1024 (number)

← 1023 1024 1025 →
List of numbers ; Integers ; ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	one thousand twenty-four
Ordinal	1024th; (one thousand twenty-fourth)
Factorization	210
Divisors	1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Greek numeral	,ΑΚΔ´
Roman numeral	MXXIV
Binary	100000000002
Ternary	11012213
Senary	44246
Octal	20008
Duodecimal	71412
Hexadecimal	40016

Last updated March 19, 2024

1024 is the natural number following 1023 and preceding 1025.

Enumeration of groups

The number of groups of order 1024 is 49487367289, up to isomorphism.^[4] An earlier calculation gave this number as 49487365422,^[5]^[6] but in 2021 this was shown to be in error.^[4]

This count is more than 99% of all the isomorphism classes of groups of order less than 2000.^[7]

Approximation to 1000

The neat coincidence that 2¹⁰ is nearly equal to 10³ provides the basis of a technique of estimating larger powers of 2 in decimal notation. Using 2^10a+b ≈ 2^b10^3a(or 2^a≈2^{a mod 10}10^floor(a/10) if "a" stands for the whole power) is fairly accurate for exponents up to about 100. For exponents up to 300, 3a continues to be a good estimate of the number of digits.

For example, 2⁵³ ≈ 8×10¹⁵. The actual value is closer to 9×10¹⁵.

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

{\begin{aligned}{\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{aligned}}

In measuring bytes, 1024 is often used in place of 1000 as the quotients of the units byte, kilobyte, megabyte, etc. In 1999, the IEC coined the term kibibyte for multiples of 1024, with kilobyte being used for multiples of 1000.

Special use in computers

In binary notation, 1024 is represented as 10000000000, making it a simple round number occurring frequently in computer applications.

1024 is the maximum number of computer memory addresses that can be referenced with ten 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 proofing. Most used languages codes in RTF are integers slightly over 1024.

1024×768 pixels and 1280×1024 pixels are common standards of display resolution.

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

Related Research Articles

The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit of memory in many computer architectures. To disambiguate arbitrarily sized bytes from the common 8-bit definition, network protocol documents such as the Internet Protocol refer to an 8-bit byte as an octet. Those bits in an octet are usually counted with numbering from 0 to 7 or 7 to 0 depending on the bit endianness.

A binary prefix is a unit prefix that indicates a multiple of a unit of measurement by an integer power of two. The most commonly used binary prefixes are kibi (symbol Ki, meaning 2¹⁰ = 1024), mebi (Mi, 2²⁰ = 1048576), and gibi (Gi, 2³⁰ = 1073741824). They are most often used in information technology as multipliers of bit and byte, when expressing the capacity of storage devices or the size of computer files.

<span class="mw-page-title-main">Gigabyte</span> Unit of digital information

The gigabyte is a multiple of the unit byte for digital information. The prefix giga means 10⁹ in the International System of Units (SI). Therefore, one gigabyte is one billion bytes. The unit symbol for the gigabyte is GB.

<span class="mw-page-title-main">Isomorphism</span> In mathematics, invertible homomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσοςisos "equal", and μορφήmorphe "form" or "shape".

The kilobyte is a multiple of the unit byte for digital information. The International System of Units (SI) defines the prefix kilo as a multiplication factor of 1000 (10³); therefore, one kilobyte is 1000 bytes. The internationally recommended unit symbol for the kilobyte is kB.

Kilo is a decimal unit prefix in the metric system denoting multiplication by one thousand (10³). It is used in the International System of Units, where it has the symbol k, in lowercase.

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$ . For example, since $1000 = 10 3$ , the logarithm base 10 of $1000$ is $3$ , or $log 10 (1000) = 3$ . The logarithm of $x$ to base $b$ is denoted as $log b (x)$ , or without parentheses, $log b x$ , or even without the explicit base, $log x$ , when no confusion is possible, or when the base does not matter such as in big O notation.

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

<span class="mw-page-title-main">Exponentiation</span> Arithmetic operation

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as $b n$ , where $b$ is the base and $n$ is the power; this is pronounced as " $b$ (raised) to the $n$ ". When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b n$ is the product of multiplying $n$ bases:

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

The tables contain the prime factorization of the natural numbers from 1 to 1000.

A power of two is a number of the form $2 n$ where $n$ is an integer, that is, the result of exponentiation with number two as the base and integer $n$ as the exponent.

An order of magnitude is usually a factor of ten. Thus, four orders of magnitude is a factor of 10,000 or 10⁴.

A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.

A unit prefix is a specifier or mnemonic that is prepended to units of measurement to indicate multiples or fractions of the units. Units of various sizes are commonly formed by the use of such prefixes. The prefixes of the metric system, such as kilo and milli, represent multiplication by positive or negative powers of ten. In information technology it is common to use binary prefixes, which are based on powers of two. Historically, many prefixes have been used or proposed by various sources, but only a narrow set has been recognised by standards organisations.

In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

693 is the natural number following 692 and preceding 694.

This timeline of binary prefixes lists events in the history of the evolution, development, and use of units of measure which are germane to the definition of the binary prefixes by the International Electrotechnical Commission (IEC) in 1998, used primarily with units of information such as the bit and the byte.

An order of magnitude is generally a factor of ten. A quantity growing by four orders of magnitude implies it has grown by a factor of 10000 or 10⁴. However, because computers are binary, orders of magnitude are sometimes given as powers of two.

References

↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 170
↑ Sloane, N. J. A. (ed.). "SequenceA002620". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-21.
↑ Denis Roegel. (2013). A reconstruction of Bürger's table of quarter-squares (1817) (Research Report). Lyons: HAL. p. 18. S2CID 202132792
1 2 Burrell, David (2021-12-08). "On the number of groups of order 1024". Communications in Algebra. 50 (6): 2408–2410. doi:10.1080/00927872.2021.2006680. MR 4413840. S2CID 244772374.
↑ "Numbers of isomorphism types of finite groups of given order". www.icm.tu-bs.de. Archived from the original on 2019-07-25. Retrieved 2017-04-05.
↑ Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2002), "A millennium project: constructing small groups", International Journal of Algebra and Computation, 12 (5): 623–644, doi:10.1142/S0218196702001115, MR 1935567, S2CID 31716675
↑ Paolo, Aluffi (2009). Algebra: Chapter 0. American Mathematical Society. ISBN 9780821847817.

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[1] Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 170

[2] Sloane, N. J. A. (ed.). "SequenceA002620". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-21.

[3] Denis Roegel. (2013). A reconstruction of Bürger's table of quarter-squares (1817) (Research Report). Lyons: HAL. p. 18. S2CID 202132792

[Burrell-4] 1 2 Burrell, David (2021-12-08). "On the number of groups of order 1024". Communications in Algebra. 50 (6): 2408–2410. doi:10.1080/00927872.2021.2006680. MR 4413840. S2CID 244772374.

[5] "Numbers of isomorphism types of finite groups of given order". www.icm.tu-bs.de. Archived from the original on 2019-07-25. Retrieved 2017-04-05.

[6] Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2002), "A millennium project: constructing small groups", International Journal of Algebra and Computation, 12 (5): 623–644, doi:10.1142/S0218196702001115, MR 1935567, S2CID 31716675

[7] Paolo, Aluffi (2009). Algebra: Chapter 0. American Mathematical Society. ISBN 9780821847817.

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