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1. Long and short scales – Thus, billion means a million millions, trillion means a million billions, and so on. Short scale Every new term greater than million is one thousand times larger than the previous term, thus, billion means a thousand millions, trillion means a thousand billions, and so on. For whole numbers less than a million the two scales are identical. From a thousand million up the two scales diverge, using the words for different numbers, this can cause misunderstanding. Countries where the scale is currently used include most countries in continental Europe and most French-speaking, Spanish-speaking. The short scale is now used in most English-speaking and Arabic-speaking countries, in Brazil, in former Soviet Union, number names are rendered in the language of the country, but are similar everywhere due to shared etymology. Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, for example the Indian numbering system. After several decades of increasing informal British usage of the scale, in 1974 the government of the UK adopted it. With very few exceptions, the British usage and American usage are now identical, the first recorded use of the terms short scale and long scale was by the French mathematician Geneviève Guitel in 1975. At and above a million the same names are used to refer to numbers differing by a factor of an integer power of 1,000. Each scale has a justification to explain the use of each such differing numerical name. The short-scale logic is based on powers of one thousand, whereas the long-scale logic is based on powers of one million, in both scales, the prefix bi- refers to 2 and tri- refers to 3, etc. However only in the scale do the prefixes beyond one million indicate the actual power or exponent. In the short scale, the prefixes refer to one less than the exponent, the word, million, derives from the Old French, milion, from the earlier Old Italian, milione, an intensification of the Latin word, mille, 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 109. However, it is unknown 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, is used on financial markets, as, unlike the term, billion, it is internationally unambiguous and phonetically distinct from million. Likewise, many long scale use the word billiard for one thousand long scale billions

2. Number – Numbers that answer the question How many. Are 0,1,2,3 and so on, when used to indicate position in a sequence they are ordinal numbers. To the Pythagoreans and Greek mathematician Euclid, the numbers were 2,3,4,5, Euclid did not consider 1 to be a number. Numbers like 3 +17 =227, expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers and these make it possible to measure such quantities as two and a quarter gallons and six and a half miles. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are incommensurable, Euclid included proofs of incommensurability of lengths arising in geometry in his Elements. In the Rhind Mathematical Papyrus, a pair of walking forward marked addition. They were the first known civilization to use negative numbers, negative numbers came into widespread use as a result of their utility in accounting. They were used by late medieval Italian bankers, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. In Maya civilization zero was a numeral with a shape as a symbol. The ancient Egyptians represented all fractions in terms of sums of fractions with numerator 1, for example, 2/5 = 1/3 + 1/15. Such representations are known as Egyptian Fractions or Unit Fractions. The earliest written approximations of π are found in Egypt and Babylon, in Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 =3.1250. In Egypt, the Rhind Papyrus, dated around 1650 BC, astronomical calculations in the Shatapatha Brahmana use a fractional approximation of 339/108 ≈3.139. Other Indian sources by about 150 BC treat π as √10 ≈3.1622 The first references to the constant e were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant and it is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, the first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Eulers Mechanica. While in the subsequent years some researchers used the letter c, e was more common, the first numeral system known is Babylonian numeric system, that has a 60 base, it was introduced in 3100 B. C. and is the first Positional numeral system known

3. Tetration – In mathematics, tetration is the next hyperoperation after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra-, Tetration is used for the notation of very large numbers. The notation n a means a a ⋅ ⋅ a, the application of exponentiation n −1 times. Shown here are the first four hyperoperations, with tetration as the fourth, multiplication a × n = a + a + ⋯ + a ⏟ n n copies of a combined by addition. Exponentiation a n = a × a × ⋯ × a ⏟ n n copies of a combined by multiplication, Tetration n a = a a ⋅ ⋅ a ⏟ n n copies of a combined by exponentiation, right-to-left. The above example is read as the nth tetration of a, each operation is defined by iterating the previous one. Tetration is not an elementary function, analogously, tetration can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, computer programmers refer to this choice as right-associative. When a and 10 are coprime, we can compute the last m decimal digits of n a using Eulers theorem, there are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale, the term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory, has gained dominance. It was also popularized in Rudy Ruckers Infinity and the Mind, the term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. It was used earlier by Ed Nelson in his book Predicative Arithmetic, the term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence, when considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is referring to tetration. The term power tower is used, in the form the power tower of order n for a a ⋅ ⋅ a ⏟ n. This is a misnomer, however, because tetration cannot be expressed with iterated power functions, the term snap is occasionally used in informal contexts, in the form a snap n for a a ⋅ ⋅ a ⏟ n. This term is not yet accepted, although it is used within select communities. It is believed to be a reference to Jounce, the derivative of position in physics

4. 1,000,000 – One million or one thousand thousand is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, thousand and it is commonly abbreviated as m or M, further MM, mm, or mn in financial contexts. In scientific notation, it is written as 1×106 or 106, physical quantities can also be expressed using the SI prefix mega, when dealing with SI units, for example,1 megawatt equals 1,000,000 watts. The meaning of the word million is common to the scale and long scale numbering systems, unlike the larger numbers. Information, Not counting spaces, the text printed on 136 pages of an Encyclopædia Britannica, length, There are one million millimeters in a kilometer, and roughly a million sixteenths of an inch in a mile. A typical car tire might rotate a million times in a 1, 200-mile trip, fingers, If the width of a human finger is 2.2 cm, then a million fingers lined up would cover a distance of 22 km. If a person walks at a speed of 4 km/h, it would take approximately five. A city lot 70 by 100 feet is about a million square inches, volume, The cube root of one million is only one hundred, so a million objects or cubic units is contained in a cube only a hundred objects or linear units on a side. A million grains of salt or granulated sugar occupies only about 64 ml. One million cubic inches would be the volume of a room only 8 1⁄3 feet long by 8 1⁄3 feet wide by 8 1⁄3 feet high. Mass, A million cubic millimeters of water would have a volume of one litre, a million millilitres or cubic centimetres of water has a mass of a million grams or one tonne. Weight, A million 80-milligram honey bees would weigh the same as an 80 kg person, landscape, A pyramidal hill 600 feet wide at the base and 100 feet high would weigh about a million tons. Computer, A display resolution of 1,280 by 800 pixels contains 1,024,000 pixels, money, A USD bill of any denomination weighs 1 gram. There are 454 grams in a pound, one million $1 bills would weigh 2,204.62 pounds, or just over 1 ton. Time, A million seconds is 11.57 days, in Indian English and Pakistani English, it is also expressed as 10 lakh or 10 Lac. Lakh is derived from laksh for 100,000 in Sanskrit

5. Orders of magnitude (numbers) – This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Mathematics – Writing, Approximately 10−183,800 is a rough first estimate of the probability that a monkey, however, taking punctuation, capitalization, and spacing into account, the actual probability is far lower, around 10−360,783. Computing, The number 1×10−6176 is equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value, Computing, The number 6. 5×10−4966 is approximately equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE floating-point value. Computing, The number 3. 6×10−4951 is approximately equal to the smallest positive non-zero value that can be represented by a 80-bit x86 double-extended IEEE floating-point value. Computing, The number 1×10−398 is equal to the smallest positive non-zero value that can be represented by a double-precision IEEE decimal floating-point value, Computing, The number 4. 9×10−324 is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value. Computing, The number 1×10−101 is equal to the smallest positive non-zero value that can be represented by a single-precision IEEE decimal floating-point value, Mathematics, The probability in a game of bridge of all four players getting a complete suit is approximately 4. 47×10−28. ISO, yocto- ISO, zepto- Mathematics, The probability of matching 20 numbers for 20 in a game of keno is approximately 2.83 × 10−19. ISO, atto- Mathematics, The probability of rolling snake eyes 10 times in a row on a pair of dice is about 2. 74×10−16. ISO, micro- Mathematics – Poker, The odds of being dealt a flush in poker are 649,739 to 1 against. Mathematics – Poker, The odds of being dealt a flush in poker are 72,192 to 1 against. Mathematics – Poker, The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4 × 10−4. ISO, milli- Mathematics – Poker, The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10−3. Mathematics – Poker, The odds of being dealt a flush in poker are 507.8 to 1 against, Mathematics – Poker, The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10−3. Physics, α =0.007297352570, the fine-structure constant, ISO, deci- Mathematics – Poker, The odds of being dealt only one pair in poker are about 5 to 2 against, for a probability of 0.42. Demography, The population of Monowi, a village in Nebraska. Mathematics, √2 ≈1.414213562373095489, the ratio of the diagonal of a square to its side length. Mathematics, φ ≈1.618033988749895848, the golden ratio Mathematics, the number system understood by most computers, human scale, There are 10 digits on a pair of human hands, and 10 toes on a pair of human feet. Mathematics, The number system used in life, the decimal system, has 10 digits,0,1,2,3,4,5,6,7,8,9

6. Megaprime – A megaprime is a prime number with at least one million decimal digits. As of September 2016,208 megaprimes are known, including 192 definitely primes and 16 probable primes. The first to be found was the Mersenne prime 26972593−1 with 2,098,960 digits, discovered in 1999 by Nayan Hajratwala, the term bevaprime has been proposed as a term for a prime with at least 1,000,000,000 digits. In fact, almost all primes are megaprimes, as the amount of primes less than a million digits is finite. However, the vast majority of known primes are not megaprimes, entries labelled Prime have been proved prime, those labelled PRP have not. All numbers from 10999999 through 10999999 +593498 are known to be composite, and there is a high probability 10999999 +593499

7. 1,000,000,000 – 1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn, in scientific notation, it is written as 1 ×109. The SI prefix giga indicates 1,000,000,000 times the base unit, one billion years may be called eon in astronomy and geology. Previously in British English, the word billion referred exclusively to a million millions, however, this is no longer as common as earlier, and the word has been used to mean one thousand million for some time. The alternative term one thousand million is used in the U. K. or countries such as Spain that uses one thousand million as one million million constitutes a billion. The worded figure, as opposed to the figure is used to differentiate between one thousand million or one billion. The term milliard can also be used to refer to 1,000,000,000, whereas milliard is seldom used in English, in the South Asian numbering system, it is known as 100 crore or 1 Arab. 1000000007 – smallest prime number with 10 digits,1023456789 – smallest pandigital number in base 10. 1026753849 – smallest pandigital square that includes 0,1073741824 –2301073807359 – 14th Kynea number. 1162261467 –3191220703125 –513 1232922769- 35113^2 Centered hexagonal number,1234567890 – pandigital number with the digits in order. 1882341361 – The least prime whose reversal is both square and triangular,1977326743 –7112147483647 – 8th Mersenne prime and the largest signed 32-bit integer. 2147483648 –2312176782336 –6122214502422 – 6th primary pseudoperfect number,2357947691 –1192971215073 – 11th Fibonacci prime. 3405691582 – hexadecimal CAFEBABE, used as a placeholder in programming,3405697037 – hexadecimal CAFED00D, used as a placeholder in programming. 3735928559 – hexadecimal DEADBEEF, used as a placeholder in programming,3486784401 –3204294836223 – 16th Carol number. 4294967291 – Largest prime 32-bit unsigned integer,4294967295 – Maximum 32-bit unsigned integer, perfect totient number, product of the five prime Fermat numbers. 4294967296 –2324294967297 – the first composite Fermat number,6103515625 –5146210001000 – only self-descriptive number in base 10. 6975757441 –1786983776800 – 15th colossally abundant number, 15th superior highly composite number 7645370045 – 27th Pell number,8589934592 –2339043402501 – 25th Motzkin number. 9814072356 – largest square pandigital number, largest pandigital pure power,9876543210 – largest number without redundant digits

8. Graham's number – Grahams number is an unimaginably large number that is a proven upper bound to the solution of a certain problem in Ramsey theory. Gardner later described the number in Scientific American in 1977, introducing it to the general public, the number was published in the 1980 Guinness Book of World Records which added to the popular interest in the number. Grahams number, although smaller than TREE, is larger than many other large numbers such as Skewes number and Mosers number. Even power towers of the form a b c ⋅ ⋅ ⋅ are insufficient for purpose, although it can be described by recursive formulas using Knuths up-arrow notation or equivalent. Though too large to be computed in full, many of the last digits of Grahams number can be derived through simple algorithms, the last 12 digits are,262464195387. Grahams number is connected to the problem in Ramsey theory. Colour each of the edges of this graph either red or blue, what is the smallest value of n for which every such colouring contains at least one single-coloured complete subgraph on four coplanar vertices. This was reduced in 2014 via upper bounds on the Hales–Jewett number to N ′ =2 ↑↑↑6, the lower bound of 6 was later improved to 11 by Geoff Exoo in 2003, and to 13 by Jerome Barkley in 2008. Thus, the best known bounds for N* are 13 ≤ N* ≤ N. Grahams number, G, is larger than N, f 64. This weaker upper bound for the problem, attributed to a work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977. The 1980 Guinness Book of World Records repeated Gardners claim, adding to the popular interest in this number, according to physicist John Baez, Graham invented the quantity now known as Grahams number in conversation with Gardner. Because the number which Graham described to Gardner is larger than the number in the paper itself, both are valid upper bounds for the solution to the problem studied by Graham and Rothschild. Equivalently, G = f 64, where f =3 ↑ n 3, and the superscript on f indicates an iteration of the function, e. g. f 4 = f. Expressed in terms of the family of hyperoperations H0, H1, H2, ⋯, the f is the particular sequence f = H n +2. First, in terms of tetration alone, g 1 =3 ↑↑↑↑3 =3 ↑↑↑ =3 ↑↑ where the number of 3s in the expression on the right is 3 ↑↑↑3 =3 ↑↑. Note that the result of calculating the third tower is the value of n, the magnitude of this first term, g1, is so large that it is practically incomprehensible, even though the above display is relatively easy to comprehend. Even n, the number of towers in this formula for g1, is far greater than the number of Planck volumes into which one can imagine subdividing the observable universe. And after this first term, still another 63 terms remain in the rapidly growing g sequence before Grahams number G = g64 is reached

9. Power of two – In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two

10. Largest known prime number – As of January 2017, the largest known prime number is 274,207,281 −1, a number with 22,338,618 digits. It was found in 2016 by the Great Internet Mersenne Prime Search, euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers. Many of the largest known primes are Mersenne primes, as of January 2017, the six largest known primes are Mersenne primes, while the seventh is the largest known non-Mersenne prime. The last 16 record primes were Mersenne primes, the fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers. The record is held by 274,207,281 −1 with 22,338,618 digits, found by GIMPS in 2015. 717774014762912462113646879425801445107393100212927181629335931494239018213879217671164956287190498687010073391086436351 The first and last 120 digits are shown above, there are several prizes offered by the Electronic Frontier Foundation for record primes. GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and larger, the record passed one million digits in 1999, earning a $50,000 prize. In 2008 the record passed ten million digits, earning a $100,000 prize, time called it the 29th top invention of 2008. Additional prizes are being offered for the first prime number found with at least one hundred million digits, both the $50,000 and the $100,000 prizes were won by participation in GIMPS. The following table lists the progression of the largest known prime number in ascending order, here Mn= 2n −1 is the Mersenne number with exponent n. The longest record-holder known was M19 =524,287, which was the largest known prime for 144 years, almost no records are known before 1456. GIMPS found the thirteen latest records on ordinary computers operated by participants around the world