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Not A Number
In computing , NAN, standing for NOT A NUMBER, is a numeric data type value representing an undefined or unrepresentable value, especially in floating-point calculations. Systematic use of NaNs was introduced by the IEEE 754 floating-point standard in 1985, along with the representation of other non-finite quantities like infinities . Two separate kinds of NaNs are provided, termed quiet NaNs and signaling NaNs. Quiet NaNs are used to propagate errors resulting from invalid operations or values, whereas signaling NaNs can support advanced features such as mixing numerical and symbolic computation or other extensions to basic floating-point arithmetic. For example, 0/0 is undefined as a real number , and so represented by NaN; the square root of a negative number is imaginary , and thus not representable as a real floating-point number, and so is represented by NaN; and NaNs may be used to represent missing values in computations
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Denormal Number
In computer science , DENORMAL NUMBERS or DENORMALIZED NUMBERS (now often called SUBNORMAL NUMBERS) fill the underflow gap around zero in floating-point arithmetic. Any non-zero number with magnitude smaller than the smallest normal number is 'subnormal'. In a normal floating-point value, there are no leading zeros in the significand ; instead leading zeros are moved to the exponent. So 0.0123 would be written as 1.23 × 10−2. Denormal numbers are numbers where this representation would result in an exponent that is below the minimum exponent (the exponent usually having a limited range). Such numbers are represented using leading zeros in the significand. The significand (or mantissa) of an IEEE floating point number is the part of a floating-point number that represents the significant digits. For a positive normalised number it can be represented as m0.m1m2m3...mp-2mp-1 (where m represents a significant digit and p is the precision, and m0 is non-zero)
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Normal Number (computing)
In computing , a NORMAL NUMBER is a non-zero number in a floating-point representation which is within the balanced range supported by a given floating-point format: it is a floating point number that can be represented without leading zeros in its significand . The magnitude of the smallest normal number in a format is given by bemin, where b is the base (radix) of the format (usually 2 or 10) and emin depends on the size and layout of the format. Similarly, the magnitude of the largest normal number in a format is given by bemax × (b − b1−p), where p is the precision of the format in digits and emax is (−emin)+1
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0 (number)
0 (ZERO; /ˈzɪəroʊ/ ) is both a number and the numerical digit used to represent that number in numerals . The number 0 fulfills a central role in mathematics as the additive identity of the integers , real numbers , and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems . Names for the number 0 in English include ZERO, NOUGHT (UK), NAUGHT (US) (/ˈnɔːt/ ), NIL, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—OH or O (/ˈoʊ/ ). Informal or slang terms for zero include ZILCH and ZIP. Ought and aught (/ˈɔːt/ ), as well as cipher, have also been used historically
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Significand
The SIGNIFICAND (also MANTISSA or COEFFICIENT) is part of a number in scientific notation or a floating-point number , consisting of its significant digits . Depending on the interpretation of the exponent , the significand may represent an integer or a fraction . The word mantissa seems to have been introduced by Arthur Burks in 1946 writing for the Institute for Advanced Study
Institute for Advanced Study
at Princeton , although this use of the word is discouraged by the IEEE floating-point standard committee as well as some professionals such as the creator of the standard William Kahan
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IEEE Floating-point
The IEEE STANDARD FOR FLOATING-POINT ARITHMETIC (IEEE 754) is a technical standard for floating-point computation established in 1985 by the Institute of Electrical and Electronics Engineers
Institute of Electrical and Electronics Engineers
(IEEE). 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 now use the IEEE 754
IEEE 754
standard
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Arithmetic Underflow
The term ARITHMETIC UNDERFLOW (or "floating point underflow", or just "underflow") is a condition in a computer program where the result of a calculation is a number of smaller absolute value than the computer can actually store in memory on its CPU. Arithmetic underflow can occur when the true result of a floating point operation is smaller in magnitude (that is, closer to zero) than the smallest value representable as a normal floating point number in the target datatype. Underflow can in part be regarded as negative overflow of the exponent of the floating point value. For example, if the exponent part can represent values from −128 to 127, then a result with a value less than −128 may cause underflow
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Arithmetic Overflow
In computer programming , an INTEGER OVERFLOW occurs when an arithmetic operation attempts to create a numeric value that is outside of the range that can be represented with a given number of bits – either larger than the maximum or lower than the minimum representable value. The most common result of an overflow is that the least significant representable bits of the result are stored; the result is said to wrap around the maximum (i.e. modulo power of two). An overflow condition gives incorrect results and, particularly if the possibility has not been anticipated, can compromise a program's reliability and security . On some processors like graphics processing units (GPUs) and digital signal processors (DSPs) which support saturation arithmetic , overflown results would be "clamped", i.e. set to the minimum or the maximum value in the representable range, rather than wrapped around
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Microsoft Binary Format
In computing , MICROSOFT BINARY FORMAT (MBF) was a format for floating point numbers used in Microsoft 's BASIC language products including MBASIC , GW-BASIC and QuickBasic prior to version 4.00. CONTENTS * 1 History * 2 Technical details * 3 Examples * 4 See also * 5 Notes and references * 6 External links HISTORYIn 1975, Bill Gates and Paul Allen were working on Altair BASIC , which they were developing at Harvard University on a PDP-10 running their Altair emulator . One thing still missing was code to handle floating point numbers, needed to support calculations with very big and very small numbers, which would be particularly useful for science and engineering. One of the proposed uses of the Altair was as a scientific calculator. Altair 8800 front panel At a dinner at Currier House , an undergraduate residential house at Harvard, Gates and Allen complained to their dinner companions about having to write this code
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IBM Floating Point Architecture
IBM
IBM
System/360
System/360
computers, and subsequent machines based on that architecture (mainframes), support a hexadecimal floating-point format. In comparison to IEEE 754
IEEE 754
floating-point, the IBM
IBM
floating-point format has a longer significand , and a shorter exponent . All IBM floating-point formats have 7 bits of exponent with a bias of 64. The normalized range of representable numbers is from 16−65 to 1663 (approx. 5.39761 × 10−79 to 7.237005 × 1075). The number is represented as the following formula: (−1)sign × 0.significand × 16exponent−64
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Arbitrary-precision Arithmetic
In computer science , ARBITRARY-PRECISION ARITHMETIC, also called BIGNUM ARITHMETIC, MULTIPLE PRECISION ARITHMETIC, or sometimes INFINITE-PRECISION ARITHMETIC, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision. Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than store values as a fixed number of binary bits related to the size of the processor register , these implementations typically use variable-length arrays of digits. Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required
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Extended Real Number Line
In mathematics , the AFFINELY EXTENDED REAL NUMBER SYSTEM is obtained from the real number system ℝ by adding two elements: + ∞ and – ∞ (read as POSITIVE INFINITY and NEGATIVE INFINITY respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis , especially in the theory of measure and integration . The affinely extended real number system is denoted R {displaystyle {overline {mathbb {R} }}} or or ℝ ∪ {–∞, +∞}. When the meaning is clear from context, the symbol +∞ is often written simply as ∞
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Indeterminate Forms
In calculus and other branches of mathematical analysis , limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits ; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an INDETERMINATE FORM. The term was originally introduced by Cauchy 's student Moigno in the middle of the 19th century. The indeterminate forms typically considered in the literature are denoted: 0 / 0 , / , 0 , , 0 0 , 1 and 0
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Complex Number
A COMPLEX NUMBER is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit , satisfying the equation i2 = −1. In this expression, a is called the real part and b is the imaginary part of the complex number. If z = a + b i {displaystyle z=a+bi} , then we write Re ( z ) = a , Im ( z ) = b . {displaystyle operatorname {Re} (z)=a,quad operatorname {Im} (z)=b.} Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary , whereas a complex number whose imaginary part is zero is a real number
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Zero To The Power Of Zero
ZERO TO THE POWER OF ZERO is a mathematical expression with no necessarily obvious value. If the expression results, e.g., from two functions f and g, both approaching 0 for some argument, then the limit of f g depends on the way in which the functions approach 0 there. In other contexts 00 unambiguously has the value 1. There is no unanimous opinion on leaving the value generally undefined. CONTENTS * 1 Discrete exponents * 2 Polynomials and power series * 3 Continuous exponents * 4 Complex exponents * 5 History of differing points of view * 6 Treatment on computers * 6.1 IEEE floating-point standard * 6.2 Programming languages * 6.3 Mathematics software * 7 References * 8 External links DISCRETE EXPONENTSThere are many widely used formulas having terms involving natural-number exponents that require 00 to be evaluated to 1. For example, regarding b0 as an empty product assigns it the value 1, even when b = 0
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Math.h
C MATHEMATICAL OPERATIONS are a group of functions in the standard library of the C programming language implementing basic mathematical functions. All functions use floating point numbers in one manner or another. Different C standards provide different, albeit backwards-compatible, sets of functions. Most of these functions are also available in the C++ standard library , though in different headers (the C headers are included as well, but only as a deprecated compatibility feature). CONTENTS* 1 Overview of functions * 1.1 Floating point environment * 1.2 Complex numbers * 1.3 Type-generic functions * 1.4 Random number generation * 2 libm * 3 See also * 4 References * 5 External links OVERVIEW OF FUNCTIONSMost of the mathematical functions are defined in math.h (cmath header in C++). The functions that operate on integers, such as abs, labs, div, and ldiv, are instead defined in the stdlib.h header (cstdlib header in C++)
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