 Show
I can understand why many of us have a hard time accepting the fact that the value of zero factorial is equal to one. It comes across as an absurd statement that there’s no way it can be true. We have a common perception of zero for being notorious because there’s something about it that can make any number associated with it either vanish or misbehave. For instance, a large number such as 1,000 multiplied by zero becomes zero. It disappears! On the other hand, a nice number such as 5 divided by zero becomes undefined. It misbehaves. So it is okay to be skeptical about why zero “suddenly” becomes one, a nice number, after treating it with some special operation. There are other ways to show why the statement is true. For this one, we will use the definition of factorial itself. To be honest, with this method the justification is simple and requires little math. Simple “Proof” Why Zero Factorial is Equal to OneLet n be a whole number, where n! is defined as the product of all whole numbers less than n and including n itself. What it means is that you first start writing the whole number n then count down until you reach the whole number 1. The general formula of factorial can be written in fully expanded form as or in partially expanded form as
We know with absolute certainty that 1!=1, where n = 1. If we substitute that value of n into the second formula which is the partially expanded form of n!, we obtain the following:
For the equation to be true, we must force the value of zero factorial to equal 1, and no other. Otherwise, 1!≠1 which is a contradiction. So yes, 0! = 1 is correct because mathematicians agreed to define it that way (nothing more and nothing less) in order to be consistent with the rest of mathematics. You might also be interested in: Factorial Notation, Formula, and Basic Examples
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Cardinal0, zero, "oh" (/oʊ/), nought, naught, nilOrdinalZeroth, noughth, 0thBinary02Ternary03Senary06Octal08Duodecimal012Hexadecimal016Arabic, Kurdish, Persian, Sindhi, Urdu٠Hindu Numerals०Chinese零, 〇Khmer០Thai๐Bangla০0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usually by 10. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures.
Common names for the number 0 in English are zero, nought, naught (/nɔːt/), nil. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is explicitly pronounced as oh or o (/oʊ/). Informal or slang terms for 0 include zilch and zip. Historically, ought, aught (/ɔːt/), and cipher, have also been used.
Etymology
The word zero came into the English language via French zéro from the Italian zero, a contraction of the Venetian zevero form of Italian zefiro via ṣafira or ṣifr.[1] In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty".[2] Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.[2] The first known English use of zero was in 1598.[3]
The Italian mathematician Fibonacci (c. 1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.[4]
Modern usage
Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" and "none" are often used. Sometimes, the word "nought" or "naught" is used.
It is often called "oh" in the context of reading out a string of digits, such as telephone numbers, street addresses, credit card numbers, military time, or years (e.g. the area code 201 would be pronounced "two oh one"; a year such as 1907 is often pronounced "nineteen oh seven"). The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (e.g. Canadian postal codes) may exclude the use of the letter O.
Slang words for zero include "zip", "zilch", "nada", and "scratch".[5]
"Nil" is used for many sports in British English. Several sports have specific words for a score of zero, such as "love" in tennis – from French l'oeuf, "the egg" – and "duck" in cricket, a shortening of "duck's egg"; "goose egg" is another general slang term used for zero.[5]
History
Ancient Near East
| nfr |
heart with trachea beautiful, pleasant, good |
|
|---|
Ancient Egyptian numerals were of base 10.[6] They used hieroglyphs for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line.[7]
By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. In a tablet unearthed at Kish (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system.[8] By 300 BC, a punctuation symbol (two slanted wedges) was co-opted to serve as this placeholder.[9][10]
The Babylonian placeholder was not a true zero because it was not used alone, nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60) looked the same, because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.[citation needed]
Pre-Columbian Americas
The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system. Many different glyphs, including the partial quatrefoil were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.[a]
Since the eight earliest Long Count dates appear outside the Maya homeland,[11] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.[12] Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.
Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influenced Old World numeral systems.
Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.
Classical antiquity
The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it.[13] According to mathematician Charles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in astronomy after 500 BC, representing it with the lowercase Greek letter ό (όμικρον) or omicron.[14] However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into Greek numerals.[14] Greeks seemed to have a philosophical opposition to using zero as a number.[14] Other scholars give the Greek partial adoption of the Babylonian zero a later date, with the scientist Andreas Nieder giving a date of after 400 BC and the mathematician Robert Kaplan dating it after the conquests of Alexander.[15][16]
Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.[17]
By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (—°)[18][19] in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.[20] This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.[21] Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system,[b] while the combination meant a zero angle. Minutes of immersion was also a continuous function 1/12 31′20″ √d(24−d) (a triangular pulse with convex sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.[22] Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none.
The earliest use of zero in the calculation of the Julian Easter occurred before AD 311, at the first entry in a table of epacts as preserved in an Ethiopic document for the years AD 311 to 369, using a Ge'ez word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek.[23] This use was repeated in AD 525 in an equivalent table, that was translated via the Latin nulla or "none" by Dionysius Exiguus, alongside Roman numerals.[24] When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725.[25]
China
The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese counting rods system enabled one to perform decimal calculations. As noted in Xiahou Yang's Suanjing (425–468 AD) that states that to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places,[27] According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero".[26] The counting rod system is considered a positional notation system.[28]
In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇"; originally meaning 'star', it subsequently[when?] came to represent zero.
Zero was not treated as a number at that time, but as a "vacant position".[29] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[30] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.[31]
India
Pingala (c. 3rd/2nd century BC[32]), a Sanskrit prosody scholar,[33] used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[34] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[32]
The concept of zero as a written digit in the decimal place value notation was developed in India, presumably as early as during the Gupta period (c. 5th century), with the oldest unambiguous evidence dating to the 7th century.[35]
A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[36] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[37][38][39]
The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimal place-value system, including a zero. In this text, śūnya ("void, empty") is also used to refer to zero.[40]
The Aryabhatiya (c. 500), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".[41][42][43]
Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:[44][45]
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Epigraphy
A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–383.[46]
There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[8]
A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.[47]
The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated 876.[48][49]
Middle Ages
Transmission to Islamic culture
The Arabic-language inheritance of science was largely Greek,[50] followed by Hindu influences.[51] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.
In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[51] and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.[52] This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started to acquire a meaning of any arithmetic based on decimals.[51]
Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.[53]
Transmission to Europe
The Hindu–Arabic numeral system (base 10) reached Western Europe in the 11th century, via Al-Andalus, through Spanish Muslims, the Moors, together with knowledge of classical astronomy and instruments like the astrolabe; Gerbert of Aurillac is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.[54][55][56]
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.
Mathematics
0 is the integer immediately preceding 1. Zero is an even number[57] because it is divisible by 2 with no remainder. 0 is neither positive nor negative,[58] or both positive and negative.[59] Many definitions[60] include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.
As a value or a number, zero is not the same as the digit zero, used in numeral systems with positional notation. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system (e.g., the number 02). In some instances, a leading zero may be used to distinguish a number.
Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).
The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (as 0 must always be one of the factors).[61] Zero is, however, even (i.e. a multiple of 2, as well as being a multiple of any other integer, rational, or real number).
The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.
- Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
- Subtraction: x − 0 = x and 0 − x = −x.
- Multiplication: x · 0 = 0 · x = 0.
- Division: 0/x = 0, for nonzero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
- Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0.
The expression 0/0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)/g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)/g(x), if it exists, must be found by another method, such as l'Hôpital's rule.
The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.
Other branches of mathematics
- In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
- Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set.
- In propositional logic, 0 may be used to denote the truth value false.
- In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
- In lattice theory, 0 may denote the bottom element of a bounded lattice.
- In category theory, 0 is sometimes used to denote an initial object of a category.
- In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions.
Related mathematical terms
- A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. This is related to zeros of a holomorphic function.
- The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
- Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
Physics
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolute temperature (as measured in kelvins), zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.
Chemistry
Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on its nucleus.
As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.
Computer science
The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to n−1. This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first.[citation needed]
There can be confusion between 0- and 1-based indexing; for example, Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.[62]
In databases, it is possible for a field not to have a value. It is then said to have a null value.[63] For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result.[64]
A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).[citation needed]
In mathematics, −0 and +0 is equivalent to 0; both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement binary form used in most modern computers), and most floating-point number representations (such as IEEE 754 and IBM S/390 floating-point formats).
In binary, 0 represents the value for "off", which means no electricity flow.[65]
Zero is the value of false in many programming languages.
The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.[66][67][68]
The Classic Mac OS epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.[69]
Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.
Programmers often use a slashed zero to avoid confusion with the letter "O".[70]
Other fields
- In comparative zoology and cognitive science, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the evolution of species.[71]
- In telephony, pressing 0 is often used for dialling out of a company network or to a different city or region, and 00 is used for dialling abroad. In some countries, dialling 0 places a call for operator assistance.
- DVDs that can be played in any region are sometimes referred to as being "region 0"
- Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
- In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship.
- On the U.S. Interstate Highway System, in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.
Symbols and representations
The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.[72] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[72]
A slashed zero (
0
/
{\displaystyle 0\!\!\!{/}}
Year label
In the BC calendar era, the year 1 BC is the first year before AD 1; there is not a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so forth.[73]
See also
- Brahmagupta
- Aryabhata
- Division by zero
- Grammatical number
- Gwalior Fort
- Mathematical constant
- Number theory
- Peano axioms
- Signed zero
Notes
- ^ No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
- ^ Each place in Ptolemy's sexagesimal system was written in Greek numerals from 0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20.
References
- ^ See:
- Douglas Harper (2011), Zero Archived 3 July 2017 at the Wayback Machine, Etymology Dictionary, Quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught";
- Menninger, Karl (1992). Number words and number symbols: a cultural history of numbers. Courier Dover Publications. pp. 399–404. ISBN 978-0-486-27096-8. Archived from the original on 25 April 2016. Retrieved 5 January 2016.;
- "zero, n." OED Online. Oxford University Press. December 2011. Archived from the original on 7 March 2012. Retrieved 4 March 2012. French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr
- ^ a b See:
- Smithsonian Institution, Oriental Elements of Culture in the Occident, p. 518, at Google Books, Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives, Quote="Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya.";
- Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., ISBN 978-0-393-04002-9, p. 26, Quote = Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.;
- Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, ISBN 978-981-4295-92-5, p. 38, Quote = The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.
- ^ Zero Archived 6 December 2017 at the Wayback Machine, Merriam Webster online Dictionary
- ^ Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 978-0-471-39340-5.
- ^ a b 'Aught' synonyms Archived 23 August 2014 at the Wayback Machine, Thesaurus.com – Retrieved April 2013.
- ^ "Egyptian numerals". mathshistory.st-andrews.ac.uk. Archived from the original on 15 November 2019. Retrieved 21 December 2019.
- ^ Joseph, George Gheverghese (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third ed.). Princeton UP. p. 86. ISBN 978-0-691-13526-7.
- ^ a b Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
- ^ "Zero". Maths History. Archived from the original on 21 September 2021. Retrieved 7 September 2021.
- ^ "Babylonian mathematics: View as single page". www.open.edu. Archived from the original on 7 September 2021. Retrieved 7 September 2021.
- ^ Diehl, p. 186
- ^ Mortaigne, Véronique (28 November 2014). "The golden age of Mayan civilisation – exhibition review". The Guardian. Archived from the original on 28 November 2014. Retrieved 10 October 2015.
- ^ Wallin, Nils-Bertil (19 November 2002). "The History of Zero". YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from the original on 25 August 2016. Retrieved 1 September 2016.
- ^ a b c Seife, Charles (1 September 2000). Zero: The Biography of a Dangerous Idea. Penguin. p. 39. ISBN 978-0-14-029647-1. OCLC 1005913932. Archived from the original on 30 April 2022. Retrieved 30 April 2022.
- ^ Nieder, Andreas (19 November 2019). A Brain for Numbers: The Biology of the Number Instinct. MIT Press. p. 286. ISBN 978-0-262-35432-5. Archived from the original on 30 April 2022. Retrieved 30 April 2022.
- ^ Kaplan, Robert (28 October 1999). The Nothing that Is: A Natural History of Zero. Oxford University Press. p. 17. ISBN 978-0-19-802945-8. OCLC 466431211. Archived from the original on 24 June 2014. Retrieved 30 April 2022.
- ^ Huggett, Nick (2019). "Zeno's Paradoxes". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2019 ed.). Metaphysics Research Lab, Stanford University. Archived from the original on 10 January 2021. Retrieved 9 August 2020.
- ^ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity. Acta Historica Scientiarum Naturalium et Medicinalium. Vol. 9 (2 ed.). Dover Publications. pp. 13–14, plate 2. ISBN 978-0-486-22332-2. PMID 14884919.
- ^ Mercier, Raymond. "Consideration of the Greek symbol 'zero'" (PDF). Home of Kairos. Archived (PDF) from the original on 5 November 2020. Retrieved 28 March 2020.
- ^ Ptolemy (1998) [1984, c.150]. Ptolemy's Almagest. Translated by Toomer, G. J. Princeton University Press. pp. 306–307. ISBN 0-691-00260-6.
- ^ O'Connor, J J; Robertson, E F. "A history of Zero". MacTutor History of Mathematics. Archived from the original on 7 April 2020. Retrieved 28 March 2020.
- ^ Pedersen, Olaf (2010) [1974]. A Survey of the Almagest. Springer. pp. 232–235. ISBN 978-0-387-84825-9.
- ^ Neugebauer, Otto (2016) [1979]. Ethiopic Astronomy and Computus (Red Sea Press ed.). Red Sea Press. pp. 25, 53, 93, 183, Plate I. ISBN 978-1-56902-440-9.. The pages in this edition have numbers six less than the same pages in the original edition.
- ^ Deckers, Michael (2003) [525]. "Cyclus Decemnovennalis Dionysii – Nineteen Year Cycle of Dionysius". Archived from the original on 15 January 2019.
- ^ C. W. Jones, ed., Opera Didascalica, vol. 123C in Corpus Christianorum, Series Latina.
- ^ a b Hodgkin, Luke (2005). A History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia to Modernity. Oxford University Press. p. 85. ISBN 978-0-19-152383-0.
- ^ O'Connor, J.J. (January 2004). "Chinese numerals". Mac Tutor. School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 14 June 2020. Retrieved 14 June 2020.
- ^ Crossley, Lun. 1999, p. 12 "the ancient Chinese system is a place notation system"
- ^ Kang-Shen Shen; John N. Crossley; Anthony W.C. Lun; Hui Liu (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford UP. p. 35. ISBN 978-0-19-853936-0. zero was regarded as a number in India ... whereas the Chinese employed a vacant position
- ^ "Mathematics in the Near and Far East" (PDF). grmath4.phpnet.us. p. 262. Archived (PDF) from the original on 10 August 2018. Retrieved 7 June 2012.
- ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
- ^ a b Plofker, Kim (2009). Mathematics in India. Princeton University Press. pp. 54–56. ISBN 978-0-691-12067-6. In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero
- ^ Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8. Archived from the original on 22 April 2017. Retrieved 21 April 2017.
- ^ "Math for Poets and Drummers" (PDF). people.sju.edu. Archived from the original (PDF) on 22 January 2019. Retrieved 20 December 2015.
- ^ Bourbaki, Nicolas Elements of the History of Mathematics (1998), p. 46. Britannica Concise Encyclopedia (2007), entry "Algebra"[clarification needed]
- ^ Weiss, Ittay (20 September 2017). "Nothing matters: How India's invention of zero helped create modern mathematics". The Conversation. Archived from the original on 12 July 2018. Retrieved 12 July 2018.
- ^ Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Archived from the original on 20 November 2017. Retrieved 14 September 2017.
- ^ Revell, Timothy (14 September 2017). "History of zero pushed back 500 years by ancient Indian text". New Scientist. Archived from the original on 25 October 2017. Retrieved 25 October 2017.
- ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 14 September 2017. Archived from the original on 14 September 2017. Retrieved 25 October 2017.
- ^ Ifrah, Georges (2000), p. 416.
- ^ Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
- ^ O'Connor, Robertson, J.J., E.F. "Aryabhata the Elder". School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 11 July 2015. Retrieved 26 May 2013.
- ^ William L. Hosch, ed. (15 August 2010). The Britannica Guide to Numbers and Measurement (Math Explained). The Rosen Publishing Group. pp. 97–98. ISBN 978-1-61530-108-9. Archived from the original on 4 August 2016. Retrieved 26 September 2016.
- ^ Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke (1817) London
- ^ Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. New York: Oxford University Press. pp. 68–75. ISBN 978-0-19-514237-2.
- ^ "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. 13 September 2017. Archived from the original on 20 November 2017. Retrieved 14 September 2017.
- ^ Cœdès, George, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
- ^ Casselman, Bill. "All for Nought". ams.org. University of British Columbia), American Mathematical Society. Archived from the original on 6 December 2015. Retrieved 20 December 2015.
- ^ Ifrah, Georges (2000), p. 400.
- ^ Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. p. 165.
- ^ a b c Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, "The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables."
- ^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0. Archived from the original on 29 February 2020. Retrieved 26 September 2016.
- ^ Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher."
- ^ Sigler, L., Fibonacci's Liber Abaci. English translation, Springer, 2003.
- ^ Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99–104.
- ^ Hansen, Alice (9 June 2008). Primary Mathematics: Extending Knowledge in Practice. SAGE. ISBN 978-0-85725-233-3. Archived from the original on 7 March 2021. Retrieved 7 November 2020.
- ^ Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. p. 34. ISBN 978-981-02-4088-2.
- ^ W., Weisstein, Eric. "Zero". mathworld.wolfram.com. Archived from the original on 1 June 2013. Retrieved 4 April 2018.
- ^ Weil, Andre (6 December 2012). Number Theory for Beginners. Springer Science & Business Media. ISBN 978-1-4612-9957-8. Archived from the original on 14 June 2021. Retrieved 6 April 2021.
- ^ Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. pp. 254–255. ISBN 978-0-486-13968-5. Archived from the original on 23 June 2016. Retrieved 5 January 2016., Extract of pp. 254–255 Archived 10 May 2016 at the Wayback Machine
- ^ Reid, Constance (1992). From zero to infinity: what makes numbers interesting (4th ed.). Mathematical Association of America. p. 23. ISBN 978-0-88385-505-8. zero neither prime nor composite.
- ^ "ResultSet (Java Platform SE 8 )". docs.oracle.com. Archived from the original on 9 May 2022. Retrieved 9 May 2022.
- ^ Wu, X.; Ichikawa, T.; Cercone, N. (25 October 1996). Knowledge-Base Assisted Database Retrieval Systems. World Scientific. ISBN 978-981-4501-75-0. Archived from the original on 31 March 2022. Retrieved 7 November 2020.
- ^ "Null values and the nullable type". IBM. 12 December 2018. Archived from the original on 23 November 2021. Retrieved 23 November 2021. In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service function.
- ^ Chris Woodford 2006, p. 9.
- ^ Paul DuBois. "MySQL Cookbook: Solutions for Database Developers and Administrators" Archived 24 February 2017 at the Wayback Machine 2014. p. 204.
- ^ Arnold Robbins; Nelson Beebe. "Classic Shell Scripting" Archived 24 February 2017 at the Wayback Machine. 2005. p. 274
- ^ Iztok Fajfar. "Start Programming Using HTML, CSS, and JavaScript" Archived 24 February 2017 at the Wayback Machine. 2015. p. 160.
- ^ Darren R. Hayes. "A Practical Guide to Computer Forensics Investigations" Archived 24 February 2017 at the Wayback Machine. 2014. p. 399
- ^ "Font Survey: 42 of the Best Monospaced Programming Fonts". codeproject.com. 18 August 2010. Archived from the original on 24 January 2012. Retrieved 22 July 2021.
- ^ Cepelewicz, Jordana Animals Count and Use Zero. How Far Does Their Number Sense Go? Archived 18 August 2021 at the Wayback Machine, Quanta, August 9, 2021
- ^ a b Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563. S2CID 294510.
- ^ Steel, Duncan (2000). Marking time: the epic quest to invent the perfect calendar. John Wiley & Sons. p. 113. ISBN 978-0-471-29827-4. In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.
Bibliography
- Aczel, Amir D. (2015). Finding Zero. New York: Palgrave Macmillan. ISBN 978-1-137-27984-2.
- Asimov, Isaac (1978). "Nothing Counts". Asimov on Numbers. New York: Pocket Books. ISBN 978-0-671-82134-0. OCLC 1105483009.
- Barrow, John D. (2001). The Book of Nothing. Vintage. ISBN 0-09-928845-1.
- Woodford, Chris (2006). Digital Technology. Evans Brothers. ISBN 978-0-237-52725-9. Archived from the original on 17 August 2019. Retrieved 24 March 2016.
Historical studies
- Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8.
- Diehl, Richard A. (2004). The Olmecs: America's First Civilization. London: Thames & Hudson.
- Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0-471-39340-1.
- Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero. Oxford University Press.
- Seife, Charles (2000). Zero: The Biography of a Dangerous Idea. Penguin USA. ISBN 0-14-029647-6.
External links
Look up zero in Wiktionary, the free dictionary.
0 at Wikipedia's sister projects
- Definitions from Wiktionary
- Media from Commons
- Quotations from Wikiquote
- Searching for the World's First Zero
- A History of Zero
- Zero Saga
- The History of Algebra
- Edsger W. Dijkstra: Why numbering should start at zero, EWD831 (PDF of a handwritten manuscript)
- Zero on In Our Time at the BBC
- Weisstein, Eric W. "0". MathWorld.
- Texts on Wikisource:
- "Zero". Encyclopædia Britannica (11th ed.). 1911.
- "Zero". Encyclopedia Americana. 1920.
Page 2
| This article's use of external links may not follow Wikipedia's policies or guidelines. (February 2022) |
| ← 9999 | 10000 | 10001 → |
- List of numbers
- Integers
← 0 10k 20k 30k 40k 50k 60k 70k 80k 90k →
Cardinalten thousandOrdinal10000th(ten thousandth)Numeral systemdecamillesimalFactorization24 × 54Divisors25 totalGreek numeral M α {\displaystyle {\stackrel {\alpha }{\mathrm {M} }}} Roman numeralXUnicode symbol(s)X, ↂGreek prefixmyria-Latin prefixdecamilli-Binary100111000100002Ternary1112011013Senary1141446Octal234208Duodecimal595412Hexadecimal271016Chinese numeral万, 萬
10,000 (ten thousand) is the natural number following 9,999 and preceding 10,001.
Name
Many languages have a specific word for this number: in Ancient Greek it is μύριοι (the etymological root of the word myriad in English), in Aramaic ܪܒܘܬܐ, in Hebrew רבבה [revava], in Chinese 萬/万 (Mandarin wàn, Cantonese maan6, Hokkien bān), in Japanese 万/萬 [man], in Khmer ម៉ឺន [meun], in Korean 만/萬 [man], in Russian тьма [t'ma], in Vietnamese vạn, in Sanskrit अयुत [ayuta], in Thai หมื่น [meun], in Malayalam പതിനായിരം [patinayiram], and in Malagasy alina.[1] In many of these languages, it often denotes a very large but indefinite number.[2]
The Greek root was used in early versions of the metric system in the form of the decimal prefix myria-.
The number 10000 can also be written 10,000 (UK and US), 10.000 (Europe mainland), 10 000 (transition metric), or 10•000 (with the dot raised to the middle of the zeroes; metric).
In mathematics
- In scientific notation, it is written as 104.
- In E notation it is also written as 1 E+4 (or as 1 E4).
- It is the square of 100.
- It is the square root of 100,000,000.
- A myriagon is a polygon with 10,000 edges.
- The classical Greeks used letters of the Greek alphabet to represent Greek numerals: they used a capital letter mu (Μ) to represent 10000, whose name in Greek is myriad.
- 1040000 = 1000010000 The value of a myriad to the power of itself, written (by the system of Apollonius of Perga) as a little M directly above a larger M.
In science
- In astronomy,
- asteroid Number: 10000 Myriostos, Provisional Designation: 1951 SY, Discovery Date: September 30, 1951, by A. G. Wilson:List of asteroids (9001-10000)
- In climate, Summary of 10000 Years is one of several pages of the Climate Timeline Tool: Exploring Weather & Climate Change Through the Powers of 10 sponsored by the National Climatic Data Center of the National Oceanic & Atmospheric Administration.[3]
- In computers, NASA built a 10000-processor Linux computer (it is actually a 10,240-processor) called Columbia[4][5]
- In geography,
- Land of 10000 Lakes is the nickname for the state of Minnesota.
- Land of 10000 Trails or 10000trails.com is an organization created in 1999 by the TN/KY Lakes Area Coalition and based in West Tennessee and West Kentucky to promote tourism by developing trails in the region.[6]
- Ten Thousand Islands National Wildlife Refuge is situated in the lower end of the Fakahatchee and Picayune Strands of Big Cypress Swamp and west of Everglades National Park in Florida.[7]
- Valley of Ten Thousand Smokes in Alaska
- In physics,
- Myria- (and myrio-[8][9][10]) is an obsolete metric prefix that denoted a factor of 10+4, ten thousand, or 10,000.
- 10,000 hertz, 10 kilohertz, or 10 kHz of the radio frequency spectrum falls in the very low frequency or VLF band and has a wavelength of 30 kilometres.
- In orders of magnitude (speed), the speed of a fast neutron is 10000 km/s.
- In acoustics, 10,000 hertz, 10 kilohertz, or 10 kHz of a sound signal at sea level has a wavelength of about 34 mm.
- In music, a 10 kilohertz sound is a E♭9 in the A440 pitch standard, a bit more than an octave higher in pitch than the highest note on a standard piano.
In time
- 10000 BC, 10000 BCE, or 10th millennium BC.
- 10000-year clock or the Clock of the Long Now is a mechanical clock designed to keep time for 10000 years.
In Arts
- In films,
- 10,000 Black Men Named George (2002, TV)
- The Phantom from 10,000 Leagues (1956)
- In Pixar's film Up The main character, Carl Fredrickson attaches 10,000 helium toy balloons to his house.
- Vietnam: The Ten Thousand Day War (1980, mini)
- In music,
- 10,000 Days is the title of the fourth studio album by Tool.
- Ten Thousand Fists is an album by Disturbed.
- 10,000 Hz Legend album by Air 2001.
- 10,000 Maniacs is a US rock band.
- "10000 Men" is a song by Bob Dylan.
- Ten Thousand Men of Harvard is a fight song of Harvard University.
- 10,000 Reasons (album) is a 2013 Christian album by Matt Redman.
- "10,000 promises" is a song by the Backstreet Boys.
- 10,000 Promises. is a Japanese popular music group.
- "10,000 Reasons (Bless the Lord)", is a single and title track of Matt Redman 2013 album 10,000 Reasons.
- "Ten Thousand Strong" is a song by American Power metal band, Iced Earth.
- "10k", a song by rapper KB from his 2020 album His Glory Alone.
- In paintings,
- Xenophon, on his Retreat with the Ten Thousand, first seeing the Sea, painting by Benjamin Haydon.
In other fields
- In currency,
- Two versions of Iraq's 10,000 dinar banknote has Abu Ali Hasan Ibn al-Haitham (also known as Alhazen) on the front and the current issue has sculptor Jawad Saleem's Freedom Monument in Baghdad on the front. Both notes have an image of Mosul's al-Hadba' Minaret on the back.[11] The first issue had an image of former Iraqi president Saddam Hussein and the Spiral Minaret - Al-Minārat Al-Malwiyyah in Samarra.[12]
- the Japanese ¥10,000 banknote depicts Fukuzawa Yukichi.
- Kazakhstan's 10,000₸ banknote.
- the Lebanese £L10,000 banknote depicts Beirut's Martyrs' Square .
- Myanmar's (Burma's) Ks.10,000/- banknote.
- the U.S. $10,000 note depicts a picture of Salmon P. Chase.
- In distances,
- 10 km, 10,000 m, or 1 E+4 m is equal to:
- 1 Scandinavian mil
- about 6.2137 English miles
- side of square with area 100 km2
- radius of a circle with area 100 π km2 ≈ 314.159 km2
- 10 km, 10,000 m, or 1 E+4 m is equal to:
- In finance, on March 29, 1999, the Dow Jones Industrial Average closed at 10006.78 which was the first time the index closed above the 10,000 mark.
- In futurology, Stewart Brand in Visions of the Future: The 10,000-Year Library proposes a museum built around a 10,000 year clock as an idea for assuring that vital information survives future crashes of civilizations.[13]
- In games,
- Ten Thousand is one name of a dice game that is also called farkle.
- In game shows, The $10,000 Pyramid ran on television from 1973 to 1974.
- In history,
- Army of 10,000 Sixty Day Troops, 1862–1863. American Civil War[14]
- The Army of the Ten Thousand were a group of Ancient Greek mercenaries who marched against Artaxerxes II of Persia.
- the Persian Immortals were also called the Ten Thousand or 10,000 Immortals, so named because their number of 10,000 was immediately re-established after every loss.
- The 10,000 Day War: Vietnam by Michael Maclear ISBN 0-312-79094-5 also alternate titles The ten thousand day war: Vietnam, 1945–1975 (10,000 days is 27.4 years)
- Tomb of Ten Thousand Soldiers – defeat of the Tang dynasty army of China in the Nanzhao kingdom in 751.
- In Islamic history, 10,000 is the number of besieging forces led by Muhammad's adversary, Abu Sufyan, during the Battle of the Trench .
- 10,000 is the number of Muhammad's soldiers during the conquest of Mecca.
- In language,
- the Chinese, Japanese, Korean, and Vietnamese phrase live for ten thousand years was used to bless emperors in East Asia.
- the words in the Interlingua–English Dictionary are all drawn from 10,000 roots.
- Μύριοι is an Ancient Greek name for 10.000 taken into the modern European languages as 'myriad' (see above). Hebrew, Chinese, Japanese and Korean have words with the same meaning.
- In literature,
- Man'yōshū (万葉集 Man'yōshū, Collection of Ten Thousand Leaves) is the oldest existing, and most highly revered, collection of Japanese poetry.
- Ten Thousand a Year 1839 by Samuel Warren.
- Ten Thousand a Year 1883?. A drama, in three acts. Adapted from the celebrated novel of the same name, by the author of the Diary of a Physician, and arranged for the stage by Richard Brinsley Peake.[15]
- Anabasis, by the Greek writer Xenophon (431–360 B.C.), about the Army of the Ten Thousand – Greek mercenaries taking part in the expedition of Cyrus the Younger, a Persian prince, against his brother, King Artaxerxes II.
- The Ten Thousand: A Novel of Ancient Greece by Michael Curtis Ford. 2001. ISBN 0-312-26946-3 Historic fiction about the Army of the Ten Thousand
- The World of the Ten Thousand Things: Poems 1980–1990 by Charles Wright ISBN 0-374-29293-0 ISBN 0-374-52326-6.
- Ten Thousand Lovers by Edeet Ravel ISBN 0-06-056562-4
- In philosophy, Lao Zi writes about ten thousand things in the Tao Te Ching In Taoism, the "10,000 Things" is a term meaning all of phenomenal reality.[16]
- In piphilology, ten thousand is the current world record for the number of digits of pi memorized by a human being.
- In psychology, Ten Thousand Dreams Interpreted, or what's in a dream: a scientific and practical, by Miller, Gustavus Hindman (1857–1929). Project Gutenberg[17]
- In religion,
- the Bible,
- has 52 references to ten thousand in the King James Version.[18]
- Revelation 5:11 And I beheld, and I heard the voice of many angels round about the throne and the beasts and the elders: and the number of them was ten thousand times ten thousand, and thousands of thousands;[19]
- hymn, Ten thousand times ten thousand.[20]
- The Ten thousand martyrs from[21]
- the Bible,
- In software,
- the Year 10,000 problem is the collective name for all potential software bugs that will emerge as the need to express years with five digits arises.
- In sports,
- In athletics, 10,000 metres, 10 kilometres, 10 km, or 10K (6.2 miles) is the final standard track event in a long-distance track event and a distance in other racing events such as running, cycling, and skiing.
- In bicycle racing, annual Tour of 10,000 Lakes Stage Race in Minneapolis[22]
- In baseball, on July 15, 2007, the Philadelphia Phillies became the first team in professional sports history to lose 10,000 games.
Selected numbers in the range 10001-19999
10001 to 10999
- 10007 = smallest five-digit prime number, twin prime with 10009.
- 10008 = palindromic in bases 5 (3100135), 22 (KEK22), 28 (CLC28) and 33 (96933) and a Harshad number in bases 2, 3, 4, 5, 6, 7, 8, 10, 13, 14 and 16.
- 10009 = twin prime with 10007.
- 10080 = highly composite number;[23] number of minutes in a week.
- 10111 = palindromic prime in bases 3 (1112121113) and 27 (DND27).
- 10176 = smallest (provable) generalized Riesel number in base 10: 10176*10n-1 is always divisible by one of the prime numbers {7, 11, 13, 37}.[24]
- 10201 = 1012, palindromic square (in the decimal system)
- 10206 = pentagonal pyramidal number.[25]
- 10223 = sixth last number to be eliminated (in 2016) by Seventeen or Bust (now a sub-project of PrimeGrid) in the Sierpiński problem.
- 10239 = Woodall number.[26]
- 10252 = Padovan number.[27]
- 10267 = cuban prime.[28]
- 10301 = palindromic prime in bases 10 (1030110), 27 (E3E27), 30 (BDB30) and 44 (5E544).
- 10333 = star prime,[29] palindromic in bases 9 (151519), 31 (ANA31) and 35 (8F835).
- 10416 = square pyramidal number.[30]
- 10425 = octahedral number.[31]
- 10430 = weird number.[32]
- 10433 = palindromic prime in base 44 (5H544).
- 10440 = 144th triangular number.
- 10499 = twin prime with 10501.
- 10500 = Harshad number in bases 2, 3, 4, 6, 7, 8, 9, 10, 11, 15 and 16.
- 10501 = palindromic prime in bases 10 (1050110) and 58 (37358).
- 10512 = Harshad number in bases 2, 3, 4, 5, 7, 8, 9, 10, 13 and 16.
- 10538 = 10538 Overture is a hit single by Electric Light Orchestra.
- 10560 = Harshad number in bases 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14 and 16.
- 10570 = weird number.[32]
- 10585 = Carmichael number.[33]
- 10601 = palindromic prime in bases 10 (1060110) and 30 (BNB30).
- 10609 = 1032, tribonacci number.[34]
- 10631 = palindromic prime in base 30 (BOB30).
- 10646 = ISO 10646 is the standard for Unicode.
- 10648 = 223, the smallest 5-digit cube.
- 10660 = tetrahedral number.[35]
- 10671 = tetranacci number.[36]
- 10700 = 10700 kHz or 10.7 MHz is a standard intermediate frequency for analog superheterodyne FM broadcast band receivers.
- 10744 = amicable number with 10856.
- 10752 = the second 16-bit word of a TIFF file if the byte order marker is misunderstood.
- 10792 = weird number.[32]
- 10800 = number of bricks used for the uttaravedi in the Agnicayana ritual.
- 10837 = star prime.[29]
- 10856 = amicable number with 10744.
- 10905 = Wedderburn–Etherington number[37]
- 10922 = repdigit in base 4 (22222224), and palindromic in base 8 (252528).
- 10946 = Fibonacci number,[38] Markov number.[39]
- 10958 = the smallest positive integer that cannot be represented by an equation using increasing order of integers from 1 to 9 and basic arithmetic operations.[40]
- 10989 = reverses when multiplied by 9.
- 10990 = weird number.[32]
11000 to 11999
- 11025 = 1052, sum of the cubes of the first 14 positive integers.
- 11083 = palindromic prime in 2 consecutive bases: 23 (KLK23) and 24 (J5J24).
- 11111 = repdigit.
- 11297 = number of planar partitions of 16[41]
- 11298 = Riordan number
- 11311 = palindromic prime.
- 11340 = Harshad number in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15 and 16.
- 11353 = star prime.[29]
- 11368 = pentagonal pyramidal number[25]
- 11410 = weird number.[32]
- 11411 = palindromic prime in base 10.
- 11424 = Harshad number in bases 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15 and 16.
- 11440 = square pyramidal number.[30]
- 11480 = tetrahedral number.[35]
- 11605 = smallest integer to start a run of five consecutive integers with the same number of divisors.
- 11690 = weird number.[32]
- 11717 = twin prime with 11719.
- 11719 = cuban prime,[28] twin prime with 11717.
- 11726 = octahedral number.[31]
- 11826 = smallest number whose square (algebra) is pandigital without zeros.
- 11953 = palindromic prime in bases 7 (465647) and 30 (D8D30).
12000 to 12999
- 12000 = 12,000 of each of the twelve tribes of Israel made up the 144,000 servants of God who were 'sealed' according to the Book of Revelation in the New Testament.[42]
- 12048 = number of non-isomorphic set-systems of weight 12.
- 12097 = cuban prime.[28]
- 12101 = Friedman prime.
- 12107 = Friedman prime.
- 12109 = Friedman prime.
- 12110 = weird number.[32]
- 12167 = 233
- 12172 = number of triangle-free graphs on 10 vertices[43]
- 12198 = semi-meandric number[44]
- 12251 = number of primes
≤
2
17
{\displaystyle \leq 2^{17}}
.[45]
- 12285 = amicable number with 14595.
- 12287 = Thabit number.
- 12289 = Proth prime, Pierpont prime.
- 12321 = 1112, Demlo number, palindromic square.
- 12341 = tetrahedral number.[35]
- 12407 = cited on QI as the smallest uninteresting positive integer in terms of arithmetical mathematics.[notes 1][46]
- 12421 = palindromic prime.
- 12496 = smallest sociable number.
- 12529 = square pyramidal number.[30]
- 12530 = weird number.[32]
- 12670 = weird number.[32]
- 12721 = palindromic prime.
- 12726 = Ruth–Aaron pair.
- 12758 = largest number that cannot be expressed as the sum of distinct cubes.
- 12765 = Finnish internet meme; the code accompanying no-prize caps in a Coca-Cola bottle top prize contest. Often spelled out yksi – kaksi – seitsemän – kuusi – viisi, ei voittoa, "one – two – seven – six – five, no prize".
- 12769 = 1132, palindromic in base 3.
- 12821 = palindromic prime.
13000 to 13999
- 13131 = octahedral number.[31]
- 13244 = tetrahedral number.[35]
- 13267 = cuban prime.[28]
- 13331 = palindromic prime.
- 13370 = weird number.[32]
- 13510 = weird number.[32]
- 13581 = Padovan number.[27]
- 13669 = cuban prime.[28]
- 13685 = square pyramidal number.[30]
- 13790 = weird number.[32]
- 13792 = largest number that is not a sum of 16 fourth powers.
- 13820 = meandric number, open meandric number.
- 13824 = 243
- 13831 = palindromic prime.
- 13860 = Pell number.[47]
- 13930 = weird number.[32]
- 13931 = palindromic prime.
- 13950 = pentagonal pyramidal number.[25]
14000 to 14999
- 14190 = tetrahedral number.[35]
- 14200 = number of n-Queens Problem solutions for n – 12.
- 14341 = palindromic prime.
- 14400 = 1202, sum of the cubes of the first 15 positive integers.
- 14595 = amicable number with 12285.
- 14641 = 1212 = 114, palindromic square (base 10).
- 14644 = octahedral number.[31]
- 14701 = Markov number.[39]
- 14741 = palindromic prime.
- 14770 = weird number.[32]
- 14884 = 1222, palindromic square in base 11.
- 14910 = square pyramidal number.[30]
15000 to 15999
- 15015 = smallest odd and square-free abundant number.[48]
- 15120 = highly composite number.[23]
- 15180 = tetrahedral number.[35]
- 15376 = 1242, pentagonal pyramidal number.[25]
- 15387 = Zeisel number.[49]
- 15451 = palindromic prime.
- 15511 = Motzkin prime.[50]
- 15551 = palindromic prime
- 15610 = weird number.[32]
- 15625 = 1252 = 253 = 56
- 15629 = Friedman prime.
- 15640 = initial number of only four-, five-, or six-digit century to contain two prime quadruples[51] (in between which lies a record prime gap of 43[52]).
- 15661 = Friedman prime.
- 15667 = second nice Friedman prime.
- 15679 = Friedman prime.
- 15793 – number of parallelogram polyominoes with 13 cells.[53]
- 15841 = Carmichael number.[33]
- 15876 = 1262, palindromic square in base 5.
- 15890 = weird number.[32]
16000 to 16999
- 16030 = weird number.[32]
- 16061 = palindromic prime.
- 16072 = logarithmic number.[54]
- 16091 = strobogrammatic prime.[55]
- 16206 = square pyramidal number.[30]
- 16269 = octahedral number.[31]
- 16310 = weird number.[32]
- 16361 = palindromic prime.
- 16381 = Friedman prime.
- 16384 = 1282 = 214, palindromic in base 15.
- 16447 = third nice Friedman prime.
- 16561 = palindromic prime.
- 16580 = Leyland number.[56]
- 16651 = cuban prime.[28]
- 16661 = palindromic prime.
- 16730 = weird number.[32]
- 16759 = Friedman prime.
- 16796 = Catalan number.[57]
- 16807 = 75
- 16843 = smallest Wolstenholme prime.[58]
- 16870 = weird number.[32]
- 16879 = Friedman prime.
- 16896 = pentagonal pyramidal number.[25]
- 16999 = number of partially ordered set with 8 unlabeled elements.[59]
17000 to 17999
- 17073 = number of free 11-ominoes.
- 17163 = the largest number that is not the sum of the squares of distinct primes.
- 17272 = weird number.[32]
- 17296 = amicable number with 18416.[60]
- 17344 = Kaprekar number.[61]
- 17389 = 2000th prime number.
- 17471 = palindromic prime.
- 17570 = weird number.[32]
- 17575 = square pyramidal number.[30]
- 17576 = 263, palindromic in base 5.
- 17689 = 1332, palindromic in base 11.
- 17711 = Fibonacci number.[38]
- 17971 = palindromic prime.
- 17990 = weird number.[32]
- 17991 = Padovan number.[27]
18000 to 18999
- 18010 = octahedral number.[31]
- 18181 = palindromic prime, strobogrammatic prime.[55]
- 18334 = number of planar partitions of 17[41]
- 18410 = weird number.[32]
- 18416 = amicable number with 17296.[62]
- 18481 = palindromic prime.
- 18496 = 1362, sum of the cubes of the first 16 positive integers.
- 18600 = harmonic divisor number.[63]
- 18620 = harmonic divisor number.[63]
- 18785 = Leyland number.[56]
- 18830 = weird number.[32]
- 18970 = weird number.[32]
19000 to 19999
- 19019 = square pyramidal number.[30]
- 19141 = unique prime in base 12.
- 19302 = number of ways to partition {1,2,3,4,5,6,7} and then partition each cell (block) into subcells.[64]
- 19390 = weird number.[32]
- 19391 = palindromic prime.
- 19441 = cuban prime.[28]
- 19455 = smallest integer that cannot be expressed as a sum of fewer than 548 ninth powers.
- 19513 = tribonacci number.[34]
- 19531 = repunit prime in base 5.
- 19600 = 1402, tetrahedral number.
- 19601/13860 ≈ √2
- 19609 = first prime followed by a prime gap of over fifty.[52]
- 19670 = weird number.[32]
- 19683 = 273, 39
- 19739 = fourth nice Friedman prime.
- 19871 = octahedral number.[31]
- 19891 = palindromic prime.
- 19927 = cuban prime.[28]
- 19991 = palindromic prime.
There are 1033 prime numbers between 10000 and 20000.
See also
- Mathematics portal
- 10,000 (disambiguation)
Notes
- ^ On the basis that it did not then (November 2011) appear in Sloane's On-Line Encyclopedia of Integer Sequences.
References
- ^ "Malagasy Dictionary and Madagascar Encyclopedia : Alina".
- ^ (Merriam-Webster's Online Dictionary)
- ^ Climate Timeline Information Tool
- ^ news
- ^ "NASA Project: Columbia". Archived from the original on 2005-04-08. Retrieved 2005-02-15.
- ^ 10000 trails web site
- ^ "Ten Thousand Islands NWR". U.S. Fish & Wildlife Service. Archived from the original on 2005-03-01. Retrieved 2005-02-14.
- ^ Brewster, David (1830). The Edinburgh Encyclopædia. Vol. 12. Edinburgh, UK: William Blackwood, John Waugh, John Murray, Baldwin & Cradock, J. M. Richardson. p. 494. Retrieved 2015-10-09.
- ^ Brewster, David (1832). The Edinburgh Encyclopaedia. Vol. 12 (1st American ed.). Joseph and Edward Parker. Retrieved 2015-10-09.
- ^ Dingler, Johann Gottfried (1823). Polytechnisches Journal (in German). Vol. 11. Stuttgart, Germany: J.W. Gotta'schen Buchhandlung. Retrieved 2015-10-09.
- ^ "Iraq Dinar Currency Photos| Banknote Series | 25000, 10000, 5000, 1000, 250, 50 Dinars". iraqi-dinar.com. Archived from the original on 2005-02-07. Retrieved 2022-08-04.
- ^ http://www.iraqsales.com/10%2C000.htm Archived 2005-02-06 at the Wayback Machine
- ^ Brand, Stewart. "The 10,000-Year Library". kurzweilai.net. Archived from the original on 2005-02-05. Retrieved 2022-08-04.
- ^ "Army of 10,000". mississippiscv.org. Archived from the original on 2002-04-01. Retrieved 2022-08-04.
- ^ "University of Michigan Digital Library - Login Options".
- ^ "Tao Te Ching, Verse 34". thebigview.com. Archived from the original on 2007-08-17. Retrieved 2022-08-04.
- ^ https://www.gutenberg.org/ebooks/926 : Ten Thousand Dreams Interpreted
- ^ http://bible.gospelcom.net/keyword/?search=ten%20thousand&version1=9&searchtype=phrase&wholewordsonly=yes , [1]
- ^ (KJV) The Apocalypse of John
- ^ [2][dead link]
- ^ The Catholic Encyclopedia
- ^ Ulmer, Jeanne. "Minnesota Cycling Team –Tour of 10,000 Lakes". tourof10000lakes.net. Archived from the original on 2005-02-21. Retrieved 2022-08-04.
- ^ a b "Sloane's A002182: Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A273987: Smallest Riesel number to base n". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-01-30.
- ^ a b c d e "Sloane's A002411: Pentagonal pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A003261: Woodall numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c "Sloane's A000931: Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
- ^ a b c d e f g h "Sloane's A002407: Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c "Sloane's A083577: Prime star numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c d e f g h "Sloane's A000330: Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c d e f g "Sloane's A005900: Octahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab "Sloane's A006037: Weird numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b "Sloane's A002997: Carmichael numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b "Sloane's A000073: Tribonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b c d e f "Sloane's A000292: Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A000078: Tetranacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A001190: Wedderburn-Etherington numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b "Sloane's A000045: Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b "Sloane's A002559: Markoff (or Markov) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ Taneja, Inder (2013). "Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9". arXiv:1302.1479 [math.HO].
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Revelation 7:4–8
- ^ Sloane, N. J. A. (ed.). "Sequence A006785 (Number of triangle-free graphs on n vertices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ "Sloane's A000682: Semimeanders". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ Sloane, N. J. A. (ed.). "Sequence A007053". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-02.
- ^ Host: Stephen Fry; Panellists: Alan Davies, Al Murray, Dara Ó Briain and Sandi Toksvig (11 November 2011). "Inland Revenue". QI. Series I. Episode 10. London, England. 19:55 minutes in. BBC. BBC Two.
- ^ "Sloane's A000129: Pell numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A112643: Odd and squarefree abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A051015: Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A001006: Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A007530: Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-01-30.
- ^ a b "Table of Known Maximal Gaps". Prime Pages.
- ^ Sloane, N. J. A. (ed.). "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A002104 (Logarithmic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b "Sloane's A007597: Strobogrammatic primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ a b "Sloane's A076980: Leyland numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A000108: Catalan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ "Sloane's A088164: Wolstenholme primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ Sloane, N. J. A. (ed.). "Sequence A000112 (Number of partially ordered sets (posets) with n unlabeled elements)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 61. ISBN 978-1-84800-000-1.
- ^ "Sloane's A006886: Kaprekar numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ Higgins, ibid.
- ^ a b "Sloane's A001599: Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
- ^ Sloane, N. J. A. (ed.). "Sequence A000258 (Expansion of e.g.f. exp(exp(exp(x)-1)-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
External links
Wikimedia Commons has media related to 10000 (number).
Look up ten thousand in Wiktionary, the free dictionary.
