Understanding of excess bit notation in computing, also known as excess notation or biased notation, is important to represent signed numbers with fixed biases. It makes it easier to handle binary arithmetic without sign bits.
Table of Contents
Introduction
Excess Notation is a method used in computing to represent signed numbers by adding a fixed bias to each value.
A fixed bias is a constant value added to each number in the range to shift the representation. It is centered on zero. The fixed bias makes it possible to represent both positive and negative numbers without a separate sign bit.
For example, in an excess-128 notation (commonly used in 8-bit formats), the bias is 128. This means that:
- To represent 0, you store 128 (0 + 128).
- To represent 1, you store 129 (1 + 128).
- To represent -1, you store 127 (-1 + 128).
Unlike standard binary, which uses a separate sign bit to indicate positive or negative values, this notation simplifies the process by encoding all numbers within a fixed range around zero.
This method is especially beneficial in hardware applications like floating-point representations. Now, we shall look at how Excess Notation works, why it is useful, and some practical examples.
Excess Notation
This notation is also known as biased notation and represents signed numbers by adding a fixed bias (e.g., 4, 8, or 16) to the actual value.
This helps with encoding both positive and negative numbers without a sign bit, making it particularly useful for certain hardware applications. In this notation, positive and negative values are distinguished by their shifted binary values relative to the bias rather than by a specific sign bit. Here is how it works:
- Excess-4 adds 4 to the number.
- Excess-8 adds 8.
- Excess-16 adds 16.
Tips for Understanding Excess Bit Notation in Computing
To encode a number x in Excess-4:
- Add the bias:
- Convert to binary.
Example
For example, to represent –3 in Excess-4:
- Add 4 to the number:
. - Decimal to binary for 1 is 0001.
Table – Decimal to Excess Notation
Here is the table in Excess-4 notation:
Point to Ponder
In Excess-4 encoding, each decimal value is shifted by +4, so:
- A decimal value of –8 becomes –4 (
), which in binary is 1100. - Similarly, a decimal value of 0 becomes [latex]0 + 4 = 4[latex], which in binary is 0100.
The presence of 1 in the MSB (most significant bit) for negative values in Excess-4 signifies that the number has been adjusted upward through the addition of the bias, indicating a shifted representation.
This approach eliminates the need for a separate sign bit, allowing for compact binary representation of signed integers.
Excess-8, Excess-16 etc. all works the same way. The only difference is, in Excess-8, decimal value is shifted by +8 and there are a total of 128 values. The same goes for the Excess-16.
Advantages, Disadvantages, and Limitations
Advantages
Simplifies Comparisons
Binary comparisons are easier without negative numbers.
Fixed Range
Provides a symmetric range around zero.
Disadvantages
Conversion Required
Converting between excess notation and standard binary requires additional steps.
Limitations:
Bias Values
Limited to predefined bias values, with fixed ranges.
How Computers Use Excess Notation
It is common in hardware implementations such as floating-point exponents and certain embedded systems. It simplifies comparisons and arithmetic in specific applications by removing the need for separate positive and negative ranges.
Conclusion
Excess Notation is a powerful method for encoding signed numbers without needing a sign bit. It is useful in various hardware implementations. By adding a fixed bias, this system enables a balanced range around zero and simplifies comparison operations in specific applications.
Despite limitations in flexibility, Excess Notation remains essential in areas like floating-point representation, where efficient binary encoding is necessary.
Frequently Asked Question (FAQs)
What is Excess Notation in computing?
It is a binary encoding method that represents signed numbers by adding a fixed bias to each value.
Why is Excess Notation used?
It simplifies comparisons and eliminates the need for a separate sign bit, which can make certain hardware processes more efficient.
How does Excess-4 notation work?
In Excess-4, each decimal value has 4 added to it before converting to binary, creating a range centered on zero.
What is the difference between Excess Notation and 2’s Complement?
This notation uses a bias to shift values, while 2’s Complement flips bits and adds one to represent negatives.
Where is Excess Notation used in computing?
It is commonly used in floating-point representation and embedded systems where a fixed, balanced range is beneficial.
How do I convert a number to Excess-8?
Add 8 to the decimal number, then convert the result to binary.
Can Excess Notation represent decimal fractions?
No, this notation typically represents only whole integers, similar to other integer encoding schemes.
Is there an Excess-16 notation?
Yes, Excess-16 notation shifts each value by adding 16, commonly used when a larger range is needed.
What is the range of numbers in Excess-4 with 4 bits?
In a 4-bit Excess-4 system, the range typically covers from -4 to +3 in decimal values.
Why is there no sign bit in Excess Notation?
This notation shifts all values positively, eliminating the need for a separate sign bit by creating a balanced range around zero.