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96-622: Fibonacci ( / ˌ f ɪ b ə ˈ n ɑː tʃ i / , also US : / ˌ f iː b -/ ; Italian: [fiboˈnattʃi] ; c.  1170 – c.  1240–50 ) 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 commonly called, Fibonacci , was made up in 1838 by the Franco-Italian historian Guillaume Libri and

192-748: A φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}} where a = U 1 − U 0 ψ 5 , b = U 0 φ − U 1 5 . {\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}} Since | ψ n 5 | < 1 2 {\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}} for all n ≥ 0 ,

288-589: A φ n − 1 + b ψ n − 1 + a φ n − 2 + b ψ n − 2 = U n − 1 + U n − 2 . {\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}} If

384-402: A = 1 φ − ψ = 1 5 , b = − a , {\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,} producing the required formula. Taking the starting values U 0 and U 1 to be arbitrary constants, a more general solution is: U n =

480-707: A cot–caught merger , which is rapidly spreading throughout the whole country. However, the South, Inland North, and a Northeastern coastal corridor passing through Rhode Island, New York City, Philadelphia, and Baltimore typically preserve an older cot–caught distinction. For that Northeastern corridor, the realization of the THOUGHT vowel is particularly marked , as depicted in humorous spellings, like in tawk and cawfee ( talk and coffee ), which intend to represent it being tense and diphthongal : [oə] . A split of TRAP into two separate phonemes , using different

576-539: A and b are chosen so that U 0 = 0 and U 1 = 1 then the resulting sequence U n must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations: { a + b = 0 φ a + ψ b = 1 {\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.} which has solution

672-605: A and b , the sequence defined by U n = a φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}} satisfies the same recurrence, U n = a φ n + b ψ n = a ( φ n − 1 + φ n − 2 ) + b ( ψ n − 1 + ψ n − 2 ) =

768-520: A pronunciations for example in gap [æ] versus gas [eə] , further defines New York City as well as Philadelphia–Baltimore accents. Most Americans preserve all historical /r/ sounds, using what is known as a rhotic accent . The only traditional r -dropping (or non-rhoticity) in regional U.S. accents variably appears today in eastern New England , New York City , and some of the former plantation South primarily among older speakers (and, relatedly, some African-American Vernacular English across

864-631: A complex phenomenon of "both convergence and divergence": some accents are homogenizing and leveling , while others are diversifying and deviating further away from one another. Having been settled longer than the American West Coast, the East Coast has had more time to develop unique accents, and it currently comprises three or four linguistically significant regions, each of which possesses English varieties both different from each other as well as quite internally diverse: New England ,

960-447: A consonant, such as in pearl , car and fort . Non-rhotic American accents, those that do not pronounce ⟨r⟩ except before a vowel, such as some accents of Eastern New England , New York City , and African-Americans , and a specific few (often older ones) spoken by Southerners , are often quickly noticed by General American listeners and perceived as sounding especially ethnic, regional, or antiquated. Rhoticity

1056-438: A field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit math problem : how many pairs will there be in one year? At the end of the n -th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2 ) plus

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1152-417: A given total duration results in the Fibonacci numbers: the number of patterns of duration m units is F m +1 . Knowledge of the Fibonacci sequence was expressed as early as Pingala ( c.  450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats ( F m +1 )

1248-591: A merger with the THOUGHT ( caught ) set. Having taken place prior to the unrounding of the cot vowel, it results in lengthening and perhaps raising, merging the more recently separated vowel into the THOUGHT vowel in the following environments: before many instances of /f/ , /θ/ , and particularly /s/ (as in Austria, cloth, cost, loss, off, often, etc.), a few instances before /ŋ/ (as in strong, long, wrong ), and variably by region or speaker in gone , on , and certain other words. Unlike American accents,

1344-636: A nice day , for sure); many are now distinctly old-fashioned (swell, groovy). Some English words now in general use, such as hijacking, disc jockey , boost, bulldoze and jazz , originated as American slang. American English has always shown a marked tendency to use words in different parts of speech and nouns are often used as verbs . Examples of nouns that are now also verbs are interview, advocate, vacuum, lobby, pressure, rear-end, transition, feature, profile, hashtag, head, divorce, loan, estimate, X-ray, spearhead, skyrocket, showcase, bad-mouth, vacation , major, and many others. Compounds coined in

1440-480: A process of extensive dialect mixture and leveling in which English varieties across the colonies became more homogeneous compared with the varieties in Britain. English thus predominated in the colonies even by the end of the 17th century's first immigration of non-English speakers from Western Europe and Africa. Additionally, firsthand descriptions of a fairly uniform American English (particularly in contrast to

1536-680: A series of other vowel shifts in the same region, known by linguists as the " Inland North ". The Inland North shares with the Eastern New England dialect (including Boston accents ) a backer tongue positioning of the GOOSE /u/ vowel (to [u] ) and the MOUTH /aʊ/ vowel (to [ɑʊ~äʊ] ) in comparison to the rest of the country. Ranging from northern New England across the Great Lakes to Minnesota, another Northern regional marker

1632-401: A stem , the fruit sprouts of a pineapple , the flowering of an artichoke , and the arrangement of a pine cone 's bracts, though they do not occur in all species. Fibonacci numbers are also strongly related to the golden ratio : Binet's formula expresses the n -th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to

1728-525: A survey, completed in 2003, polling English speakers across the United States about their specific everyday word choices, hoping to identify regionalisms. The study found that most Americans prefer the term sub for a long sandwich, soda (but pop in the Great Lakes region and generic coke in the South) for a sweet and bubbly soft drink , you or you guys for the plural of you (but y'all in

1824-453: A variation of American English in these islands. In 2021, about 245 million Americans, aged 5 or above, spoke English at home: a majority of the United States total population of roughly 330 million people. The United States has never had an official language at the federal level, but English is commonly used at the federal level and in states without an official language. 32 of the 50 states, in some cases as part of what has been called

1920-481: Is asymptotic to φ n / 5 {\displaystyle \varphi ^{n}/{\sqrt {5}}} , the number of digits in F n is asymptotic to n log 10 ⁡ φ ≈ 0.2090 n {\displaystyle n\log _{10}\varphi \approx 0.2090\,n} . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits. More generally, in

2016-501: Is extended to negative integers using the Fibonacci rule F n = F n + 2 − F n + 1 . {\displaystyle F_{n}=F_{n+2}-F_{n+1}.} Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of 5 x 2 + 4 {\displaystyle 5x^{2}+4} or 5 x 2 − 4 {\displaystyle 5x^{2}-4}

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2112-486: Is a perfect square . This is because Binet's formula, which can be written as F n = ( φ n − ( − 1 ) n φ − n ) / 5 {\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}} , can be multiplied by 5 φ n {\displaystyle {\sqrt {5}}\varphi ^{n}} and solved as

2208-654: Is also associated with the United States, perhaps mostly in the Midwest and the South. American accents that have not undergone the cot–caught merger (the lexical sets LOT and THOUGHT ) have instead retained a LOT – CLOTH split : a 17th-century distinction in which certain words (labeled as the CLOTH lexical set ) separated away from the LOT set. The split, which has now reversed in most British English, simultaneously shifts this relatively recent CLOTH set into

2304-664: Is also home to a creole language known commonly as Hawaiian Pidgin , and some Hawaii residents speak English with a Pidgin-influenced accent. American English also gave rise to some dialects outside the country, for example, Philippine English , beginning during the American occupation of the Philippines and subsequently the Insular Government of the Philippine Islands ; Thomasites first established

2400-421: Is also the case that ψ 2 = ψ + 1 {\displaystyle \psi ^{2}=\psi +1} and it is also the case that ψ n = F n ψ + F n − 1 . {\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.} These expressions are also true for n < 1 if the Fibonacci sequence F n

2496-544: 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 , and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings , such as branching in trees, the arrangement of leaves on

2592-538: Is common in most American accents despite being now rare in England because, during the 17th-century British colonization, nearly all dialects of English were rhotic, and most North American English simply remained that way. The preservation of rhoticity in North America was also supported by continuing waves of rhotic-accented Scotch-Irish immigrants, most intensely during the 18th century (and moderately during

2688-507: Is obtained by adding one [S] to the F m cases and one [L] to the F m −1 cases. Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135): Variations of two earlier meters [is

2784-420: Is often identified by Americans as a "country" accent, and is defined by the /aɪ/ vowel losing its gliding quality : [aː] , the initiation event for a complicated Southern vowel shift, including a " Southern drawl " that makes short front vowels into distinct-sounding gliding vowels . The fronting of the vowels of GOOSE , GOAT , MOUTH , and STRUT tends to also define Southern accents as well as

2880-699: Is short for filius Bonacci ('son of Bonacci'). However, even earlier, in 1506, a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci". Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci ( Book of Calculation ) and also introduced Europe to the sequence of Fibonacci numbers , which he used as an example in Liber Abaci . Fibonacci

2976-1043: Is the golden ratio , and ψ is its conjugate : ψ = 1 − 5 2 = 1 − φ = − 1 φ ≈ − 0.61803 39887 … . {\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .} Since ψ = − φ − 1 {\displaystyle \psi =-\varphi ^{-1}} , this formula can also be written as F n = φ n − ( − φ ) − n 5 = φ n − ( − φ ) − n 2 φ − 1 . {\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.} To see

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3072-672: Is the common language at home, in public, and in government. Fibonacci number The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci , who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there

3168-469: Is the variable fronting of /ɑ/ before /r/ , for example, appearing four times in the stereotypical Boston shibboleth Park the car in Harvard Yard . Several other phenomena serve to distinguish regional U.S. accents. Boston , Pittsburgh , Upper Midwestern , and Western U.S. accents have fully completed a merger of the LOT vowel with the THOUGHT vowel ( /ɑ/ and /ɔ/ , respectively):

3264-475: Is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio φ : {\displaystyle \varphi \colon } lim n → ∞ F n + 1 F n = φ . {\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .} This convergence holds regardless of

3360-474: Is valid for n > 2 . The first 20 Fibonacci numbers F n are: The Fibonacci sequence appears in Indian mathematics , in connection with Sanskrit prosody . In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with

3456-482: The English-only movement , have adopted legislation granting official or co-official status to English. Typically only "English" is specified, not a particular variety like American English. (From 1923 to 1969, the state of Illinois recognized its official language as "American", meaning American English.) Puerto Rico is the largest example of a United States territory in which another language – Spanish –

3552-652: The Indians ), today known as the Hindu–Arabic numeral system , with ten digits including a zero and positional notation . The book showed the practical use and value of this by applying the numerals to commercial bookkeeping , converting weights and measures, calculation of interest, money-changing, and other applications. The book was well-received throughout educated Europe and had a profound impact on European thought. Replacing Roman numerals, its ancient Egyptian multiplication method, and using an abacus for calculations,

3648-622: The Mid-Atlantic states (including a New York accent as well as a unique Philadelphia–Baltimore accent ), and the South . As of the 20th century, the middle and eastern Great Lakes area , Chicago being the largest city with these speakers, also ushered in certain unique features, including the fronting of the LOT /ɑ/ vowel in the mouth toward [a] and tensing of the TRAP /æ/ vowel wholesale to [eə] . These sound changes have triggered

3744-551: The Native American languages . Examples of such names are opossum , raccoon , squash , moose (from Algonquian ), wigwam , and moccasin . American English speakers have integrated traditionally non-English terms and expressions into the mainstream cultural lexicon; for instance, en masse , from French ; cookie , from Dutch ; kindergarten from German , and rodeo from Spanish . Landscape features are often loanwords from French or Spanish, and

3840-461: The Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo) 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. Fibonacci is thought to have died between 1240 and 1250, in Pisa. In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of

3936-426: The base b representation, the number of digits in F n is asymptotic to n log b ⁡ φ = n log ⁡ φ log ⁡ b . {\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.} Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges . He wrote that "as 5

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4032-1188: The floor function gives the largest index of a Fibonacci number that is not greater than F : n l a r g e s t ( F ) = ⌊ log φ ⁡ 5 ( F + 1 / 2 ) ⌋ ,   F ≥ 0 , {\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,} where log φ ⁡ ( x ) = ln ⁡ ( x ) / ln ⁡ ( φ ) = log 10 ⁡ ( x ) / log 10 ⁡ ( φ ) {\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )} , ln ⁡ ( φ ) = 0.481211 … {\displaystyle \ln(\varphi )=0.481211\ldots } , and log 10 ⁡ ( φ ) = 0.208987 … {\displaystyle \log _{10}(\varphi )=0.208987\ldots } . Since F n

4128-722: The francophile tastes of the 19th century Victorian era Britain (for example they preferred programme for program , manoeuvre for maneuver , cheque for check , etc.). AmE almost always uses -ize in words like realize . BrE prefers -ise , but also uses -ize on occasion (see: Oxford spelling ). There are a few differences in punctuation rules. British English is more tolerant of run-on sentences , called " comma splices " in American English, and American English prefers that periods and commas be placed inside closing quotation marks even in cases in which British rules would place them outside. American English also favors

4224-764: The 18th century; apartment , shanty in the 19th century; project, condominium , townhouse , mobile home in the 20th century; and parts thereof ( driveway , breezeway, backyard ) . Industry and material innovations from the 19th century onwards provide distinctive new words, phrases, and idioms through railroading (see further at rail terminology ) and transportation terminology, ranging from types of roads ( dirt roads , freeways ) to infrastructure ( parking lot , overpass , rest area ), to automotive terminology often now standard in English internationally. Already existing English words—such as store , shop , lumber —underwent shifts in meaning; others remained in

4320-546: The 20th century. The use of English in the United States is a result of British colonization of the Americas . The first wave of English-speaking settlers arrived in North America during the early 17th century, followed by further migrations in the 18th and 19th centuries. During the 17th and 18th centuries, dialects from many different regions of England and the British Isles existed in every American colony, allowing

4416-627: The British form is a back-formation , such as AmE burglarize and BrE burgle (from burglar ). However, while individuals usually use one or the other, both forms will be widely understood and mostly used alongside each other within the two systems. While written American English is largely standardized across the country and spoken American English dialects are highly mutually intelligible, there are still several recognizable regional and ethnic accents and lexical distinctions. The regional sounds of present-day American English are reportedly engaged in

4512-413: The East Coast (perhaps in imitation of 19th-century London speech), even the East Coast has gradually begun to restore rhoticity, due to it becoming nationally prestigious in the 20th century. The pronunciation of ⟨r⟩ is a postalveolar approximant [ ɹ̠ ] or retroflex approximant [ ɻ ] , but a unique "bunched tongue" variant of the approximant r sound

4608-598: The Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity , the Fibonacci search technique , and the Pisano period . Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis . American English American English , sometimes called United States English or U.S. English , is the set of varieties of the English language native to

4704-537: The Fibonacci recursion. In other words, φ n = φ n − 1 + φ n − 2 , ψ n = ψ n − 1 + ψ n − 2 . {\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}} It follows that for any values

4800-571: The Inland North. Rather than one particular accent, General American is best defined as an umbrella covering an American accent that does not incorporate features associated with some particular region, ethnicity, or socioeconomic group. Typical General American features include rhoticity , the father–bother merger , Mary–marry–merry merger , pre-nasal "short a " tensing , and other particular vowel sounds . General American features are embraced most by Americans who are highly educated or in

4896-573: The South), sneakers for athletic shoes (but often tennis shoes outside the Northeast), and shopping cart for a cart used for carrying supermarket goods. American English and British English (BrE) often differ at the levels of phonology, phonetics, vocabulary, and, to a much lesser extent, grammar and orthography. The first large American dictionary, An American Dictionary of the English Language , known as Webster's Dictionary ,

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4992-528: The U.S. Several verbs ending in -ize are of U.S. origin; for example, fetishize, prioritize, burglarize, accessorize, weatherize , etc.; and so are some back-formations (locate, fine-tune, curate, donate, emote, upholster and enthuse). Among syntactic constructions that arose are outside of, headed for, meet up with, back of, etc. Americanisms formed by alteration of some existing words include notably pesky, phony, rambunctious, buddy, sundae , skeeter, sashay and kitty-corner. Adjectives that arose in

5088-570: The U.S. are for instance foothill , landslide (in all senses), backdrop , teenager , brainstorm , bandwagon , hitchhike , smalltime, and a huge number of others. Other compound words have been founded based on industrialization and the wave of the automobile: five-passenger car, four-door sedan, two-door sedan, and station-wagon (called an estate car in British English). Some are euphemistic ( human resources , affirmative action , correctional facility ). Many compound nouns have

5184-676: The U.S. are, for example, lengthy, bossy, cute and cutesy, punk (in all senses), sticky (of the weather), through (as in "finished"), and many colloquial forms such as peppy or wacky . A number of words and meanings that originated in Middle English or Early Modern English and that have been in everyday use in the United States have since disappeared in most varieties of British English; some of these have cognates in Lowland Scots . Terms such as fall ("autumn"), faucet ("tap"), diaper ("nappy"; itself unused in

5280-530: The U.S. while changing in Britain. Science, urbanization, and democracy have been important factors in bringing about changes in the written and spoken language of the United States. From the world of business and finance came new terms ( merger , downsize , bottom line ), from sports and gambling terminology came, specific jargon aside, common everyday American idioms, including many idioms related to baseball . The names of some American inventions remained largely confined to North America ( elevator [except in

5376-427: The U.S.), candy ("sweets"), skillet , eyeglasses , and obligate are often regarded as Americanisms. Fall for example came to denote the season in 16th century England, a contraction of Middle English expressions like "fall of the leaf" and "fall of the year." Gotten ( past participle of get ) is often considered to be largely an Americanism. Other words and meanings were brought back to Britain from

5472-597: The U.S., especially in the second half of the 20th century; these include hire ("to employ"), I guess (famously criticized by H. W. Fowler ), baggage , hit (a place), and the adverbs overly and presently ("currently"). Some of these, for example, monkey wrench and wastebasket , originated in 19th century Britain. The adjectives mad meaning "angry", smart meaning "intelligent", and sick meaning "ill" are also more frequent in American (and Irish) English than British English. Linguist Bert Vaux created

5568-538: The United States and the United Kingdom suggest that, while spoken American English deviated away from period British English in many ways, it is conservative in a few other ways, preserving certain features 21st-century British English has since lost. Full rhoticity (or "R-fulness") is typical of American accents, pronouncing the phoneme /r/ (corresponding to the letter ⟨r⟩ ) in all environments, including in syllable-final position or before

5664-524: The United States. English is the most widely spoken language in the United States ; the de facto common language used in government, education, and commerce; and an official language in 32 of the 50 U.S. states . Since the late 20th century, American English has become the most influential form of English worldwide. Varieties of American English include many patterns of pronunciation, vocabulary, grammar and particularly spelling that are unified nationwide but distinct from other English dialects around

5760-605: The West and Midwest, and New York Latino English , spoken in the New York metropolitan area . Additionally, ethnic varieties such as Yeshiva English and " Yinglish " are spoken by some American Orthodox Jews , Cajun Vernacular English by some Cajuns in southern Louisiana , and Pennsylvania Dutch English by some Pennsylvania Dutch people. American Indian Englishes have been documented among diverse Indian tribes. The island state of Hawaii , though primarily English-speaking,

5856-555: The accents spoken in the " Midland ": a vast band of the country that constitutes an intermediate dialect region between the traditional North and South. Western U.S. accents mostly fall under the General American spectrum. Below, ten major American English accents are defined by their particular combinations of certain vowel sounds: In 2010, William Labov noted that Great Lakes, Philadelphia, Pittsburgh, and West Coast accents have undergone "vigorous new sound changes" since

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5952-505: The aeronautical sense ], gasoline ) as did certain automotive terms ( truck , trunk ). New foreign loanwords came with 19th and early 20th century European immigration to the U.S.; notably, from Yiddish ( chutzpah , schmooze, bupkis, glitch ) and German ( hamburger , wiener ). A large number of English colloquialisms from various periods are American in origin; some have lost their American flavor (from OK and cool to nerd and 24/7 ), while others have not ( have

6048-406: The country), though the vowel-consonant cluster found in "bird", "work", "hurt", "learn", etc. usually retains its r pronunciation, even in these non-rhotic American accents. Non-rhoticity among such speakers is presumed to have arisen from their upper classes' close historical contact with England, imitating London's r -dropping, a feature that has continued to gain prestige throughout England from

6144-614: The diverse regional dialects of British English) became common after the mid-18th century, while at the same time speakers' identification with this new variety increased. Since the 18th century, American English has developed into some new varieties, including regional dialects that retain minor influences from waves of immigrant speakers of diverse languages, primarily European languages. Some racial and regional variation in American English reflects these groups' geographic settlement, their de jure or de facto segregation, and patterns in their resettlement. This can be seen, for example, in

6240-742: The double quotation mark ("like this") over the single ('as here'). Vocabulary differences vary by region. For example, autumn is used more commonly in the United Kingdom, whereas fall is more common in American English. Some other differences include: aerial (United Kingdom) vs. antenna, biscuit (United Kingdom) vs. cookie/cracker, car park (United Kingdom) vs. parking lot, caravan (United Kingdom) vs. trailer, city centre (United Kingdom) vs. downtown, flat (United Kingdom) vs. apartment, fringe (United Kingdom) vs. bangs, and holiday (United Kingdom) vs. vacation. AmE sometimes favors words that are morphologically more complex, whereas BrE uses clipped forms, such as AmE transportation and BrE transport or where

6336-517: The following two centuries) when this ethnic group eventually made up one-seventh of the colonial population. Scotch-Irish settlers spread from Delaware and Pennsylvania throughout the larger Mid-Atlantic region, the inland regions of both the South and North, and throughout the West: American dialect areas that were all uninfluenced by upper-class non-rhoticity and that consequently have remained consistently rhotic. While non-rhoticity spread on

6432-630: The golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers , which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences . The Fibonacci numbers may be defined by the recurrence relation F 0 = 0 , F 1 = 1 , {\displaystyle F_{0}=0,\quad F_{1}=1,} and F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} for n > 1 . Under some older definitions,

6528-546: The golden ratio satisfies the equation φ 2 = φ + 1 , {\displaystyle \varphi ^{2}=\varphi +1,} this expression can be used to decompose higher powers φ n {\displaystyle \varphi ^{n}} as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of φ {\displaystyle \varphi } and 1. The resulting recurrence relationships yield Fibonacci numbers as

6624-404: The growing banking industry. The book also 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 later known as Fibonacci numbers . Although Fibonacci's Liber Abaci contains the earliest known description of

6720-944: The hospital , BrE to hospital ; contrast, however, AmE actress Elizabeth Taylor , BrE the actress Elizabeth Taylor ). Often, these differences are a matter of relative preferences rather than absolute rules; and most are not stable since the two varieties are constantly influencing each other, and American English is not a standardized set of dialects. Differences in orthography are also minor. The main differences are that American English usually uses spellings such as flavor for British flavour , fiber for fibre , defense for defence , analyze for analyse , license for licence , catalog for catalogue and traveling for travelling . Noah Webster popularized such spellings in America, but he did not invent most of them. Rather, "he chose already existing options on such grounds as simplicity, analogy or etymology." Other differences are due to

6816-674: The influence of 18th-century Protestant Ulster Scots immigrants (known in the U.S. as the Scotch-Irish ) in Appalachia developing Appalachian English and the 20th-century Great Migration bringing African-American Vernacular English to the Great Lakes urban centers. Any phonologically unmarked North American accent falls under an umbrella known as General American. This section mostly refers to such General American features. Studies on historical usage of English in both

6912-407: The last is the number ... of the next mātrā-vṛtta." The Fibonacci sequence first appears in the book Liber Abaci ( The Book of Calculation , 1202) by Fibonacci where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized ( biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in

7008-438: The late 18th century onwards, but which has conversely lost prestige in the U.S. since at least the early 20th century. Non-rhoticity makes a word like car sound like cah or source like sauce . New York City and Southern accents are the most prominent regional accents of the country, as well as the most stigmatized and socially disfavored. Southern speech, strongest in southern Appalachia and certain areas of Texas,

7104-1129: The linear coefficients : φ n = F n φ + F n − 1 . {\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.} This equation can be proved by induction on n ≥ 1 : φ n + 1 = ( F n φ + F n − 1 ) φ = F n φ 2 + F n − 1 φ = F n ( φ + 1 ) + F n − 1 φ = ( F n + F n − 1 ) φ + F n = F n + 1 φ + F n . {\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.} For ψ = − 1 / φ {\displaystyle \psi =-1/\varphi } , it

7200-752: 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 ( Book of Abacus or The Book of Calculation ), which popularized Hindu–Arabic numerals in Europe. Fibonacci was a guest of Emperor Frederick II , who enjoyed mathematics and science. A member of Frederick II's court, John of Palermo , posed several questions based on Arab mathematical works for Fibonacci to solve. In 1240,

7296-427: The mid-nineteenth century onwards, so they "are now more different from each other than they were 50 or 100 years ago", while other accents, like of New York City and Boston, have remained stable in that same time-frame. However, a General American sound system also has some debated degree of influence nationwide, for example, gradually beginning to oust the regional accent in urban areas of the South and at least some in

7392-617: The most formal contexts, and regional accents with the most General American native features include North Midland, Western New England, and Western accents. Although no longer region-specific, African-American Vernacular English , which remains the native variety of most working- and middle-class African Americans , has a close relationship to Southern dialects and has greatly influenced everyday speech of many Americans, including hip hop culture . Hispanic and Latino Americans have also developed native-speaker varieties of English. The best-studied Latino Englishes are Chicano English , spoken in

7488-523: 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 set 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

7584-505: The number F n is the closest integer to φ n 5 {\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}} . Therefore, it can be found by rounding , using the nearest integer function: F n = ⌊ φ n 5 ⌉ ,   n ≥ 0. {\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.} In fact,

7680-491: The number of pairs alive last month (month n – 1 ). The number in the n -th month is the n -th Fibonacci number. The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas . Like every sequence defined by a homogeneous linear recurrence with constant coefficients , the Fibonacci numbers have a closed-form expression . It has become known as Binet's formula , named after French mathematician Jacques Philippe Marie Binet , though it

7776-402: The past forms of a few verbs (for example, AmE/BrE: learned / learnt , burned / burnt , snuck/sneaked , dove/dived ) although the purportedly "British" forms can occasionally be seen in American English writing as well; different prepositions and adverbs in certain contexts (for example, AmE in school, BrE at school ); and whether or not a definite article is used, in very few cases (AmE to

7872-409: The relation between the sequence and these constants, note that φ and ψ are both solutions of the equation x 2 = x + 1 {\textstyle x^{2}=x+1} and thus x n = x n − 1 + x n − 2 , {\displaystyle x^{n}=x^{n-1}+x^{n-2},} so the powers of φ and ψ satisfy

7968-485: The rounding error quickly becomes very small as n grows, being less than 0.1 for n ≥ 4 , and less than 0.01 for n ≥ 8 . This formula is easily inverted to find an index of a Fibonacci number F : n ( F ) = ⌊ log φ ⁡ 5 F ⌉ ,   F ≥ 1. {\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.} Instead using

8064-427: The same convergence towards the golden ratio. In general, lim n → ∞ F n + m F n = φ m {\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}} , because the ratios between consecutive Fibonacci numbers approaches φ {\displaystyle \varphi } . Since

8160-407: 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" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to

8256-521: The starting values U 0 {\displaystyle U_{0}} and U 1 {\displaystyle U_{1}} , unless U 1 = − U 0 / φ {\displaystyle U_{1}=-U_{0}/\varphi } . This can be verified using Binet's formula . For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows

8352-570: The traditional standard accent of (southern) England, Received Pronunciation (RP), has evolved a trap–bath split . Moreover, American accents preserve /h/ at the start of syllables, while perhaps a majority of the regional dialects of England participate in /h/ dropping , particularly in informal contexts. However, General American is also innovative in a number of its own ways: The process of coining new lexical items started as soon as English-speaking British-American colonists began borrowing names for unfamiliar flora, fauna, and topography from

8448-411: The value F 0 = 0 {\displaystyle F_{0}=0} is omitted, so that the sequence starts with F 1 = F 2 = 1 , {\displaystyle F_{1}=F_{2}=1,} and the recurrence F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}}

8544-402: The variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations]. Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before

8640-417: The verb-and-preposition combination: stopover, lineup, tryout, spin-off, shootout , holdup, hideout, comeback, makeover , and many more. Some prepositional and phrasal verbs are in fact of American origin ( win out, hold up, back up/off/down/out, face up to and many others). Noun endings such as -ee (retiree), -ery (bakery), -ster (gangster) and -cian (beautician) are also particularly productive in

8736-775: The word corn , used in England to refer to wheat (or any cereal), came to denote the maize plant, the most important crop in the U.S. Most Mexican Spanish contributions came after the War of 1812 , with the opening of the West, like ranch (now a common house style ). Due to Mexican culinary influence, many Spanish words are incorporated in general use when talking about certain popular dishes: cilantro (instead of coriander), queso, tacos, quesadillas, enchiladas, tostadas, fajitas, burritos, and guacamole. These words usually lack an English equivalent and are found in popular restaurants. New forms of dwelling created new terms ( lot , waterfront) and types of homes like log cabin , adobe in

8832-609: The world. Any American or Canadian accent perceived as lacking noticeably local, ethnic, or cultural markers is known in linguistics as General American ; it covers a fairly uniform accent continuum native to certain regions of the U.S. but especially associated with broadcast mass media and highly educated speech. However, historical and present linguistic evidence does not support the notion of there being one single mainstream American accent . The sound of American English continues to evolve, with some local accents disappearing, but several larger regional accents having emerged in

8928-657: Was already known by Abraham de Moivre and Daniel Bernoulli : F n = φ n − ψ n φ − ψ = φ n − ψ n 5 , {\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},} where φ = 1 + 5 2 ≈ 1.61803 39887 … {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }

9024-513: Was an advance in making business calculations easier and faster, which assisted the growth of banking and accounting in Europe. The original 1202 manuscript is not known to exist. In a 1228 copy of the manuscript, the first section introduces the numeral system and compares it with others, such as Roman numerals , and methods to convert numbers to it. The second section explains uses in business, for example converting different currencies, and calculating profit and interest, which were important to

9120-628: Was born around 1170 to Guglielmo, an Italian merchant and customs official. Guglielmo directed a trading post in Bugia (Béjaïa) , in modern-day Algeria. Fibonacci travelled with him as a young boy, and it was in Bugia (Algeria) where he was educated 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

9216-469: Was written by Noah Webster in 1828, codifying several of these spellings. Differences in grammar are relatively minor, and do not normally affect mutual intelligibility; these include: typically a lack of differentiation between adjectives and adverbs, employing the equivalent adjectives as adverbs he ran quick / he ran quickly ; different use of some auxiliary verbs ; formal (rather than notional) agreement with collective nouns ; different preferences for

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