Floating point decimal representation, floating point number representation

Floating-point numbersIt is a formulaic expression that is used to approximate real numbers, and can be used to make a trade-off between the range of representations and the precision of representations (hence the name floating-point numbers).

Floating-point numbers are usually expressed as:

n=m×r^e

For example, m(mantissa) is called the mantissa of floating-point numbers, r(radix) is called the cardinality of the order code, and e(exponent) is called the order code of the order. In computers, r is generally specified as or 16, which is a definite constant and does not need to be explicitly expressed in floating-point numbers.

Therefore, under a known standard, floating-point numbers are to be represented.

One is to give the value of the mantissa m, which is usually expressed in the form of a fixed-point decimal, which determines the representation accuracy of the floating-point number, that is, the number of digits of the significant digit that can be given.

The second is to give the order code, which is usually expressed in the form of fixed-point integers, which indicates the position of the decimal point in the data, which determines the representation range of the floating-point number. Therefore, in computers, floating-point numbers are usually expressed in the following format: (assuming a 32-bit floating-point number, with a base of 2 and the highest bit being a sign bit).

A normalized representation of floating-point numbers

Following the exponential notation above, a floating-point number will have a different representation:

In order to improve the accuracy of data representation and ensure the uniqueness of data representation, floating-point numbers need to be standardized.

In the computer, the absolute value of the mantissa is required for floating-point numbers with non-zero values.

Must be greater than the reciprocal of the cardinality, i.e. m|≥1/r。

That is, the most significant bit of the mantissa field should be 1, and the floating-point number that satisfies this representation requirement is called a normalized representation: the operation process of turning the mantissa that does not meet this representation requirement into a mantissa that meets this requirement is called the normalization of floating-point number, which is realized by mantissa shift and modifying the order code.

A floating-point number is a numerical representation of a number that belongs to a specific subset of rational numbers, and is used in computers to approximate any real number

Specifically, this real number is obtained by multiplying an integer or fixed-point number (i.e., mantissa number) by the integer power of a certain base (usually 2 in a computer), similar to the scientific notation of the base 10.

Awareness of rational numbers:Rational numbers are a collective term for integers (positive integers, 0s, negative integers) and fractions. Positive integers and positive fractions are collectively called positive rational numbers, and negative integers and negative fractions are collectively referred to as negative rational numbers.

Therefore, the number of rational numbers in the set of rational numbers can be divided into positive rational numbers, negative rational numbers, and zeros. Since any integer or fraction can be reduced to a decimal cyclic decimal, and conversely, every decimal cyclic decimal can also be reduced to an integer or fraction, therefore, a rational number can also be defined as a decimal cyclic decimal.

It is known that the C C++ compiler is based on the IEEE (Institute of Electrical and Electronics Engineers) IEEE floating-point notation. This structure is a scientific representation, represented by symbols (+ or -), exponents, and mantissas, with the base being determined as 2. So in IEEE floating-point notation, a floating-point number is the mantissa multiplied by 2 to the exponential power plus the sign.

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<> list three decimal (floating-point) operations.

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Hello <> dear. The specific method of the order is to first find the difference between the two floating-point order codes, that is, e=e x-e y, add e to the small-order code to make it equal to the large-order code, and at the same time shift the mantissa of the floating-point number corresponding to the small-order code to the right by the corresponding number of digits to ensure that the value of the floating-point number remains unchanged.

The floating-point number a is represented by two numbers m and e: a = m b e. <>

Floating-point numbers are expressed as follows:

Floating-point representationA representation of computer data. The decimal point position of the data is not fixed, but floating.

In floating-point notation, the number n can be expressed as n = d x 2tp, where d is called the mantissa (or numeric value) of n, and p is called the order (or exponent) of the number nd is a pure decimal and p is an integer. The symbol of p indicates the direction in which the decimal point floats.

p is positive, indicating that the decimal point is shifted to the right by p digits; p is negative, indicating that the decimal point has shifted to the left by p digits.

Floating-point numbers use exponents to achieve the effect of "floating" decimal points. Thus it is possible to flexibly express a larger range of numbers, such as Lingran

The position of the decimal point is not fixed. However, there are many expressions for the same floating-point number, which can be expressed as:

Due to its multi-speculative nature, many computer manufacturers have devised their own rules for representing floating-point numbers, as well as the details of floating-point arithmetic. Diverse rules are detrimental to both the reliability and portability of the program.