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Ruler Function
In number theory, the ruler function of an integer n can be either of two closely related functions. One of these functions counts the number of times n can be evenly divided by two, which for the numbers 1, 2, 3, ... is Alternatively, the ruler function can be defined as the same numbers plus one, which for the numbers 1, 2, 3, ... produces the sequence As well as being related by adding one, these two sequences are related in a different way: the second one can be formed from the first one by removing all the zeros, and the first one can be formed from the second one by adding zeros at the start and between every pair of numbers. For either definition of the ruler function, the rising and falling patterns of the values of this function resemble the lengths of marks on rulers with traditional units such as inches. These functions should be distinguished from Thomae's function, a function on real numbers which behaves similarly to the ruler function when restricted to the dyadic r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurement Unit
A unit of measurement, or unit of measure, is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement. For example, a length is a physical quantity. The metre (symbol m) is a unit of length that represents a definite predetermined length. For instance, when referencing "10 metres" (or 10 m), what is actually meant is 10 times the definite predetermined length called "metre". The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to the present. A multitude of System of measurement, systems of units used to be very common. Now there is a global standard, the International System of Units (SI), the modern form of the metric system. In trade, weights and measures are often a su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruler
A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instrument is rigid and the edge itself is a straightedge ("ruled straightedge"), which additionally allows one to draw straighter lines. Rulers are an important tool in geometry, geography and mathematics. They have been used since at least 2650 BC. Variants Rulers have long been made from different materials and in multiple sizes. Historically, they were mainly wood but plastics have also been used. They can be created with length markings instead of being wikt:scribe, scribed. Metal is also used for more durable rulers for use in the workshop; sometimes a metal edge is embedded into a wooden desk ruler to preserve the edge when used for straight-line cutting. Typically in length, though some can go up to 100 cm, it is useful for a ruler to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inch
The inch (symbol: in or prime (symbol), ) is a Units of measurement, unit of length in the imperial units, British Imperial and the United States customary units, United States customary System of measurement, systems of measurement. It is equal to yard or of a foot (unit), foot. Derived from the Uncia (unit), Roman uncia ("twelfth"), the word ''inch'' is also sometimes used to translate similar units in other measurement systems, anthropic units, usually understood as deriving from the width of the human thumb. Standards for the exact length of an inch have varied in the past, but since the adoption of the international yard during the 1950s and 1960s the inch has been based on the metric system and defined as exactly 25.4Millimetre, mm. Name The English word "inch" () was an early borrowing from Latin ' ("one-twelfth; Roman inch; Roman ounce"). The vowel change from Latin to Old English (which became Modern English ) is known as Germanic umlaut, umlaut. The consonant c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomae's Function
Thomae's function is a real-valued function of a real variable that can be defined as: f(x) = \begin \frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\ 0 &\textx \text \end It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), the Riemann function, or the Stars over Babylon (John Horton Conway's name). Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration. Since every rational number has a unique representation with coprime (also termed relatively prime) p \in \mathbb Z and q \in \mathbb N, the function is well-defined. Note that q = +1 is the only number in \mathbb N that is coprime to p = 0. It is a modification of the Dirichlet function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Rational Number
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number. The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted \Z tfrac12/math>. In advanced mathematics, the dyadic rational numbers are central to the cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides . It is denoted \nu_p(n). Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of n. The P-adic number, -adic valuation is a Valuation (algebra), valuation and gives rise to an analogue of the usual absolute value. Whereas the Complete metric space, completion of the rational numbers with respect to the usual absolute value results in the real numbers \mathbb, the completion of the rational numbers with respect to the p-adic absolute value results in the p-adic number, numbers \mathbb_p. Definition and properties Let be a prime number. Integers The -adic valuation of an integer n is defined to be : \nu_p(n)= \begin \mathrm\ & \text n \neq 0\\ \infty & \text n=0, \end where \mathbb_0 denotes the set of natural numbers (including zero) and m \mid n denotes divisor, divisibility of n by m. In particular, \nu_p is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into Sequence#Increasing_and_decreasing, increasing sequences, to which the lexicographical order is applied. A generalization defines an order on an ''n''-ary Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. Definition The words in a lexicon (the set of words u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-free Word
In combinatorics, a square-free word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form , where is not empty. Thus, a square-free word can also be defined as a word that avoids the pattern . Finite square-free words Binary alphabet Over a binary alphabet \, the only square-free words are the empty word \epsilon,0,1,01,10,010, and 101. Ternary alphabet Over a ternary alphabet ''\'', there are infinitely many square-free words. It is possible to count the number c(n) of ternary square-free words of length . This number is bounded by c(n) = \Theta(\alpha^n) , where 1.3017597 in \Sigma_ uniformly at random set \chi_w to ''w /math>'' followed by all other letters of \Sigma_ in increasing order set the number of iterations to 0 while , w, to the end of update \chi_w shifting the first elements to the right and setting \chi_w = a increment by if ends with a square of rank then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray (researcher), Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be "", and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two. Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some DOCSIS, cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. Function Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
