View source for 1024 (number)

{{infobox number
| number        = 1024
| divisor       = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
| unicode       = 
| greek prefix  = 
| latin prefix  = 
}}
[[File:Leibniz binary system 1697.jpg|thumb|The number 1024 in a treatise on [[binary number]]s by [[Gottfried Wilhelm Leibniz|Leibniz]] (1697)]]
'''1024''' is the [[natural number]] following [[1023 (number)|1023]] and preceding [[1025 (number)|1025]].

1024 is a [[power of two]]: 2{{sup|10}} (2 to the tenth power).<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 170</ref> It is the nearest power of two from [[decimal]] [[1000 (number)|1000]] and [[senary]] 10000{{sub|6}} (decimal [[1296 (number)|1296]]). It is the 64th quarter square.<ref>{{Cite OEIS |A002620 |Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4). |access-date=2024-01-21 }}</ref><ref>{{Cite report |author=Denis Roegel. |title=A reconstruction of Bürger's table of quarter-squares (1817). |url=https://inria.hal.science/hal-00880832/document |type=Research Report |year=2013 |publisher=[[HAL (open archive)|HAL]] |location=Lyons |page=18 }} {{s2cid|202132792}}</ref>

1024 is the smallest number with exactly 11 [[divisor]]s (but there are smaller numbers with more than 11 divisors; e.g., [[60 (number)|60]] has 12 divisors) {{OEIS|A005179}}.

== Enumeration of groups ==
The number of [[group (algebra)|groups]] of [[order of a group|order]] 1024 is {{gaps|49|487|367|289}}, [[up to isomorphism]].<ref name="Burrell">{{cite journal |last1=Burrell |first1=David |title=On the number of groups of order 1024 |journal=Communications in Algebra |date=2021-12-08 |volume=50 |issue=6 |pages=2408–2410 |doi=10.1080/00927872.2021.2006680 |s2cid=244772374 |mr=4413840 |url=https://www.tandfonline.com/doi/full/10.1080/00927872.2021.2006680}}</ref> An earlier calculation gave this number as {{gaps|49|487|365|422}},<ref>{{Cite web|url=http://www.icm.tu-bs.de/ag_algebra/software/small/number.html|title=Numbers of isomorphism types of finite groups of given order|website=www.icm.tu-bs.de|archive-url=https://web.archive.org/web/20190725032846/http://www.icm.tu-bs.de/ag_algebra/software/small/number.html|language=en|access-date=2017-04-05|archive-date=2019-07-25}}</ref><ref>{{Citation | last1=Besche | first1=Hans Ulrich | last2=Eick | first2=Bettina | last3=O'Brien | first3=E. A. | title=A millennium project: constructing small groups | mr=1935567 | year=2002 | journal=International Journal of Algebra and Computation | volume=12 | issue=5 | pages=623–644 | doi=10.1142/S0218196702001115| s2cid=31716675 }}</ref> but in 2021 this was shown to be in error.<ref name="Burrell"/>

This count is more than 99% of all the isomorphism classes of groups of order less than 2000.<ref>{{cite book |last1=Paolo |first1=Aluffi |title=Algebra: Chapter 0 |date=2009 |publisher=American Mathematical Society |isbn=9780821847817}}</ref>

== Approximation to 1000 ==
{{see also|Binary prefix}}
The neat coincidence that 2<sup>10</sup> is nearly equal to [[1000 (number)|10<sup>3</sup>]] provides the basis of a technique of estimating larger powers of 2 in decimal notation. Using 2<sup>10''a''+''b''</sup> ≈ 2<sup>''b''</sup>10<sup>3''a''</sup>(or 2<sup>a</sup>≈2<sup>a mod 10</sup>10<sup>floor(a/10)</sup> if "a" stands for the whole power) is fairly accurate for exponents up to about 100. For exponents up to 300, 3''a'' continues to be a good estimate of the number of digits.

For example, 2<sup>53</sup> ≈ 8×10<sup>15</sup>. The actual value is closer to 9×10<sup>15</sup>.

In the case of larger exponents, the relationship becomes increasingly inaccurate, with errors exceeding an order of magnitude for ''a'' ≥ 97. For example:

:<math>\begin{align}
\frac{2^{1000}}{10^{300}}
&= \exp \left( \ln \left( \frac{2^{1000}}{10^{300}} \right) \right) \\
&= \exp \left( \ln \left( 2^{1000}\right) - \ln\left(10^{300}\right)\right)\\
&\approx \exp\left(693.147-690.776\right)\\
&\approx \exp\left(2.372\right)\\
&\approx 10.72
\end{align}</math>

In measuring [[byte]]s, 1024 is often used in place of 1000 as the quotients of the units [[byte]], [[kilobyte]], [[megabyte]], etc. In 1999, the [[International Electrotechnical Commission|IEC]] coined the term [[kibibyte]] for multiples of 1024, with kilobyte being used for multiples of 1000.

== Special use in computers ==
In [[binary number|binary notation]], 1024 is represented as 10000000000, making it a simple [[round number]] occurring frequently in [[Computer science|computer]] [[Application software|applications]].

1024 is the maximum number of computer [[memory address]]es that can be referenced with ten [[bit|binary switches]]. This is the origin of the organization of computer memory into 1024-byte chunks or [[kibibytes]].

In the [[Rich Text Format]] (RTF), language code 1024 indicates the text is not in any language and should be skipped over when [[spell checker|proofing]]. Most used languages codes in RTF are integers slightly over 1024.

1024×[[768 (number)|768]] pixels and [[1280 (number)|1280]]×1024 pixels are common [[Display resolution#Current standards|standards of display resolution]].

1024 is the lowest non-system and non-reserved port number in [[Internet_protocol_suite|TCP/IP networking]]. Ports above this number can usually be opened for listening by non-superusers.

== See also ==
* [[Powers of 1024]]

== References ==
{{Reflist}}<!--added above categories/infobox footers by script-assisted edit-->
{{Integers|10}}
[[Category:Integers]]