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Digital Root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, in base 10, the digital root of the number 12345 is 6 because the sum of the digits in the number is 1 + 2 + 3 + 4 + 5 = 15, then the addition process is repeated again for the resulting number 15, so that the sum of 1 + 5 equals 6, which is the digital root of that number. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0), which allows it to be used as a divisibility rule. Formal definition Let n be a natural number. For base b > 1, we define the digit sum F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \sum_^ d_i where k = \lfloor \lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Digital Root
In number theory, the multiplicative digital root of a natural number n in a given number base b is found by multiplying the digits of n together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n. The multiplicative digital root for the first few positive integers are: :0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. Multiplicative digital roots are the multiplicative equivalent of digital roots. Definition Let n be a natural number. We define the digit product for base b > 1 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \prod_^ d_i where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, and :d_i = \frac is the value of each digit of the number. A natural number n is a multiplicative digital root if it is a fixed point for F_, which occurs if F_(n) = n. For example, in base b = 10, 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as '' nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Logarithm
In computer science, the iterated logarithm of n, written n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recurrence relation: : \log^* n := \begin 0 & \mbox n \le 1; \\ 1 + \log^*(\log n) & \mbox n > 1 \end On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: :\log^* n = \lceil \mathrm _e(n) \rceil i.e. the base ''b'' iterated logarithm is \log^* n = y if n lies within the interval ^b on the ''x''-axis. In computer science, is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base 2) instead of the natural logarithm (with base ''e''). Mathematically, the iterated logarithm is well-defined for any base greater than e^ \approx 1.444667, not only for base 2 and base ''e''. Analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string, or the digit sum of the binary representation of a given number and the ''ℓ''₁ norm of a bit vector. In this binary case, it is also called the population count, popcount, sideways sum, or bit summation. History and usage The Hamming weight is named after Richard Hamming although he did not originate the notion. The Hamming weight of binary numbers was already used in 1899 by James W. L. Glaisher to give a formula for the number of odd binomial coefficients in a single row of Pascal's triangle. Irving S. Reed introduced a concept, equivalent to Hamming weight in the binary case, in 1954. Hamming weight is used in several disciplines including information theory, coding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisibility Rule
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in ''Scientific American''. Divisibility rules for numbers 1–30 The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last ''n'' digits) the result must be examined by other means. For divisors with multiple ru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digit Sum
In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. We define the digit sum for base b > 1 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \sum_^ d_i where k = \lfloor \log_ \rfloor is the number of digits in the number in base b, and :d_i = \frac is the value of each digit of the number. For example, in base 10, the digit sum of 84001 is F_(84001) = 8 + 4 + 0 + 0 + 1 = 13. For any two bases 2 \leq b_1 < b_2 and for sufficiently large natural numbers , :.. The sum of the base 10 digits of the integers 0, 1, 2, ... is given by in the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casting Out Nines
Casting out nines is any of three arithmetical procedures: *Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller than the original whenever the original has more than one digit, leaves the same remainder as the original after division by nine, and may be obtained from the original by subtracting a multiple of 9 from it. The name of the procedure derives from this latter property. *Repeated application of this procedure to the results obtained from previous applications until a single-digit number is obtained. This single-digit number is called the "digital root" of the original. If a number is divisible by 9, its digital root is 9. Otherwise, its digital root is the remainder it leaves after being divided by 9. *A sanity test in which the above-mentioned procedures are used to check for errors in arithmetical calculations. The test is carried o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base 9
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 or senary 1405 corresponds to binary 101101101 (nine digits) and to ternary 111112 (six digits). However, they are still far less compact than the corresponding representations in bases such as decimalsee below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27). As for rational numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. ''Global arithmetic dynamics'' is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while ''local arithmetic dynamics'', also called p-adic or nonarchimedean dynamics, is an analogue of classical dynamics in which one replaces the complex numbers by a -adic field such as or and studies chaotic behavior and the Fatou and Julia sets. The following table describes a rough correspondence between Diophantine equations, espec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nine Hours, Nine Persons, Nine Doors
''Nine Hours, Nine Persons, Nine Doors'' is a visual novel and adventure video game developed by Chunsoft. It is the first installment in the ''Zero Escape'' series, and was released in Japan in December 2009 and in North America in November 2010 for the Nintendo DS. The story follows Junpei, a college student who is abducted along with eight other people and forced to play the "Nonary Game", which puts its participants in a life-or-death situation, to escape from a sinking cruise liner. The gameplay alternates between two types of sections: Escape sections, where the player completes puzzles in escape-the-room scenarios; and Novel sections, where the player reads the game's narrative and makes decisions that influence the story toward one of six different endings. Development of the game began after Kotaro Uchikoshi joined Chunsoft to write a visual novel for them that could reach a wider audience; Uchikoshi suggested adding puzzle elements that are integrated with the game's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
