średniowiecze: Decimal

Evidence: 100 • authority: 72

A decimal system (also called base-ten, denary or decenary) is a numeral system that uses ten as its radix (base). Decimal systems are the global standard for denoting integer and non-integer numbers. The way of denoting numbers in a decimal system is often referred to as decimal notation. Presently, the most common decimal system is the Hindu–Arabic numeral system, which is a positional numeral system. However, there are also non-positional base-ten systems, such as Roman or Chinese numerals. A decimal numeral (also often just decimal or, less correctly, decimal number), generally refers to the notation of a number in a decimal numeral system. Non-integral decimals may be identified by a d…

Evidence: 52 • authority: 88

ON THE TRANSFINITE LENGTH OF THE SEQUENCE OF DIGITS IN THE EXPANSION OF A TRANSCENDENTAL NUMBER AFTER THE DECIMAL POINT. We prove that (if we assume as consistent Cantor's diagonal process) among the transcendental numbers there are different transfinite cardinalities in the total of digits that compose them in their expansion after the decimal point. Some transcendental can be made up of a total of digits ℵ0 after the decimal point, while another by ℵ1 or ℵ2, or ℵ𝛂 figures. So, we proceed to mathematically construct such cardinalities. Which in turn forces us to also deduce the existence, within the class R of the real numbers, of Rα infinite subsets of these numbers with ℵα cardinalities…

Evidence: 38 • authority: 72

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A simple fraction (examples: ⁠1/2⁠ and ⁠17/3⁠) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction ⁠3/4⁠, the numerator 3 i…

Evidence: 38 • authority: 72

Positional notation, also known as place-value notation, is the property of a numeral system that the value represented by each symbol in a written numeral depends not only on its appearance but also on its position. Each symbol fits in a specific place or position, representing a power of a fixed base. The most common numeral system used today, the Hindu–Arabic numeral system, is a positional system in base ten; each of ten numerical digits is a distinct symbol representing one of the numbers zero through nine, and in the context of the full numeral, each symbol's value is the digit multiplied by a power of ten. Most early numeral systems, such as Roman numerals, are essentially based on t…

Evidence: — • authority: 88

Primality test and repeating decimal digits. This paper addresses the topic of prime numbers, in particular the challenge to determine whether a number is prime or composite. For that matter, we propose a new type of deterministic primality test inspired by Fermat’s little theorem and based on the properties of repeating decimal digits.

Evidence: — • authority: 88

High-speed binary coded decimal digit multipliers with multiple error detection. Abstract Decimal arithmetic in the form of binary coded decimal (BCD) numbers is preferred in many financial and commercial applications. BCD multipliers are introduced as a key hardware unit to support both integer and floating-point decimal arithmetic operations. However, due to the increasing sensitivity of VLSI-based digital designs to the environmental effects, BCD multipliers are also prone to faults and errors similar to other arithmetic circuits. In addition, multiple error occurrence is possible in current digital systems which motivates to reach multiple error detection/correction in addition to singl…

Evidence: — • authority: 72

Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions (common in human-entered data, such as measurements or financial information) and binary (base-2) fractions. The advantage of decimal floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values. For example, while a fixed-point representation that allocates 8 decimal digits and 2 decimal places can represent the numbers 123456.78, 8765.43, 123.00, and…

Evidence: — • authority: 72

In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for a sign or other indications (e.g. error or overflow). In byte-oriented systems (i.e. most modern computers), the term unpacked BCD usually implies a full byte for each digit (often including a sign), whereas packed BCD typically encodes two digits within a single byte by taking advantage of the fact that four bits are enough to represent the range 0 to 9. The precise four-bit encoding, however, may vary for technical reasons (e.g. excess-3). The ten stat…

Evidence: — • authority: 72

Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega). The record of manual…

Evidence: — • authority: 72

In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is…

Evidence: — • authority: 72

Square root algorithms compute the non-negative square root S {\displaystyle {\sqrt {S}}} of a positive real number S {\displaystyle S} . Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed to some finite precision: these algorithms typically construct a series of increasingly accurate approximations. Most square root computation methods are iterative: after choosing a suitable initial estimate of S {\displaystyle {\sqrt {S}}} , an iterative refinement is performed until some termination criterion is met. One refinement option is Heron's method, a special case of Newton's method. If division is much more costly…

Evidence: — • authority: 72

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the 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 use the IEEE 754 standard. The standard defines: arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs) interchange formats: encodings (bit strings) that may be used…