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495 (number)
495 (four hundred ndninety-five) is the natural number following 494 and preceding 496. It is a pentatope number (and so a binomial coefficient \tbinom 4 ). The maximal number of pieces that can be obtained by cutting an annulus with 30 cuts. Kaprekar transformation The Kaprekar's routine algorithm is defined as follows for three-digit numbers: # Take any three-digit number, other than repdigits such as 111. Leading zeros are allowed. # Arrange the digits in descending and then in ascending order to get two three-digit numbers, adding leading zeros if necessary. # Subtract the smaller number from the bigger number. # Go back to step 2 and repeat. Repeating this process will always reach 495 in a few steps. Once 495 is reached, the process stops because 954 – 459 = 495. Example For example, choose 495: :495 The only three-digit numbers for which this function does not work are repdigits such as 111, which give the answer 0 after a single iteration. All other three- ... [...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 number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, 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 Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
494 (number)
400 (four hundred) is the natural number following 399 and preceding 401. Mathematical properties 400 is the square of 20. 400 is the sum of the powers of 7 from 0 to 3, thus making it a repdigit in base 7 (1111). A circle is divided into 400 grads, which is equal to 360 degrees and 2π radians. (Degrees and radians are the SI accepted units). 400 is a self number in base 10, since there is no integer that added to the sum of its own digits results in 400. On the other hand, 400 is divisible by the sum of its own base 10 digits, making it a Harshad number. Other fields Four hundred is also * The Four Hundred (oligarchy) of ancient Athens. * An HTTP status code for a bad client request. * The Four Hundred (sometimes The Four Hundred Club) a phrase meaning the wealthiest, most famous, or most powerful social group (see, e.g., Ward McAllister), leading to the generation of such lists as the Forbes 400. * The Atari 400 home computer. * A former limited stop bus route wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
496 (number)
496 (four hundred ndninety-six) is the natural number following 495 and preceding 497. In mathematics 496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case. A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500. There is no solution to the equation φ(''x'') = 496, making 496 a nontotient. ''E''8 has real dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentatope Number
A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row , either from left to right or from right to left. The first few numbers of this kind are: : 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns. Formula The formula for the th pentatope number is represented by the 4th rising factorial of divided by the factorial of 4: :P_n = \frac = \frac . The pentatope numbers can also be represented as binomial coefficients: :P_n = \binom , which is the number of distinct quadruples that can be selected from objects, and it is read aloud as " plus three choose four". Properties Two of every three pentatope numbers are also pentagonal numbers. To be precise, the th pentatope number is always the th pentagonal number and the th pentatope number is always the th pentagonal number. The th pent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar's Routine
In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts the second from the first to yield the natural number for the next iteration. It is named after its inventor, the Indian mathematician D. R. Kaprekar. Kaprekar showed that in the case of four-digit numbers in base 10, if the initial number has at least two distinct digits, after seven iterations this process always yields the number 6174, which is now known as Kaprekar's constant. Definition and properties The algorithm is as follows: # Choose any natural number n in a given number base b. This is the first number of the sequence. # Create a new number \alpha by sorting the digits of n in descending order, and another new number \beta by sorting the digits of n in ascending order. These numbers may have leading zeros, which are discarded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Repdigits are the representation in base B of the number x\frac where 0 1 and ''n'', ''m'' > 2 : **(''p'', ''x'', ''y'', ''m'', ''n'') = (31, 5, 2, 3, 5) corresponding to 31 = 111112 = 1115, and, **(''p'', ''x'', ''y'', ''m'', ''n'') = (8191, 90, 2, 3, 13) corresponding to 8191 = 11111111111112 = 11190, with 11111111111 is the repunit with thirteen digits 1. *For each sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0 (number)
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usually by 10. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures. Common names for the number 0 in English are ''zero'', ''nought'', ''naught'' (), ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' (). Informal or slang terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' (), and ''cipher'', have also been used. Etymology The word ''zero'' came into the English language via French from the Italian , a contraction of the Venetian form of Italian via ''ṣafira'' or ''ṣifr''. In pre-Islamic time the word (Arabic ) had the meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6174 (number)
6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is renowned for the following rule: # Take any four-digit number, using at least two different digits (leading zeros are allowed). # Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. # Subtract the smaller number from the bigger number. # Go back to step 2 and repeat. The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Hanover 2017, p. 1, Overview. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1459: :9541 – 1459 = 8082 :8820 – 0288 = 8532 :8532 – 2358 = 6174 :7641 – 1467 = 6174 The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collatz Conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the problem, the conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem. The sequence of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
