"१ (संख्या)" च्या विविध आवृत्यांमधील फरक

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बराच मोठ्या प्रमाणात असलेला परभाषिक मजकूर वगळला.
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== आकडा म्हणून ==
One, sometimes referred to as '''unity''', is the [[integer]] before [[2 (number)|two]] and after [[0 (number)|zero]]. One is the first non-zero number in the [[natural number]]s as well as the first [[odd number]] in the natural numbers.
 
Any number multiplied by one is the number, as one is the [[Identity element|identity]] for multiplication. As a result, one is its own [[factorial]], its own [[Square (algebra)|square]], its own [[Cube (algebra)|cube]], and so on. One is also the [[empty product]], as any number multiplied by one is itself, which produces the same result as multiplying by no numbers at all.
 
== अंक म्हणून ==
[[चित्र:Evolution1glyph.svg|thumb|left]]
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 [[Indian subcontinent|Indians]], who wrote 1 as a horizontal line, much like the [[Written Chinese|Chinese]] character [[一]]. The [[Gupta script|Gupta]] wrote it as a curved line, and the [[Nagari]] sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the [[Gujarati language|Gujarati]] and [[Punjabi language|Punjabi]] scripts). The [[Nepali language|Nepali]] also rotated it to the right but kept the circle small.<ref>Georges Ifrah, ''The Universal History of Numbers: From Prehistory to the Invention of the Computer'' transl. David Bellos et al. London: The Harvill Press (1998): 392, Fig. 24.61</ref> This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the [[Roman numeral]] <math>\mathrm{I}</math>. In some European countries (e.g., [[Germany]]), the little 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 for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.
 
[[चित्र:TextFigs148.svg]].
While the shape of the 1 character has an [[Ascender (typography)|ascender]] in most modern [[typeface]]s, in typefaces with [[text figures]], the character usually is of [[x-height]], as, for example, in [[चित्र:TextFigs148.svg]].
 
[[चित्र:Clock 24 J.jpg|thumb|The 24-hour tower clock in [[Venice]], using ''J'' as a symbol for 1]]
Many older typewriters do not have a separate symbol for ''1'' and use the lowercase ''l'' instead. It is possible to find cases when the uppercase ''J'' is used, while it may be for decorative purposes.
 
== गणित ==
Mathematically, 1 is
* in [[arithmetic]] ([[algebra]]) and [[calculus]], the [[natural number]] that follows [[0 (number)|0]] and precedes [[2 (number)|2]] and the multiplicative [[identity (mathematics)|identity]] of the [[integer]]s, [[real number]]s and [[complex number]]s;
* more generally, in [[abstract algebra]], the multiplicative identity ("unity"), usually of a [[ring (mathematics)|ring]].
 
One cannot be used as the base of a positional [[numeral system]]; sometimes [[tally mark|tallying]] is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation.
 
The [[logarithm]]s base 1 are undefined, since the function 1<sup>''x''</sup> always equals 1 and so has no unique [[inverse function|inverse]].
 
In the real-number system, 1 can be represented in two ways as a [[recurring decimal]]: as 1.000... and as [[0.999...]] (''q.v.'').
 
Formalizations of the natural numbers have their own representations of 1:
* in the [[Peano axioms]], 1 is the successor of 0;
* in [[Principia Mathematica]], 1 is defined as the set of all [[singleton (mathematics)|singletons]] (sets with one element);
* in the [[Von Neumann cardinal assignment]] of natural numbers, 1 is defined as the [[Set (mathematics)|set]] {0}.
 
In a multiplicative [[group (mathematics)|group]] or [[monoid]], the [[identity element]] is sometimes denoted ''1'', but ''e'' (from the German ''Einheit'', "unity") is more 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 (algebra)|characteristic]] ''n'' not equal to 0, the element called 1 has the property that ''n''1 = 1''n'' = 0 (where this 0 is the additive identity of the ring). Important examples are general [[Field (mathematics)|fields]].
 
One 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 equations, numeric values are typically ''normalized''<!--need a good link for this--> to fall within the [[unit interval]] from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters.
 
Because of the multiplicative identity, if ''f''(''x'') is a [[multiplicative function]], then ''f''(1) must equal 1.
 
It is also the first and second numbers in the [[Fibonacci number|Fibonacci]] sequence and is the first number in many other mathematical sequences. As a matter of convention, Sloane's early ''Handbook of Integer Sequences'' added an initial 1 to any sequence that did not already have it and considered these initial 1's in its lexicographic ordering. Sloane's later ''Encyclopedia of Integer Sequences'' and its Web counterpart, the ''[[On-Line Encyclopedia of Integer Sequences]]'', ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.
 
One is neither a [[prime number]] nor a [[composite number]], but a [[unit (ring theory)|unit]], like -1 and, in the [[Gaussian integers]], ''[[imaginary unit|i]]'' and -''i''. The [[fundamental theorem of arithmetic]] guarantees [[factorization|unique factorization]] over the integers only up to units (e.g., 4 = 2<sup>2</sup> = (-1)<sup>6</sup>×1<sup>23</sup>×2<sup>2</sup>).
 
The definition of a [[field (mathematics)|field]] requires that 1 must not be equal to [[zero|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.
 
One 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). One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units. The last professional [[mathematician]] to publicly label 1 a prime number was [[Henri Lebesgue]] in 1899.
 
One is one of three possible values of the [[Möbius function]]: it takes the value one for [[square-free integer]]s with an even number of distinct prime factors.
 
One is the only odd number in the range of [[Euler's totient function]] φ(''x''), in the cases ''x'' = 1 and ''x'' = 2.
 
One is the only 1-perfect number (see [[multiply perfect number]]).
 
By definition, 1 is the [[magnitude (mathematics)|magnitude]] or [[absolute value]] of a [[unit vector]] and a [[identity matrix|unit matrix]] (more usually called an identity matrix). Note that the term ''unit matrix'' is sometimes used to mean something [[Matrix of ones|quite different]].
 
By definition, 1 is the [[probability]] of an event that is [[almost certain]] to occur.
 
One is the most common leading digit in many sets of data, a consequence of [[Benford's law]].
 
The [[ancient Egypt]]ians represented all fractions (with the exception of 2/3 and 3/4) in terms of sums of fractions with [[numerator]] 1 and distinct [[denominator]]s. For example, <math>\frac{2}{5} = \frac{1}{3} + \frac{1}{15}</math>. Such representations are popularly known as [[Egyptian Fractions]] or [[Unit Fractions]]. <!-- Historical. -->
 
The [[Generating Function]] that has all coefficients 1 is given by
 
<math>\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots</math>.
 
This power series converges and has finite value [[if and only if]], <math>| x | < 1</math>. <!-- Probably not needed, really should look at convergence theorem as there are other series that converge iff |x| < 1 -->
 
=== आकडेमोडीचे मूळ कोष्टक ===
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Line ३४४ ⟶ २८८:
== तंत्रज्ञानात ==
[[चित्र:U+2673 DejaVu Sans.svg|50px|right|1 as a resin identification code, used in recycling]]
* The [[resin identification code]] used in recycling to identify [[polyethylene terephthalate]]
* Used in [[binary code]] along with [[0 (number)|0]]
 
== शास्त्र ==
"https://mr.wikipedia.org/wiki/१_(संख्या)" पासून हुडकले