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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
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
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Sodium Azide
Sodium
Sodium
azide is the inorganic compound with the formula NaN3. This colorless salt is the gas-forming component in many car airbag systems. It is used for the preparation of other azide compounds. It is an ionic substance, is highly soluble in water, and is very acutely toxic.[5]Contents1 Structure 2 Preparation2.1 Laboratory methods3 Chemical reactions 4 Applications4.1 Automobile airbags and airplane escape chutes 4.2 Organic and inorganic synthesis 4.3 Biochemistry and biomedical uses 4.4 Agricultural uses5 Safety considerations 6 References 7 External linksStructure[edit] Sodium
Sodium
azide is an ionic solid. Two crystalline forms are known, rhombohedral and hexagonal.[1][6] Both adopt layered structures. The azide anion is very similar in each form, being centrosymmetric with N–N distances of 1.18 Å. The Na+ ion has octahedral geometry
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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 (IEEE)
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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.[1][2][3] 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
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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
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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 may give results leading to unintended behavior. In particular, if the possibility has not been anticipated, overflow can compromise a program's reliability and security. For some applications, such as timers and clocks, wrapping on overflow can be desirable
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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 represent 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
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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 bybemax × (b − b1−p),where p is the precision of the format in digits and emax is (−emin)+1. In the IEEE 754
IEEE 754
binary and decimal format
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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
IEEE floating point
number is the part of a floating-point number that represents the significant digits
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0 (number)
0 (zero; /ˈzɪəroʊ/) is both a number[1] 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) (/nɔːt/), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (/oʊ/)
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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
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Associative Property
In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider the following equations: ( 2 + 3 ) + 4 = 2 + ( 3 + 4 ) = 9 displaystyle (2+3)+4=2+(3+4)=9, 2 × ( 3 × 4 ) = ( 2 × 3 ) × 4 = 24. displaystyle 2times (3times 4)=(2times 3)times 4=24
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Minifloat
In computing, minifloats are floating-point values represented with very few bits. Predictably, they are not well suited for general-purpose numerical calculations. They are used for special purposes, most often in computer graphics, where iterations are small and precision has aesthetic effects. Additionally, they are frequently encountered as a pedagogical tool in computer-science courses to demonstrate the properties and structures of floating-point arithmetic and IEEE 754
IEEE 754
numbers. Minifloats with 16 bits are half-precision numbers (opposed to single and double precision). There are also minifloats with 8 bits or even fewer. Minifloats can be designed following the principles of the IEEE 754 standard. In this case they must obey the (not explicitly written) rules for the frontier between subnormal and normal numbers and must have special patterns for infinity and NaN. Normalized numbers are stored with a biased exponent
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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
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Zero To The Power Of Zero
Zero to the power of zero, denoted by 00, is a mathematical expression with no necessarily obvious value. Possibilities include 0, 1, or leaving the expression undefined altogether, and there is no consensus as to which approach is best. Several justifications exist for each of the possibilities, and they are outlined below. In algebra, combinatorics, or set theory, the generally agreed upon answer is 00 = 1, whereas in analysis, the expression is generally left undefined. Computer programs also have differing ways of handling this expression.Contents1 Discrete exponents 2 Polynomials and power series 3 Continuous exponents 4 Complex exponents 5 History of differing points of view 6 Treatment on computers6.1 IEEE floating-point standard 6.2 Programming languages 6.3 Mathematics software7 References 8 External linksDiscrete exponents[edit] There are many widely used formulas having terms involving natural-number exponents that require 00 to be evaluated to 1
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