987 in decimal becomes MCMLXXXVII in Roman numerals—long and difficult to interpret.
- Zero-inclusive. Unlike the Roman numeral system, modern number systems include zero, a critical component for both arithmetic and placeholding.