Deoxyribonucleic acid is a molecule composed of two chains that coil around each other to form a double helix carrying the genetic instructions used in the growth, development and reproduction of all known organisms and many viruses. DNA and ribonucleic acid are nucleic acids; the two DNA strands are known as polynucleotides as they are composed of simpler monomeric units called nucleotides. Each nucleotide is composed of one of four nitrogen-containing nucleobases, a sugar called deoxyribose, a phosphate group; the nucleotides are joined to one another in a chain by covalent bonds between the sugar of one nucleotide and the phosphate of the next, resulting in an alternating sugar-phosphate backbone. The nitrogenous bases of the two separate polynucleotide strands are bound together, according to base pairing rules, with hydrogen bonds to make double-stranded DNA; the complementary nitrogenous bases are divided into two groups and purines. In DNA, the pyrimidines are cytosine. Both strands of double-stranded DNA store the same biological information.
This information is replicated as and when the two strands separate. A large part of DNA is non-coding, meaning that these sections do not serve as patterns for protein sequences; the two strands of DNA are thus antiparallel. Attached to each sugar is one of four types of nucleobases, it is the sequence of these four nucleobases along the backbone. RNA strands are created using DNA strands as a template in a process called transcription. Under the genetic code, these RNA strands specify the sequence of amino acids within proteins in a process called translation. Within eukaryotic cells, DNA is organized into long structures called chromosomes. Before typical cell division, these chromosomes are duplicated in the process of DNA replication, providing a complete set of chromosomes for each daughter cell. Eukaryotic organisms store most of their DNA inside the cell nucleus as nuclear DNA, some in the mitochondria as mitochondrial DNA, or in chloroplasts as chloroplast DNA. In contrast, prokaryotes store their DNA only in circular chromosomes.
Within eukaryotic chromosomes, chromatin proteins, such as histones and organize DNA. These compacting structures guide the interactions between DNA and other proteins, helping control which parts of the DNA are transcribed. DNA was first isolated by Friedrich Miescher in 1869, its molecular structure was first identified by Francis Crick and James Watson at the Cavendish Laboratory within the University of Cambridge in 1953, whose model-building efforts were guided by X-ray diffraction data acquired by Raymond Gosling, a post-graduate student of Rosalind Franklin. DNA is used by researchers as a molecular tool to explore physical laws and theories, such as the ergodic theorem and the theory of elasticity; the unique material properties of DNA have made it an attractive molecule for material scientists and engineers interested in micro- and nano-fabrication. Among notable advances in this field are DNA origami and DNA-based hybrid materials. DNA is a long polymer made from repeating units called nucleotides.
The structure of DNA is dynamic along its length, being capable of coiling into tight loops and other shapes. In all species it is composed of two helical chains, bound to each other by hydrogen bonds. Both chains are coiled around the same axis, have the same pitch of 34 angstroms; the pair of chains has a radius of 10 angstroms. According to another study, when measured in a different solution, the DNA chain measured 22 to 26 angstroms wide, one nucleotide unit measured 3.3 Å long. Although each individual nucleotide is small, a DNA polymer can be large and contain hundreds of millions, such as in chromosome 1. Chromosome 1 is the largest human chromosome with 220 million base pairs, would be 85 mm long if straightened. DNA does not exist as a single strand, but instead as a pair of strands that are held together; these two long strands coil in the shape of a double helix. The nucleotide contains both a segment of the backbone of a nucleobase. A nucleobase linked to a sugar is called a nucleoside, a base linked to a sugar and to one or more phosphate groups is called a nucleotide.
A biopolymer comprising multiple linked nucleotides is called a polynucleotide. The backbone of the DNA strand is made from alternating sugar residues; the sugar in DNA is 2-deoxyribose, a pentose sugar. The sugars are joined together by phosphate groups that form phosphodiester bonds between the third and fifth carbon atoms of adjacent sugar rings; these are known as the 3′-end, 5′-end carbons, the prime symbol being used to distinguish these carbon atoms from those of the base to which the deoxyribose forms a glycosidic bond. When imagining DNA, each phosphoryl is considered to "belong" to the nucleotide whose 5′ carbon forms a bond therewith. Any DNA strand therefore has one end at which there is a phosphoryl attached to the 5′ carbon of a ribose and another end a
In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The value of 0! is 1, according to the convention for an empty product. The factorial operation is encountered in many areas of mathematics, notably in combinatorics and mathematical analysis, its most basic use counts the possible distinct sequences -- the permutations -- of n distinct objects: there are n!. The factorial function can be extended to non-integer arguments while retaining its most important properties. Factorials were used to count permutations at least as early as the 12th century, by Indian scholars. In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells. After describing a recursive approach, Stedman gives a statement of a factorial: The notation n! was introduced by the French mathematician Christian Kramp in 1808. The factorial function is defined by the product n!
= 1 ⋅ 2 ⋅ 3 ⋯ ⋅ ⋅ n, for integer n ≥ 1. This may be written in the Pi product notation as n! = ∏ i = 1 n i. From these formulas, one may derive the recurrence relation n! = n ⋅!. For example, one has 5! = 5 ⋅ 4! 6! = 6 ⋅ 5! 50! = 50 ⋅ 49! and so on. The factorial of 0, 0!, is 1. There are several motivations for this definition: For n = 0, the definition of n! as a product involves the product of no numbers at all, so is an example of the broader convention that the product of no factors is equal to the multiplicative identity. There is one permutation of zero objects, it makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is given by the binomial coefficient = 0! 0! 0! = 1. More the number of ways to choose all n elements among a set of n is = n! N! 0! = 1. It allows for the compact expression of many formulae, such as the exponential function, as a power series: e x = ∑ n = 0 ∞ x n n!. It extends the recurrence relation to 0; the factorial function can be defined for non-integer values using more advanced mathematics, detailed in the section below.
This more generalized definition is used by advanced calculators and mathematical software such as Maple, Mathematica, or APL. Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n! Different ways of arranging n distinct objects into a sequence, the permutations of those objects. Factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements. One can obtain such a combination by choosing a k-permutation: successively selecting and removing one element of the set, k times, for a total of ⋯ = n!! = n k _ possibilities. This however produces the k-combinations in a particular order. Different ways, the correct number of k-combinations is n ( n
Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is called, "Fibonacci", was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci, he is known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano. Fibonacci popularized the Hindu–Arabic numeral system in the Western World through his composition in 1202 of Liber Abaci, he introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci was born around 1170 to an Italian merchant and customs official. Guglielmo directed a trading post in Algeria. Fibonacci travelled with him as a young boy, it was in Bugia that he learned about the Hindu–Arabic numeral system. Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic, he soon realised the many advantages of the Hindu-Arabic system which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system.
In 1202, he completed the Liber Abaci. Fibonacci became a guest of Emperor Frederick II. In 1240, the Republic of Pisa honored Fibonacci by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens; the date of Fibonacci's death is not known, but it has been estimated to be between 1240 and 1250, most in Pisa. In the Liber Abaci, Fibonacci introduced the so-called modus Indorum, today known as the Hindu–Arabic numeral system; the book advocated numeration with the digits place value. The book showed the practical use and value of the new Hindu-Arabic numeral system by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, other applications; the book had a profound impact on European thought. No copies of the 1202 edition are known to exist; the 1228 edition, first section introduces the Hindu-Arabic numeral system and compares the system with other systems, such as Roman numerals, methods to convert the other numeral systems into Hindu-Arabic numerals.
Replacing the Roman numeral system, its ancient Egyptian multiplication method, using an abacus for calculations, with a Hindu-Arabic numeral system was an advance in making business calculations easier and faster, which led to the growth of banking and accounting in Europe. The second section explains the uses of Hindu-Arabic numerals in business, for example converting different currencies, calculating profit and interest, which were important to the growing banking industry; the book discusses irrational numbers and prime numbers. Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions; the solution, generation by generation, was a sequence of numbers known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century. In the Fibonacci sequence, each number is the sum of the previous two numbers.
Fibonacci omitted the "0" included today and began the sequence with 1, 1, 2.... He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377. Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was raised in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis. Liber Abaci, a book on calculations Practica Geometriae, a compendium of techniques in surveying, the measurement and partition of areas and volumes, other topics in practical geometry.
Flos, solutions to problems posed by Johannes of Palermo Liber quadratorum on Diophantine equations, dedicated to Emperor Frederick II. See in particular congruum and the Brahmagupta–Fibonacci identity. Di minor guisa Commentary on Book X of Euclid's Elements Fibonacci numbers in popular culture Republic of Pisa Adelard of Bath Footnotes Citations Devlin, Keith; the Man of Numbers: Fibonacci's Arithmetic Revolution. Walker Books. ISBN 978-0802779083. Goetzmann, William N. and Rouwenhorst, K. Geert, The Origins of Value: The Financial Innovations That Created Modern Capital Markets, ISBN 0-19-517571-9. Goetzmann, William N. Fibonacci and the Financial Revolution, Yale School of Management International Center for Finance Working Paper No. 03–28 Grimm, R. E. "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99–104. Horadam, A. F. "Eight hundred years young," The Australian Mathematics Teacher 31 (1975