Property of number 1
This is the property for the number 1. Go back to the main page to read the whole property and other parts of the page.
Properties
[edit]Formal definitions
[edit]Mathematically, 1 is:
- in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
- more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.
Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.
In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.
By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.
By definition, 1 is the probability of an event that is absolutely or almost certain to occur.
In category theory, 1 is sometimes used to denote the terminal object of a category.
In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.
Normal curios
[edit]- Not a prime or composite
- Is 1 when written in all bases
- An element of unity, along with -1
- The first number
- No level of factorization
- 1^n=1 with all n
Prime curios
[edit]- Any integer greater than one is called a prime number if (and only if) its only positive divisors (factors) are one and itself.
- The number 1 is considered neither prime or composite but in a class of its own. It is the multiplicative identity, so it is also a unit and a divisor of unity.
- Every natural number is the period length of at least 1 prime.
- The number of factors of an integer can be found by adding 1 to the exponent of each prime factor and then calculating the product. For example, the prime factorization of 12 is 22 * 3 and 12 has (2 1)(1 1) = 6 factors.
- The chance of a random integer x being prime is about 1/log(x).
- Unlike the original proof of the Prime Number Theorem from 1896 by Hadamard and Poussin, Erdös and Selberg's proof in 1949 did not employ the square root of -1.
- Henry Ernest Dudeney's 3-by-3 magic square contains "1" non-prime:
| 1|43|- | 37|61|- | 73|7 |
- Bertrand's postulate asserts the existence of at least one prime between n and 2n.
- Johann H. Lambert (1728-1777), announced without proof that every prime number has at least one primitive root.
- The only proper divisor of primes. [Beedassy]
- Ernst Gabor Strauss (1922--1983) was said to have replied to a student's question about why 1 is not a prime: "The primes are the building bricks for arithmetic, and 1 is just not a brick!"
- Mersenne primes can be written as unbroken strings of consecutive 1s in binary form.
- The only number with exactly one positive divisor. [Gupta]
- The only number whose concatenation with itself can yield primes in many cases. [Murthy]
- There is only 1 "Prime Street" in England. It is in Stoke-on-Trent, Staffordshire. [Croll]
- Bertrand's Postulate guarantees that in every base there is at least one prime of any given length beginning with the digit 1, and Benford's Law tells us that primes with leading digit 1 occur more often than primes beginning with any other digit in all bases. [Rupinski]
- Carl Sagan included the number 1 in an example of prime numbers in his book Cosmos.
- The smallest number n such that 10n 1, 10n 3, 10n 7 and 10n 9 are all primes. [Firoozbakht]
- π(1) = !1, where !1 denotes subfactorial 1. [Gupta]
- The only number that is exactly 1/2 prime. [McAlee]
- If primes were called pints, then we could say, "1 is a half-pint." [McAlee]
- George Bernard Riemann extended Euler's Zeta function to include the sans' simple pole at s = 1. [McAlee]
- 1 is the only positive integer whose primal code characteristic is 1. [Awbrey]
- The number 1 is an "extinct" prime since it was once thought to be prime by many and now is no longer considered to be prime. [Hilliard]
- The remainder of division of the Mersenne numbers 2p - 1 by p is always equal to 1. [Capelle]
- The number of primes between two squares is never equal to 1. [Capelle]
- Henri Lebesgue (1875-1941) is said to be the last professional mathematician to call 1 prime.
- In his Elements of Algebra, Euler did not consider 1 a prime. [Waterhouse]
- The first orbit of an atom is the only orbit having the maximum possible number of electrons equal to a prime number. [Gudipati]
- The Egyptian fraction 1/6 1/10 1/14 1/15 1/21 1/22 1/26 1/33 1/34 1/35 1/38 1/39 1/46 1/51 1/55 1/57 1/58 1/62 1/65 1/69 1/77 1/82 1/85 1/86 1/87 1/91 1/93 1/95 1/115 1/119 1/123 1/133 1/155 1/187 1/203 1/209 1/215 1/221 1/247 1/265 1/287 1/299 1/319 1/323 1/391 1/689 1/731 1/901 = 1. Note that each denominator is semiprime. Found by ALLAN Wm. JOHNSON Jr. of Washington, D.C.
- 1 = 69709^3 - 56503^3 - 54101^3, where all integers used (other than the unit), are prime numbers. [Rivera]
- Every Mersenne prime is represented in binary as a string of 1's. [Brink]
- Why isn't 1 a prime number?
Primality
[edit]1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i.
The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations".
1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).
1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.
Table of basic calculations
[edit]| Multiplication | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 50 | 100 | 1000 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 × x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 50 | 100 | 1000 |
| Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 ÷ x | 1 | 0.5 | 0.3 | 0.25 | 0.2 | 0.16 | 0.142857 | 0.125 | 0.1 | 0.1 | 0.09 | 0.083 | 0.076923 | 0.0714285 | 0.06 | |
| x ÷ 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| Exponentiation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1x | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| x1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
1 in popular culture
[edit]Etymology
[edit]The word one can be used as a noun, an adjective, and a pronoun.[1]
It comes from the English word an,[1] which comes from the Proto-Germanic root *ainaz .[1] The Proto-Germanic root *ainaz
comes from the Proto-Indo-European root *oi-no-.[1]
Compare the Proto-Germanic root *ainaz
to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn.
Compare the Proto-Indo-European root *oi-no- (which means "one, single"[1]) to Greek oinos (which means "ace" on dice[1]), Latin unus (one[1]), Old Persian aivam , Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one[1]).
As a number
[edit]One, sometimes referred to as unity,[2][3] is the first non-zero natural number. It is thus the integer after zero.
Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).
As a digit
[edit]The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.
In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.
While the shape of the character for the digit 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in 
.
Many older typewriters do not have a separate symbol for 1, and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used, while it may be for decorative purposes.
In technology
[edit]
- The resin identification code used in recycling to identify polyethylene terephthalate.[4]
- The ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean.
- A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.
- In many physical devices, 1 represents the value for "on", which means that electricity is flowing.[5][6]
- The numerical value of true in many programming languages.
- 1 is the ASCII code of "Start of Header".
In science
[edit]- Dimensionless quantities are also known as quantities of dimension one.
- 1 is the atomic number of hydrogen.
- 1 is the electric charge of positrons and protons.
- Group 1 of the periodic table consists of the alkali metals.
- Period 1 of the periodic table consists of just two elements, hydrogen and helium.
- The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered.
- The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.
In philosophy
[edit]In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.[7] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]).
The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.[8]
In literature
[edit]- Number One is a character in the book series Lorien Legacies by Pittacus Lore.
- Number 1 is also a character in the series Artemis Fowl by Eoin Colfer.
In music
[edit]- In a 1968 song by Harry Nilsson and recorded by Three Dog Night, the number one is identified as "the loneliest number".
- We Are Number One is a 2014 song from the children's TV show LazyTown, which gained popularity as a meme.
- 1 (Beatles album), a compilation album by the Beatles.
- One, a 1991 song by Irish rock band U2.
In comics
[edit]- A character in the Italian comic book Alan Ford (authors Max Bunker and Magnus), very old disabled man, the supreme leader of the group TNT.
- A character in the Italian comic series PKNA and its sequels, an artificial intelligence as an ally of the protagonist Paperinik.
In sports
[edit]- In baseball scoring, the number 1 is assigned to the pitcher.
- In association football (soccer) the number 1 is often given to the goalkeeper.
- In most competitions of rugby league (though not the Super League, which uses static squad numbering), the starting fullback wears jersey number 1.
- In rugby union, the starting loosehead prop wears the jersey number 1.
- 1 is the lowest number permitted for use by players of the National Hockey League (NHL); the league prohibited the use of 00 and 0 in the late 1990s (the highest number permitted being 98).
- 1 is the lowest number permitted for use at most levels of American football. Under National Football League policy, it cannot be used by offensive linemen or defensive linemen.
- In Formula One, the previous year's world champion is allowed to use the number 1.
In other fields
[edit]- Number One is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew.
- 1 is the value of an ace in many playing card games, such as cribbage.
- List of highways numbered 1
- List of public transport routes numbered 1
- 1 is often used to denote the Gregorian calendar month of January.
- 1 CE, the first year of the Common Era
- 01, the former dialing code for Greater London
- For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number.
- PRS One, a German paraglider design
- 1 is the code for international telephone calls to countries in the North American Numbering Plan.
- ^ a b c d e f g h Online Etymology Dictionary. Douglas Harper.
- ^ Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758.
- ^ Cite error: The named reference
:0was invoked but never defined (see the help page). - ^ Plastic Packaging Resins. American Chemistry Council. Archived 2011-07-21 at the Wayback Machine
- ^ Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9
- ^ Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8
- ^ Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.
{{cite book}}: CS1 maint: location missing publisher (link) - ^ British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. Retrieved May 16, 2021.