What are fixed point numbers in a computer system? Why are negative fixed point numbers represented in complement form? Give examples of fixed point numbers (positive as well as negative) assuming the size of the notation to be 8 bits (including sign bit)?
Answers
In real life, we deal with real numbers -- numbers with fractional part. Most modern computer have native (hardware) support for floating point numbers. However, the use of floating point is not necessarily the only way to represent fractional numbers. This article describes the fixed point representation of real numbers. The use of fixed point data type is used widely in digital signal processing (DSP) and game applications, where performance is sometimes more important than precision. As we will see later, fixed point arithmetic is much faster than floating point arithmetic.
Fixed point numbers: A number representation that is a real data type for a number that has a fixed number of digits after the radix point is called a fixed point number.
A fixed-point data type is defined by the length of the word in bits, the position of the radix point and stating whether it is signed or unsigned.
It can also be seen as a more complicated floating-point number representation.
Negative fixed point numbers represented in complement form because, for a scaling factor whose exponent is negative, the number represented is smaller than the integer number that was stored and the number being represented has a fractional component. So, it has to be complemented.
Hence, negative fixed point numbers represented in complement form.
Eg:
8-bit signed binary number 000110112 that is stored in memory using 8-bits of storage.
Firstly, let us also assume this number was stored as a signed fixed-point representation with a scale factor of 22.
As the scale factor is greater than 1, after translation the bits stored in memory into the number we are actually representing, we move the radix point two places to the right.
This gives 11011002
Secondly, the same binary number in memory but this time we’ll assume that it is stored as a signed fixed-point representation with a scale factor of 2-3. As the exponent is negative we move the radix point three places to the left.
This gives 00011.0112
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