1 (број)

1
−2 · −1 · 0 · 1 · 2 · 3 · 4
Кардинални број	један
Редни број	први
Делиоци	1
Факторизација	/
Римски	I
Бинарно	1
Октално	1
Дуодецимално	1
Хексадецимално	1
φ(1)	0
σ(1)	1
π(1)	0
μ(1)	1
M(1)	1

1 (један) је број, нумерал и име глифа који представља тај број. То је природни број који се јавља после броја 0, а претходи броју 2.

Стари Хелени један нису сматрали бројем. То је за њих била монада, једнота, нераздељива целина. Сматрали су да се јединица не може раздељивати без губљења својства једне целине, једноте. Тек је два било мноштво, те је представљало број.

Број један се не сматра ни простим ни сложеним бројем, мада постоје мишљења да би га требало сматрати простим бројем. У 20. веку дефинитивно је уклоњен из тог списка, а последњи математичар који га јесте сматрао простим је Анри Лебеск (1875—1941).

Број 1 не може бити основа позиционог бројног система јер су сви степени броја 1 и даље 1.

Као број

One, sometimes referred to as unity, 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).

Еволуција развоја графичког исписа знака

Графички облик који се данас користи за испис бројке 1, усправна линија, често са малим серифом на врху и понекад са кратком линијом у подножју, вуче корене од Брахмана у Индији који су писали 1 са једном положеном линијом. У кинеском језику и данас се тако пише број 1. Гупте су исписивали ту црту закривљено, а Нагари су понекад додавали мали кружић са леве стране. Овај знак, који мало личи на положену бројку 9 се данас може наћи у Гуџарати и Пенџаби писму. У непалском писмо је црта заокренута надесно али са истим кружићем на врху. Овај кружић је постао цртица на врху усправне линије, а доња линија која се понекад исписује је потекла од исписа римске бројке I. У неким језицима (немачки) се мала коса црта претвара у велику хоризонталну, што понекад може довести до замене са бројком 7 код других народа. Тамо где се 1 пише са великом хоризонталном цртом 7 се пише са још једном хоризонталном линијом преко вертикалне.

У фонтовима са словима и цифрама, 1 је типично исте висине као мало слово X, на пример, .

У математици

За сваки број x важи:

x·1 = 1·x = x

x/1 = x

x¹ = x, 1^x = 1

x⁰ = 1, ако је x различито од 0

x↑↑1 = x и 1↑↑x = 1

У десетичном бројном систему важи следећа тврдња:

$1=0.999\dots =0.{\dot {9}}$

где тачка изнад 9 означава да се 9 понавља бесконачан број пута.

Definitions

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.

Properties

Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.

Since the base 1 exponential function (1^x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.

It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

1 is the most common leading digit in many sets of data, a consequence of Benford's law.

1 is the only known Tamagawa number for a simply connected algebraic group over a number field.

The generating function that has all coefficients 1 is given by

${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots$

This power series converges and has finite value if and only if $|x|<1$.

Table of basic calculations

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

In technology

1 as a resin identification code, used in recycling

• The resin identification code used in recycling to identify polyethylene terephthalate.
• 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.
• The numerical value of true in many programming languages.
• 1 is the ASCII code of "Start of Header".

У хемији

Један је редни број и атомски број хемијског елемента водоника.

In philosophy

In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence. 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.

Види још

• 1. година нове ере
• 1. година пре нове ере

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Спољашње везе

• The Number 1
• The Positive Integer 1
• Prime curiosities: 1