 # Fixed And Floating Point Number Representation Pdf

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Published: 02.06.2021  Digital Computers use Binary number system to represent all types of information inside the computers. Alphanumeric characters are represented using binary bits i. Digital representations are easier to design, storage is easy, accuracy and precision are greater.

Human beings use decimal base 10 and duodecimal base 12 number systems for counting and measurements probably because we have 10 fingers and two big toes. Computers use binary base 2 number system, as they are made from binary digital components known as transistors operating in two states - on and off.

In computing , a fixed-point number representation is a real data type for a number that has a fixed number of digits after and sometimes also before the radix point after the decimal point '. Fixed-point number representation can be compared to the more complicated and more computationally demanding floating-point number representation. Fixed-point numbers are useful for representing fractional values , usually in base 2 or base 10, when the executing processor has no floating point unit FPU as is the case for older or low-cost embedded microprocessors and microcontrollers , if fixed-point provides improved performance or accuracy for the application at hand, or if their use is more natural for the problem such as for the representation of angles. A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type.

## Fixed-point arithmetic

In computing , a fixed-point number representation is a real data type for a number that has a fixed number of digits after and sometimes also before the radix point after the decimal point '. Fixed-point number representation can be compared to the more complicated and more computationally demanding floating-point number representation.

Fixed-point numbers are useful for representing fractional values , usually in base 2 or base 10, when the executing processor has no floating point unit FPU as is the case for older or low-cost embedded microprocessors and microcontrollers , if fixed-point provides improved performance or accuracy for the application at hand, or if their use is more natural for the problem such as for the representation of angles. A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type.

For example, the value 1. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation. The scaling factor is usually a power of 10 for human convenience or a power of 2 for computational efficiency. However, other scaling factors may be used occasionally, e. The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type multiplied by the scaling factor; and similarly for the minimum value.

Thus, for example, to convert the value 1. If S does not divide R in particular, if the new scaling factor S is greater than the original R , the new integer will have to be rounded.

The rounding rules and methods are usually part of the language's specification. To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow occurs i. If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.

To multiply two fixed-point numbers, it suffices to multiply the two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. This operation involves no rounding. If the two operands belong to the same fixed-point type, and the result is also to be represented in that type, then the product of the two integers must be explicitly multiplied by the common scaling factor; in this case the result may have to be rounded, and overflow may occur.

To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors.

The first division involves rounding in general. If both operands and the desired result all have the same scaling factor, then the quotient of the two integers must be explicitly multiplied by that common scaling factor. The two most common classes of fixed-point types are decimal and binary. Decimal fixed-point types have a scaling factor that is a power of ten; for binary fixed-point types it is a power of two. Binary fixed-point types are most commonly used, because the rescaling operations can be implemented as fast bit shifts.

Binary fixed-point numbers can represent fractional powers of two exactly, but, like binary floating-point numbers, cannot exactly represent fractional powers of ten. If exact fractional powers of ten are desired, then a decimal format should be used. For example, one-tenth 0. These representations may be encoded in many ways, including binary-coded decimal BCD.

Note how since there are 3 decimal places we show the trailing zeros. This works equivalently if we choose a different base, notably base 2 for computing, since a bit shift is the same as a multiplication or division by an order of 2. Three decimal digits is equivalent to about 10 binary digits, so we should round 0. The closest approximation is then 0. There are various notations used to represent word length and radix point in a binary fixed-point number.

In the following list, f represents the number of fractional bits, m the number of magnitude or integer bits, s the number of sign bits, and b the total number of bits. Because fixed point operations can produce results that have more digits than the operands , information loss is possible. For instance, the result of fixed point multiplication could potentially have as many digits as the sum of the number of digits in the two operands. In order to fit the result into the same number of digits as the operands, the answer must be rounded or truncated.

If this is the case, the choice of which digits to keep is very important. For simplicity, many fixed-point multiply procedures use the same result format as the operands. This has the effect of keeping the middle digits; the I -number of least significant integer digits, and the Q -number of most significant fractional digits.

Fractional digits lost below this value represent a precision loss which is common in fractional multiplication. If any integer digits are lost, however, the value will be radically inaccurate. Some model-based fixed-point packages  allow specifying a result format different from the input formats, enabling the user to maximize precision and avoid overflow. Some operations, like divide, often have built-in result limiting so that any positive overflow results in the largest possible number that can be represented by the current format.

Likewise, negative overflow results in the largest negative number represented by the current format. This built in limiting is often referred to as saturation. Some processors support a hardware overflow flag that can generate an exception on the occurrence of an overflow, but it is usually too late to salvage the proper result at this point.

Very few computer languages include built-in support for fixed point values other than with the radix point immediately to the right of the least significant digit i. Floating-point representations are easier to use than fixed-point representations, because they can handle a wider dynamic range and do not require programmers to specify the number of digits after the radix point.

A common use of fixed-point BCD numbers is for storing monetary values, where the inexact values of binary floating-point numbers are often a liability. The Ada programming language includes built-in support for both fixed-point binary and decimal and floating-point. Fixed-point support is implemented in GCC.

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Categories : Computer arithmetic Data types Primitive types. Hidden categories: Articles with short description Short description matches Wikidata Articles that may contain original research from September All articles that may contain original research Articles with too many examples from August All articles with too many examples Wikipedia articles with style issues from August Namespaces Article Talk. ## A Tutorial on Data Representation

Fixed point representation is used to store integers , the positive and negative whole numbers: … -3, -2, -1, 0, 1, 2, 3, …. This is called unsigned integer format, and a simplified example is shown in Fig. Conversion between the bit pattern and the number being represented is nothing more than changing between base 2 binary and base 10 decimal. The disadvantage of unsigned integer is that negative numbers cannot be represented. Offset binary is similar to unsigned integer, except the decimal values are shifted to allow for negative numbers. In the 4 bit example of Fig. In this same manner, a 16 bit representation would use 32, as an offset, resulting in a range between , and 32,

The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE standard. The first standard for floating-point arithmetic, IEEE , was published in It covered only binary floating-point arithmetic. The binary formats in the original standard are included in this new standard along with three new basic formats, one binary and two decimal. To conform to the current standard, an implementation must implement at least one of the basic formats as both an arithmetic format and an interchange format.

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### Fixed-point arithmetic

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PDF | In Chapters , we dealt with various methods for representing fixed-point numbers. Such representations suffer from limited range and/or | Find, read.

## Telecommunication switching system and networks thyagarajan viswanath pdf A fixed-point representation of a number may be thought to consist of 3 parts: the sign field, integer field, and fractional field. One way to store a number using a 32​-. United nations convention on the rights of the child 1991 pdf devote yourself to an ideal jaret grossman pdf free In computing , floating-point arithmetic FP is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. In my previous post we learnt the fundamental concepts of how binary could be used to represent real numbers i. You may use these HTML tags and attributes: ```<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong> ```