A fixed-point number is a number representation system where the decimal (or binary) point is fixed at a predetermined position. This method is frequently used in both mathematical computations and computer science for representing numerical values with a specific number of digits after the decimal point.
Key Characteristics of Fixed-Point Numbers
Definition
A fixed-point number can be represented as \( N = f \cdot 10^{-d} \), where:
- \( N \) is the fixed-point number.
- \( f \) is an integer representation.
- \( d \) is the number of digits to the right of the decimal point.
For example, the population of a city can be represented as 25,000 with no digits to the right of the decimal point. An amount of money in U.S. currency can be represented as $10.50, with two digits to the right of the decimal point.
Types of Fixed-Point Numbers
Integer Fixed-Point:
- No fractional digits (e.g., 25,000).
Fractional Fixed-Point:
- Specific number of fractional digits (e.g., $10.50).
Special Considerations
- Precision: Fixed-point numbers have a limited precision based on the preset number of digits after the decimal point.
- Range: The range of representable numbers is constrained by the total number of digits and the position of the fixed point.
Applications of Fixed-Point Numbers
In Computing
Fixed-point arithmetic is beneficial in systems where memory and processing power are limited, such as embedded systems and digital signal processors (DSPs). The deterministic execution time of fixed-point calculations is advantageous for real-time computing.
In Finance
Fixed-point representation is crucial in financial calculations where precise decimal places (like two decimal places for cents) are necessary.
Historical Context
The use of fixed-point numbers dates back to early computing systems where hardware and memory limitations necessitated a simpler and more consistent representation of numbers. Over time, it has evolved to cater to specific application needs, most notably in real-time systems and precision-critical computations.
Comparison with Floating-Point Numbers
Floating-Point Numbers
Floating-point numbers represent real numbers in a way that can accommodate a wide range of values. The decimal point’s position can ‘float,’ hence the name. This allows for the representation of very large or very small numbers with significant precision.
Key Differences
- Precision: Fixed-point has a consistent precision based on its fixed position, whereas floating-point precision can vary.
- Range: Floating-point numbers can represent a broader range of values than fixed-point numbers.
- Performance: Fixed-point arithmetic is generally faster and requires less computational power compared to floating-point arithmetic.
Related Terms
- Integer: A whole number without a fractional component.
- Binary Number: A number expressed in the base-2 numeral system.
- Decimal Point: A dot used to separate the integer part from the fractional part of a number.
FAQs
What is a fixed-point number used for?
How do you convert a decimal number to a fixed-point representation?
Why are fixed-point numbers preferred in embedded systems?
References
- Goldberg, D. (1991). “What Every Computer Scientist Should Know About Floating-Point Arithmetic.” ACM Computing Surveys, 23(1), 5-48.
- Hennessy, J. L., & Patterson, D. A. (2011). “Computer Organization and Design: The Hardware/Software Interface.” Morgan Kaufmann.
Summary
Fixed-point numbers provide a consistent and efficient way to represent numerical values with a fixed number of decimal places. This system is crucial in various applications, particularly in financial computations and real-time embedded systems. Understanding the differences between fixed-point and floating-point representations helps in choosing the appropriate numerical method for specific applications.
