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37-493: The "billion" in bya is the 10 "billion" of the short scale of the U.S. , not the long-scale 10 "billion" of some European usage. Billion by this convention (10) is often called a "thousand million" in the UK and a "milliard" in some other countries. For this reason, there is potential for some confusion, and some scientists prefer the unit Gya, while others prefer Ga (Giga-annum), however, bya remains in more widespread use. In 1974,

74-445: A regular number which are most useful in the context of powers of 60 , the smallest number with twelve divisors: 1728 is also an untouchable number since there is no number whose sum of proper divisors is 1728. Many relevant calculations involving 1728 are computed in the duodecimal number system, in-which it is represented as "1000". 1728 occurs in the algebraic formula for the j -invariant of an elliptic curve , as

111-756: A different spelling to the Brazilian Portuguese variant, but in Brazil referring to short scale) all refer to 10 , being long-scale terms. Therefore, each of these words translates to the American English or post-1974 British English word: trillion (10 in the short scale), and not billion (10 in the short scale). On the other hand, the pre-1961 former French word billion , pre-1994 former Italian word bilione , Brazilian Portuguese word bilhão , and Welsh word biliwn all refer to 10 , being short scale terms. Each of these words translates to

148-498: A function over a complex variable on the upper half-plane H : { τ ∈ C ,   I m ( τ ) > 0 } {\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}} , Inputting a value of 2 i {\displaystyle 2i} for τ {\displaystyle \tau } , where i {\displaystyle i}

185-472: A long scale n -illion equals 10 . In some languages, the long scale uses additional names for the intermediate multipliers, replacing the ending -ion with -iard ; for example, the next multiplier after million is milliard (10 ); after a billion it is billiard (10 ). Hence, a long scale n -iard equals 10 . This table displays the naming conventions in the two English numbering systems, with conflicting terms shown in bold. The relationship between

222-527: A million is worth a thousand thousand units, and a byllion is worth a thousand thousand millions, and tryllion is worth a thousand thousand byllions, and a quadrillion is worth a thousand thousand tryllions, and so on for the others. And an example of this follows, a number divided up and punctuated as previously described, the whole number being 745324 tryllions, 804300 byllions 700023 millions 654321. Example: 745324'8043000'700023'654321 ... [ sic ] The extract from Chuquet's manuscript,

259-567: A million millions .. milliart/ofte duysent millioenen.. Translation : ..milliart / also thousand millions.. It should be remembered that "billion" does not mean in American use (which follows the French) what it means in British. For to us it means the second power of a million, i.e. a million millions ( 1,000,000,000,000 ); for Americans it means a thousand multiplied by itself twice, or

296-482: A thousand millions ( 1,000,000,000 ), what we call a milliard. Since billion in our sense is useless except to astronomers, it is a pity that we do not conform. Although American English usage did not change, within the next 50 years French usage changed from short scale to long and British English usage changed from long scale to short. Mr. Maxwell-Hyslop asked the Prime Minister whether he would make it

333-543: A thousand. That is, a million is a big thousand , much as a great gross is a dozen gross or 12 × 144 = 1728 . The word milliard , or its translation, is found in many European languages and is used in those languages for 10 . However, it is not found in American English, which uses billion , and not used in British English, which preferred to use thousand million before the current usage of billion . The financial term yard , which derives from milliard ,

370-427: Is also a Jordan–Pólya number such that it is a product of factorials : 2 ! × ( 3 ! ) 2 × 4 ! = 1728 {\displaystyle 2!\times (3!)^{2}\times 4!=1728} . 1728 has twenty-eight divisors , which is a perfect count (as with 12, with six divisors). It also has a Euler totient of 576 or 24 , which divides 1728 thrice over. 1728

407-456: Is an abundant and semiperfect number, as it is smaller than the sum of its proper divisors yet equal to the sum of a subset of its proper divisors. It is a practical number as each smaller number is the sum of distinct divisors of 1728, and an integer-perfect number where its divisors can be partitioned into two disjoint sets with equal sum. 1728 is 3-smooth , since its only distinct prime factors are 2 and 3. This also makes 1728

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444-406: Is now used internationally to mean 1,000 million and it would be confusing if British Ministers were to use it in any other sense. I accept that it could still be interpreted in this country as 1 million million and I shall ask my colleagues to ensure that, if they do use it, there should be no ambiguity as to its meaning. The BBC and other UK mass media quickly followed the government's lead within

481-455: Is the imaginary number , yields another cubic integer : In moonshine theory , the first few terms in the Fourier q -expansion of the normalized j -invariant exapand as, The Griess algebra (which contains the friendly giant as its automorphism group ) and all subsequent graded parts of its infinite-dimensional moonshine module hold dimensional representations whose values are

518-410: Is used on financial markets, as, unlike the term billion , it is internationally unambiguous and phonetically distinct from million . Likewise, many long scale countries use the word billiard (or similar) for one thousand long scale billions (i.e., 10 ), and the word trilliard (or similar) for one thousand long scale trillions (i.e., 10 ), etc. Although this situation has been developing since

555-413: The forint was introduced at a rate of 1 forint = 400 quadrilliard (long scale) pengő = 4 × 10 pengő = 400 octillion (short scale) pengő. Great gross 1728 is the natural number following 1727 and preceding 1729 . It is a dozen gross , or one great gross (or grand gross ). It is also the number of cubic inches in a cubic foot . 1728 is the cube of 12 , and therefore equal to

592-575: The 1200s, the first recorded use of the terms short scale ( French : échelle courte ) and long scale ( French : échelle longue ) was by the French mathematician Geneviève Guitel in 1975. The short scale was never widespread before its universal adoption in the United States. It has been taught in American schools since the early 1800s. It has since become common in other English-speaking nations and several other countries. For most of

629-898: The 13th is worth a bymillion, the 14th is worth ten bymillions, the 15th is worth one [hundred] bymillions, the 16th is worth one thousand bymillions, the 17th is worth ten thousand bymillions, the 18th is worth hundred thousand bymillions, the 19th is worth a trimillion, the 20th is worth ten trimillions ... ...[preder s'] Item l'on doit savoir que ung million vault mille milliers de unitez, et ung byllion vault mille milliers de millions, et [ung] tryllion vault mille milliers de byllions, et ung quadrillion vault mille milliers de tryllions et ainsi des aultres : Et de ce en est pose ung exemple nombre divise et punctoye ainsi que devant est dit, tout lequel nombre monte 745324 tryllions 804300 byllions 700023 millions 654321. Exemple : 745324'8043000'700023'654321 ... [ sic ] Translation : ...likewise, one should know that

666-660: The 19th and 20th centuries, the United Kingdom largely used the long scale, whereas the United States used the short scale, so that the two systems were often referred to as British and American in the English language . After several decades of increasing informal British usage of the short scale, in 1974 the government of the UK adopted it, and it is used for all official purposes. The British usage and American usage are now identical. The existence of

703-937: The American English or post-1974 British English word billion (10 in the short scale). The term billion originally meant 10 when introduced. Coueyte not his goodes For millions of moneye Translation : Covet not his goods for millions of money ... item noctes que le premier greton dembas vault ung, le second vault dix, le trois vault cent, le quart vult [ sic ] mille, le Ve vault dix M, le VIe vault cent M, le VIIe vault Milion, Le VIIIe vault dix Million, Le IXe vault cent Millions, Le Xe vault Mil Millions, Le XIe vault dix mil Millions, Le XIIe vault Cent mil Millions, Le XIIIe vault bymillion, Le XIIIIe vault dix bymillions, Le XVe vault cent mil [ sic ] bymillions, Le XVIe vault mil bymillions, Le XVIIe vault dix Mil bymillions, Le XVIIIe vault cent mil bymillions, Le XIXe vault trimillion, Le XXe vault dix trimillions ... Translation : ... Likewise, note that

740-471: The Fourier coefficients in this q -expansion. The number of directed open knight's tours in 5 × 5 {\displaystyle 5\times 5} minichess is 1728. 1728 is one less than the first taxicab or Hardy–Ramanujan number 1729 , which is the smallest number that can be expressed as sums of two positive cubes in two ways. Regarding strings of digits of 1728, 1728

777-623: The German populace. The Mark or Papiermark was replaced at the end of 1923 by the Rentenmark at an exchange rate of 1 Rentenmark = 1 billion (long scale) Papiermark = 10 Papiermark = 1 trillion (short scale) Papiermark Hyperinflation in Hungary in 1946 led to the introduction of the 10 pengő banknote. 100 million b-pengő (long scale) = 100 trillion (long scale) pengő = 10 pengő = 100 quintillion (short scale) pengő. On 1 August 1946,

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814-581: The UK switched from the long scale to the short scale. Related units are mya ("million years ago"), and byr ("billion years"). These are traditionally written in lowercase . Ga or Gya has a capitalized first letter instead. This standards - or measurement -related article is a stub . You can help Misplaced Pages by expanding it . This time -related article is a stub . You can help Misplaced Pages by expanding it . Long and short scales The long and short scales are two of several naming systems for integer powers of ten which use some of

851-489: The UK. During the last quarter of the 20th century, most other English-speaking countries (Ireland, Australia, New Zealand, South Africa, Zimbabwe, etc.) either also followed this lead or independently switched to the short scale use. However, in most of these countries, some limited long scale use persists and the official status of the short scale use is not clear. As large numbers in natural sciences are usually represented by metric prefixes, scientific notation or otherwise,

888-488: The United Kingdom officially uses the short scale, but France and Italy use the long scale. The pre-1974 former British English word billion , post-1961 current French word billion , post-1994 current Italian word bilione , Spanish billón , German Billion , Dutch biljoen , Danish billion , Swedish biljon , Finnish biljoona , Slovenian bilijon , Polish bilion , and European Portuguese word bilião (with

925-414: The base-10 exponent of three, i.e. for each integer n in the sequence of multipliers 10 . Both systems use the same names for certain multipliers, including those for all numbers smaller than 10 . The differences arise from the assignment of identical names to specific values of n , for numbers starting with 10 , for which n =3. In the short scale system, the identical names are for n =3, 4, 5, ..., while

962-458: The different scales means that care must be taken when comparing large numbers between languages or countries, or when interpreting old documents in countries where the dominant scale has changed over time. For example, British English, French, and Italian historical documents can refer to either the short or long scale, depending on the date of the document, since each of the three countries has used both systems at various times in its history. Today,

999-428: The first counter from the bottom is worth one, the 2nd is worth ten, the 3rd is worth one hundred, the 4th is worth one thousand, the 5th is worth ten thousand, the 6th is worth one hundred thousand, the 7th is worth a million, the 8th is worth ten millions, the 9th is worth one hundred millions, the 10th is worth one thousand millions, the 11th is worth ten thousand millions, the 12th is worth one hundred thousand million,

1036-427: The long scale places them at n =4, 6, 8, etc. In the short scale, a billion (in many countries called a "milliard") means a thousand millions ( 1,000,000,000 , which is 10 ), a trillion means one thousand billions (10 ), and so on. Thus, a short scale n -illion equals 10 . In the long scale, a billion means one million millions (10 ) and a trillion means one million billions (10 ), and so on. Therefore,

1073-432: The most commonplace occurrence of large numbers represented by long or short scale terms is in finance. The following table includes some historic examples related to hyper-inflation and other financial incidents. German hyperinflation in the 1920s Weimar Republic caused 'Eintausend Mark' (1000 Mark = 10 Mark) German banknotes to be over-stamped as 'Eine Milliarde Mark' (10 Mark). This introduced large-number names to

1110-523: The numeric values and the corresponding names in the two scales can be described as: The relationship between the names and the corresponding numeric values in the two scales can be described as: The root mil in million does not refer to the numeral, 1 . The word, million , derives from the Old French, milion , from the earlier Old Italian, milione , an intensification of the Latin word, mille ,

1147-416: The practice of his administration that when Ministers employ the word 'billion' in any official speeches, documents, or answers to Parliamentary Questions , they will, to avoid confusion, only do so in its British meaning of 1 million million and not in the sense in which it is used in the United States of America, which uses the term 'billion' to mean 1,000 million. The Prime Minister: No. The word 'billion'

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1184-928: The product of the six divisors of 12 ( 1 , 2 , 3 , 4 , 6 , 12). It is also the product of the first four composite numbers (4, 6, 8 , and 9 ), which makes it a compositorial . As a cubic perfect power , it is also a highly powerful number that has a record value ( 18 ) between the product of the exponents (3 and 6) in its prime factorization. 1728 = 3 3 × 4 3 = 2 3 × 6 3 = 12 3 1728 = 6 3 + 8 3 + 10 3 1728 = 24 2 + 24 2 + 24 2 {\displaystyle {\begin{aligned}1728&=3^{3}\times 4^{3}=2^{3}\times 6^{3}={\mathbf {12^{3}}}\\1728&=6^{3}+8^{3}+10^{3}\\1728&=24^{2}+24^{2}+24^{2}\\\end{aligned}}} It

1221-591: The same terms for different magnitudes . Some languages, particularly in East Asia and South Asia , have large number naming systems that are different from both the long and short scales, such as the Indian numbering system and the Chinese , Japanese , or Korean numerals . Much of the remainder of the world adopted either the short scale or the long scale for everyday counting powers of ten. Countries with

1258-441: The transcription and translation provided here all contain an original mistake: one too many zeros in the 804300 portion of the fully written out example: 745324'804300 0  '700023'654321 ... .. hoc est decem myriadum myriadas:quod vno verbo nostrates abaci studiosi Milliartum appellant:quasi millionum millionem Translation : .. this is ten myriad myriads, which in one word our students of numbers call Milliart, as if

1295-513: The two systems differ. For identical names, the long scale proceeds by powers of one million, whereas the short scale proceeds by powers of one thousand. For example, the short scale "one billion" (in many languages other than English called "one milliard", even on the short scale) means one thousand million (1,000,000,000), whereas in the long scale, "one billion" always means one million million (1,000,000,000,000). The long scale system employs additional terms for interleaved values, typically replacing

1332-474: The usage of the long scale include most countries in continental Europe and most that are French-speaking , German-speaking and Spanish-speaking . Usage of the short scale is found in Brazil and in most English-speaking and Arabic -speaking countries. For whole numbers smaller than 1,000,000,000 (10 ), such as one thousand or one million, the two scales are identical. For larger numbers, starting with 10 ,

1369-544: The word ending "-ion" by "-iard". For numbers from 10 and up, all the names in the short scale have different meanings than they have in the long scale. To avoid confusion resulting from the coexistence of the two scales, the International System of Units (SI) recommends using the metric prefix to indicate orders of magnitude, associated with physical quantities . In both short and long scale naming, names are given each multiplication step for increments of

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