# one

one
1
presentation
Roman I.
dual 1
Octal 1
Duodecimal 1
Morse code - - - -
Arabic ١
Chinese 一 / 弌 / 壹
Indian
Mathematical properties
sign positive
parity odd
Factorization ${\ displaystyle 1}$ (no prime number)
Divider 1

The one (1) is the natural number between zero and two . It's odd, a square, and a cube number .

## etymology

The germ. Numeral mhd. , Ahd. One goes with the same significant other words IE. Languages on IE. Oi-no-s back.

## Mathematical properties

The number 1 is not a prime number , but it is a divisor of every natural number. It is often taken as the smallest natural number (however, some authors count natural numbers from zero). Your prime factorization is the empty product with 0 factors, which by definition has the value 1. The 1 is often referred to as one of the five most important constants in analysis (next to 0 , π , e and i ). The Euler's identity

${\ displaystyle {\ begin {matrix} e ^ {\ mathrm {i} \, \ pi} + 1 = 0 \ end {matrix}}}$

establishes a simple relationship between these constants.

The 1 is also used in other meanings in mathematics , such as the neutral element in multiplication in a ring , called the one element . In these systems other calculation rules can apply, so that 1 + 1 has different meanings and can produce different results. In linear algebra , 1 also denotes one- vectors and one-matrices , the elements of which are all equal to the one-element, and the identical mapping .

The number one is a Størmer number .

The number one is a “ neutral element ” in the point calculation : If you divide a number by 1 (each number is divisible by 1), or multiply or raise it to the power of 1, the value of the number remains unchanged.

If a number is raised to the power of 0 that is not equal to 0, the result is by definition 1. The number 0 raised to the power of itself either remains undefined or, if more appropriate, is also defined as 1.

## Importance in computer science

In computer science, one is very important because, together with zero, it forms the dual system (binary system). In machine language, it stands for “on” and can be found in programming languages as a Boolean variable data type (1 = true = true, 0 = false = false).

In data modeling (especially in the entity-relationship model ), in which the relationships and frequencies of entities to one another are clarified and described, the to-1 relationship plays an important role, as it determines the uniqueness of an assignment. For example, the “vehicle” entity has an N-to-1 relationship with the “owner” entity: an owner can have multiple vehicles, but each vehicle must have exactly one owner.

## Spellings

### The symbol 1

A handwritten 1 with upstroke, as a vertical line and with upstroke and underscore

The symbol 1 is used as a number in the place value system. If the number 1 is on its own, it means “number one” according to the usual interpretation . It is the largest digit in the dual system .

In Germany, the number 1 is handwritten in two strokes in accordance with the numerical spelling of the Latin original script : a smaller slash from bottom left to top right and, starting from this, a longer smear without separating. This notation corresponds to the Austrian school script (both versions from 1969 and 1995) and the Swiss Schnürlischrift . In the English-speaking culture and in areas influenced by it, a 1 is drawn as a vertical line. The continental European spelling can therefore be misinterpreted as 7 there. Some people in the Anglophone world write a 1 with an up and an underscore.

When writing Roman and binary numbers, the 1 is also drawn as a line in Germany, Austria and Switzerland.

### Periodic decimal fraction

In addition to the usual spelling as 1, the number one has a periodic decimal fraction as . ${\ displaystyle 0 {,} {\ bar {9}} = 0 {,} 999 \ ldots}$

This statement can be proven in several ways:

#### Traced back to a known infinite decimal fraction

${\ displaystyle {\ frac {1} {3}} = 0 {,} 333 \ ldots}$
${\ displaystyle 3 \ cdot {\ frac {1} {3}} = 3 \ cdot 0 {,} 333 \ ldots}$
${\ displaystyle 1 = 0 {,} 999 \ ldots}$

This evidence is widespread - but it should be kept in mind:

1. The first line is assumed here, but should actually be proven by means similar to the statement itself.
2. The transition from the second to the third line uses a property of limit values on the right-hand side , namely that the multiplication by a constant (here 3) can be interchanged with the limit value formation.

#### Arrangement of the real numbers

The equality is a consequence of the fact that two real numbers x and y are only different if there is a real number z between them , for which x <z <y or y <z <x applies. The existence of such a number z is not possible in this case according to the definition of the decimal fraction expansion. This argumentation uses the fact that every real number has a decimal fraction expansion. A fact that of course would have to be proven beforehand.

#### Limit of a sequence of numbers

${\ displaystyle 0 {,} 999 \ ldots}$is the limit of the sequence of numbers The general term of this sequence is . The difference between and is . For every little thing, you can find an n with for everyone . So according to the definition of the limit value . ${\ displaystyle 0 {,} 9 \ \ 0 {,} 99 \ \ 0 {,} 999 \ \ 0 {,} 9999 \ \ \ ldots}$
${\ displaystyle a_ {n}}$${\ displaystyle a_ {n} = 0 {,} \ underbrace {99 \ ldots 9} _ {n}}$${\ displaystyle 1}$${\ displaystyle a_ {n}}$${\ displaystyle {\ frac {1} {10 ^ {n}}}}$${\ displaystyle \ varepsilon> 0}$${\ displaystyle {\ frac {1} {10 ^ {m}}} <\ varepsilon}$${\ displaystyle m> n}$${\ displaystyle 0 {,} 999 \ ldots = 1}$

#### Geometric series

The following applies to periodic decimal fractions${\ displaystyle 0 {,} {\ overline {9}}}$

${\ displaystyle 0 {,} {\ overline {9}} = \ sum _ {n = 1} ^ {\ infty} {\ frac {9} {10 ^ {n}}} = \ sum _ {n = 0 } ^ {\ infty} {\ frac {9} {10}} \ cdot {\ frac {1} {10 ^ {n}}}}$.

This is an infinite geometric series of shape . Such series are for convergent and have the value . With and the total value results as . ${\ displaystyle \ sum _ {i = 0} ^ {\ infty} a \ cdot q ^ {i}}$${\ displaystyle | q | <1 \;}$ ${\ displaystyle a \ cdot {\ frac {1} {1-q}}}$${\ displaystyle a = {\ frac {9} {10}}}$${\ displaystyle q = {\ frac {1} {10}} <1}$${\ displaystyle {\ frac {9} {10}} \ cdot {\ frac {1} {1 - {\ frac {1} {10}}}} = {\ frac {9} {10}} \ cdot { \ frac {1} {\ frac {9} {10}}} = 1}$

#### Other value systems

In other priority systems, the number 9 is replaced by the highest number of the respective system. In the binary system , 1 is the same , in the hexadecimal system is 0, FFF ..., correspondingly in other systems. ${\ displaystyle 0 {,} 111 \ ldots}$

### Other numerals

The roman number for one is  I. In the Hebrew script , the letter aleph ( א) the numerical value of one, in Arabic script its equivalent, the Alif ( ا). The Arabic character for the number one is١; in Bengalî the number is also written ۱ , in Devanagari , in Malayalam and in Chinese一, in Armenian the letter Ա stands for 1.

## Other meanings

שְׁמַ ע יִשְׂרָאֵל יְהוָה אֱלֹהֵינוּ יְהוָה אֶחָד shəma yisrael adonai elohenu adonai echad
“Hear Israel! Adonai (is) our God ; Adonai (is) one "

(Deut 6,4; see Talmud Sukkot 42a and Berachot 13b).

## Linguistic

• Words that express uniqueness can begin with the Greek prefix mono , such as “ monocle ” or “ monograph ”, or they are derived from the Latin singularis or solus , such as “ singular ” or “ solo ”.
• Words that express uniformity can be derived from the Latin unus , for example " Union " or " Uniform ". Words that represent uniqueness such as " unique " or "plain colors" are derived from the Latin unus .
• If reference is made to the ranking or sequence, the Latin stem prim- is used, for example for " primus " or " prime number ".