1024 (number)

Enumeration of groups

The number of groups of order 1024 is , up to isomorphism. An earlier calculation gave this number as , but in 2021 this was shown to be in error.

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

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 210a+b ≈ 2b103a(or 2a≈2a mod 1010floor(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{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}

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.

References

References

1. Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 170
2. {{Cite OEIS. A002620
3. Denis Roegel.. (2013). "A reconstruction of Bürger's table of quarter-squares (1817).". [[HAL (open archive).
4. (2021-12-08). "On the number of groups of order 1024". Communications in Algebra.
5. "Numbers of isomorphism types of finite groups of given order".
6. (2002). "A millennium project: constructing small groups". International Journal of Algebra and Computation.
7. (2009). "Algebra: Chapter 0". American Mathematical Society.