In mathematics, the Fibonacci numbers denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. One has F2 = 1. In some books, in old ones, F0, the "0" is omitted, the Fibonacci sequence starts with F1 = F2 = 1; the beginning of the sequence is thus: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa known as Fibonacci, they appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
Fibonacci numbers appear unexpectedly in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts. Fibonacci numbers are closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. Lucas numbers are intimately connected with the golden ratio; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short syllables of 1 unit duration.
Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number... of the next mātrā-vṛtta."
Outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month; this is the nth Fibonacci number. The name "Fibonacci sequence" was first used by the 19th
The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom. It is identical to the charge number of the nucleus; the atomic number uniquely identifies a chemical element. In an uncharged atom, the atomic number is equal to the number of electrons; the sum of the atomic number Z and the number of neutrons, N, gives the mass number A of an atom. Since protons and neutrons have the same mass and the mass defect of nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units, is within 1% of the whole number A. Atoms with the same atomic number Z but different neutron numbers N, hence different atomic masses, are known as isotopes. A little more than three-quarters of occurring elements exist as a mixture of isotopes, the average isotopic mass of an isotopic mixture for an element in a defined environment on Earth, determines the element's standard atomic weight, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century.
The conventional symbol Z comes from the German word Zahl meaning number, before the modern synthesis of ideas from chemistry and physics denoted an element's numerical place in the periodic table, whose order is but not consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was the nuclear charge and a physical characteristic of atoms, did the word Atomzahl come into common use in this context. Loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, so they can be numbered in order. Dmitri Mendeleev claimed. However, in consideration of the elements' observed chemical properties, he changed the order and placed tellurium ahead of iodine; this placement is consistent with the modern practice of ordering the elements by proton number, Z, but that number was not known or suspected at the time. A simple numbering based on periodic table position was never satisfactory, however.
Besides the case of iodine and tellurium several other pairs of elements were known to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not obvious, led to inconsistency and uncertainty in the periodic numbering of elements at least from lutetium onward. In 1911, Ernest Rutherford gave a model of the atom in which a central core held most of the atom's mass and a positive charge which, in units of the electron's charge, was to be equal to half of the atom's atomic weight, expressed in numbers of hydrogen atoms; this central charge would thus be half the atomic weight. In spite of Rutherford's estimation that gold had a central charge of about 100, a month after Rutherford's paper appeared, Antonius van den Broek first formally suggested that the central charge and number of electrons in an atom was equal to its place in the periodic table.
This proved to be the case. The experimental position improved after research by Henry Moseley in 1913. Moseley, after discussions with Bohr, at the same lab, decided to test Van den Broek's and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fitted the Bohr theory's postulation that the frequency of the spectral lines be proportional to the square of Z. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube; the square root of the frequency of these photons increased from one target to the next in an arithmetic progression. This led to the conclusion that the atomic number does correspond to the calculated electric charge of the nucleus, i.e. the element number Z. Among other things, Moseley demonstrated that the lanthanide series must have 15 members—no fewer and no more—which was far from obvious from the chemistry at that time.
After Moseley's death in 1915, the atomic numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, corresponding to atomic numbers 43, 61, 72, 75, 85, 87 and 91. From 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had been discovered, so that the periodic table was complete with no gaps as far as curium. In 1915 the rea
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than itself; every positive integer is composite, prime, or the unit 1, so the composite numbers are the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7; the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150.
Every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 × 23, the composite number 360 can be written as 23 × 32 × 5; this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, without revealing the factorization of a composite input. One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a 2-almost prime. A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an number of distinct prime factors. For the latter μ = 2 x = 1, while for the former μ = 2 x + 1 = − 1. However, for prime numbers, the function returns −1 and μ = 1. For a number n with one or more repeated prime factors, μ = 0. If all the prime factors of a number are repeated it is called a powerful number.
If none of its prime factors are repeated, it is called squarefree. For example, 72 = 23 × 32, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are. A number n that has more divisors than any x < n is a composite number. Composite numbers have been called "rectangular numbers", but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers, yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed number. Such numbers are called rough numbers, respectively. Canonical representation of a positive integer Integer factorization Sieve of Eratosthenes Table of prime factors Fraleigh, John B. A First Course In Abstract Algebra, Reading: Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N.
Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 Long, Calvin T. Elementary Introduction to Number Theory, Lexington: D. C. Heath and Company, LCCN 77-171950 McCoy, Neal H. Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225 Pettofrezzo, Anthony J.. Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 Lists of composites with prime factorization Divisor Plot
The system of imperial units or the imperial system is the system of units first defined in the British Weights and Measures Act of 1824, refined and reduced. The Imperial units replaced the Winchester Standards, which were in effect from 1588 to 1825; the system came into official use across the British Empire. By the late 20th century, most nations of the former empire had adopted the metric system as their main system of measurement, although some imperial units are still used in the United Kingdom and other countries part of the British Empire; the imperial system developed from what were first known as English units, as did the related system of United States customary units. The Weights and Measures Act of 1824 was scheduled to go into effect on 1 May 1825. However, the Weights and Measures Act of 1825 pushed back the date to 1 January 1826; the 1824 Act allowed the continued use of pre-imperial units provided that they were customary known, marked with imperial equivalents. Apothecaries' units are mentioned neither in the act of 1824 nor 1825.
At the time, apothecaries' weights and measures were regulated "in England and Berwick-upon-Tweed" by the London College of Physicians, in Ireland by the Dublin College of Physicians. In Scotland, apothecaries' units were unofficially regulated by the Edinburgh College of Physicians; the three colleges published, at infrequent intervals, the London and Dublin editions having the force of law. Imperial apothecaries' measures, based on the imperial pint of 20 fluid ounces, were introduced by the publication of the London Pharmacopoeia of 1836, the Edinburgh Pharmacopoeia of 1839, the Dublin Pharmacopoeia of 1850; the Medical Act of 1858 transferred to The Crown the right to publish the official pharmacopoeia and to regulate apothecaries' weights and measures. Metric equivalents in this article assume the latest official definition. Before this date, the most precise measurement of the imperial Standard Yard was 0.914398415 metres. In 1824, the various different gallons in use in the British Empire were replaced by the imperial gallon, a unit close in volume to the ale gallon.
It was defined as the volume of 10 pounds of distilled water weighed in air with brass weights with the barometer standing at 30 inches of mercury at a temperature of 62 °F. In 1963, the gallon was redefined as the volume of 10 pounds of distilled water of density 0.998859 g/mL weighed in air of density 0.001217 g/mL against weights of density 8.136 g/mL, which works out to 4.546096 l or 277.4198 cu in. The Weights and Measures Act of 1985 switched to a gallon of 4.54609 L. These measurements were in use from 1826, when the new imperial gallon was defined, but were abolished in the United Kingdom on 1 January 1971. In the US, though no longer recommended, the apothecaries' system is still used in medicine in prescriptions for older medications. In the 19th and 20th centuries, the UK used three different systems for weight. Troy weight, used for precious metals; the distinction between mass and weight is not always drawn. A pound is a unit of mass, although it is referred to as a weight; when a distinction is necessary, the term pound-force may be used to refer to a unit of force rather than mass.
The troy pound was made the primary unit of mass by the 1824 Act. The Weights and Measures Act 1855 made the avoirdupois pound the primary unit of mass. In all the systems, the fundamental unit is the pound, all other units are defined as fractions or multiples of it. Although the 1824 act defined the yard and pound by reference to the prototype standards, it defined the values of certain physical constants, to make provision for re-creation of the standards if they were to be damaged. For the yard, the length of a pendulum beating seconds at the latitude of Greenwich at Mean Sea Level in vacuo was defined as 39.01393 inches. For the pound, the mass of a cubic inch of distilled water at an atmospheric pressure of 30 inches of mercury and a temperature of 62° Fahrenheit was defined as 252.458 grains, with there being 7,000 grains per pound. However, following the destruction of the original prototypes in the 1834 Houses of Parliament fire, it proved impossible to recreate the standards from these definitions, a new Weights and Measures Act was passed in 1855 which permitted the recreation of the prototypes from recognized secondary standards.
The imperial system is one of many systems of English units. Although most of the units are defined in more than one system, some subsidiary units were used to a much greater extent, or for different purposes, in one area rather than the other; the distinctions between these systems are not drawn precisely. One such distinction is that between these systems and older British/English units/systems or newer additions; the term imperial should not be applied to English units that were outlawed in the Weights and Measures Act 1824 or earlier, or which had fallen out of use by that time, nor to post-imperial inventions, such as the slug or poundal. The US customary system is derived from the English units that were in use at the time of settlement; because the United States was independent at the time, these units were unaffected b
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, employ seven symbols, each with a fixed integer value, as follows: The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; the original pattern for Roman numerals used the symbols I, V, X as simple tally marks. Each marker for 1 added a unit value up to 5, was added to to make the numbers from 6 to 9: I, II, III, IIII, V, VI, VII, VIII, VIIII, X; the numerals for 4 and 9 proved problematic, are replaced with IV and IX. This feature of Roman numerals is called subtractive notation; the numbers from 1 to 10 are expressed in Roman numerals as follows: I, II, III, IV, V, VI, VII, VIII, IX, X.
The system being decimal and hundreds follow the same underlying pattern. This is the key to understanding Roman numerals: Thus 10 to 100: X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C. Note that 40 and 90 follow the same subtractive pattern as 4 and 9, avoiding the confusing XXXX. 100 to 1000: C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M. Again - 400 and 900 follow the standard subtractive pattern, avoiding CCCC. In the absence of standard symbols for 5,000 and 10,000 the pattern breaks down at this point - in modern usage M is repeated up to three times; the Romans had several ways to indicate larger numbers, but for practical purposes Roman Numerals for numbers larger than 3,999 are if used nowadays, this suffices. M, MM, MMM. Many numbers include hundreds and tens; the Roman numeral system being decimal, each power of ten is added in descending sequence from left to right, as with Arabic numerals. For example: 39 = "Thirty nine" = XXXIX. 246 = "Two hundred and forty six" = CCXLVI. 421 = "Four hundred and twenty one" = CDXXI.
As each power of ten has its own notation there is no need for place keeping zeros, so "missing places" are ignored, as in Latin speech, thus: 160 = "One hundred and sixty" = CLX 207 = "Two hundred and seven" = CCVII 1066 = "A thousand and sixty six" = MLXVI. Roman numerals for large numbers are nowadays seen in the form of year numbers, as in these examples: 1776 = MDCCLXXVI. 1954 = MCMLIV 1990 = MCMXC. 2014 = MMXIV (the year of the games of the XXII Olympic Winter Games The current year is MMXIX. The "standard" forms described above reflect typical modern usage rather than an unchanging and universally accepted convention. Usage in ancient Rome varied and remained inconsistent in medieval times. There is still no official "binding" standard, which makes the elaborate "rules" used in some sources to distinguish between "correct" and "incorrect" forms problematic. "Classical" inscriptions not infrequently use IIII for "4" instead of IV. Other "non-subtractive" forms, such as VIIII for IX, are sometimes seen, although they are less common.
On the numbered gates to the colosseum, for instance, IV is systematically avoided in favour of IIII, but other "subtractives" apply, so that gate 44 is labelled XLIIII. Isaac Asimov speculates that the use of "IV", as the initial letters of "IVPITER" may have been felt to have been impious in this context. Clock faces that use Roman numerals show IIII for four o'clock but IX for nine o'clock, a practice that goes back to early clocks such as the Wells Cathedral clock of the late 14th century. However, this is far from universal: for example, the clock on the Palace of Westminster, Big Ben, uses a "normal" IV. XIIX or IIXX are sometimes used for "18" instead of XVIII; the Latin word for "eighteen" is rendered as the equivalent of "two less than twenty" which may be the source of this usage. The standard forms for 98 and 99 are XCVIII and XCIX, as described in the "decimal pattern" section above, but these numbers are rendered as IIC and IC originally from the Latin duodecentum and undecentum.
Sometimes V and L are not used, with instances such as IIIIII and XXXXXX rather than VI or LX. Most non-standard numerals other than those described above - such as VXL for 45, instead of the standard XLV are modern and may be due to error rather than being genuine variant usage. In the early years of the 20th century, different representations of 900 appeared in several inscribed dates. For instance, 1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, while on the north entrance to the Saint Louis Art Museum, 1903 is inscribed as MDCDIII rather than MCMIII. Although Roman numerals came to be written with letters
80 is the natural number following 79 and preceding 81. 80 is: the sum of Euler's totient function φ over the first sixteen integers. A semiperfect number, since adding up some subsets of its divisors gives 80. A ménage number. Palindromic in bases 3, 6, 9, 15, 19 and 39. A repdigit in bases 3, 9, 15, 19 and 39; the Pareto principle states that, for many events 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves; the atomic number of mercury According to Exodus 7:7, Moses was 80 years old when he spoke to Pharaoh on behalf of his people. Today, 80 years of age is the upper age limit for cardinals to vote in papal elections. Eighty is also: used in the classic book title Around the World in Eighty Days the length of the Eighty Years' War or Dutch revolt the standard TCP/IP port number for HTTP connections the 80A, 80B and 80C photographic filters correct for excessive redness under tungsten lighting The year AD 80, 80 BC, or 1980 Eighty shilling ale The older four-pin-base version of the 5Y3GT rectifier tube A common limit for the characters per line, in computing, derived from the number of columns in IBM cards American band Green Day has a song called "80" A fictional alien superhero named Ultraman 80 List of highways numbered 80 wiktionary:eighty for 80 in other languages
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. 100 is the square of 10. The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages. 100 is the sum of the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g. 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. 100 is the sum of the cubes of the first four integers. This is related by Nicomachus's theorem to the fact that 100 equals the square of the sum of the first four integers: 100 = 102 = 2.26 + 62 = 100, thus 100 is a Leyland number.100 is an 18-gonal number. It is divisible by 25, the number of primes below it, it can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors.
100 is a Harshad number in base 10, in base 4, in that base it is a self-descriptive number. There are 100 prime numbers whose digits are in ascending order. 100 is the smallest number. One hundred is the atomic number of fermium, an actinide and the first of the heavy metals that cannot be created through neutron bombardment. On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level; the Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is used to define the boundary between Earth's atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of the Jewish New Year. A religious Jew is expected to utter at least 100 blessings daily. In the Hindu book of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas; the United States Senate has 100 Senators. Most of the world's currencies are divided into 100 subunits; the 100 Euro banknotes feature a picture of a Rococo gateway on the obverse and a Baroque bridge on the reverse.
The U. S. hundred-dollar bill has Benjamin Franklin's portrait. S. bill in print. American savings bonds of $100 have Thomas Jefferson's portrait, while American $100 treasury bonds have Andrew Jackson's portrait. One hundred is also: The number of years in a century; the number of pounds in an American short hundredweight. In Greece, India and Nepal, 100 is the police telephone number. In Belgium, 100 is the firefighter telephone number. In United Kingdom, 100 is the operator telephone number; the HTTP status code indicating that the client should continue with its request. The 100 The age at which a person becomes a centenarian; the number of yards in an American football field. The number of runs required for a cricket batsman to score a significant milestone; the number of points required for a snooker player to score a century break, a significant milestone. The record number of points scored in one NBA game by a single player, set by Wilt Chamberlain of the Philadelphia Warriors on March 2, 1962.
1 vs. 100 AFI's 100 Years... Hundred Hundred Hundred Days Hundred Years' War List of highways numbered 100 Top 100 Greatest 100 Wells, D; the Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group.: 133 Chisholm, Hugh, ed.. "Hundred". Encyclopædia Britannica. Cambridge University Press. On the Number 100