| 2048 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| All Numbers | |||||||||
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2,048 is a positive integer following 2,047 and preceding 2,049. It is the 11th power of 2, and equal to f2(8) and f3(2) in the fast-growing hierarchy.
It is also the largest known power of 2 where all digits are even.[1]
Furthermore, it is the smallest number, for which the number of groups with this order is currently unknown.
It is also called “giga” by André Joyce.[2]
HaydenTheGoogologist2009 coined What about me?! for this number, and it is the 4th number of the Meet yourself in 105 degrees Celsius series.[3]
Properties[]
- Its factors are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 and 2048, making it a composite number.[4][5][6]
- 2048 is an even number[7][8] .
- 2048 is an unhappy number.[9][10]
- 2048 is deficient.[11]
- Its prime factorization is 211.
- 2048 is the name of a web game where you slide tiles representing powers of 2 across a board and try to combine them to reach higher and higher powers of 2.[12] The largest tile you can reach is 131,072.[13]
Approximations[]
| Notation | Lower bound | Upper bound | |
|---|---|---|---|
| Up-arrow notation | 45 ↑ 2 | ||
| Scientific notation | 2.048 x 103 | 2.049 x 103 | |
| Slow-growing hierarchy | \(g_{\omega^{\omega}}(4)\) | \(g_{\omega^{\omega}}(5)\) | |
| Copy notation | 20[2] | 21[2] | |
| Chained arrow notation | 45 → 2 | ||
| Bowers' Exploding Array Function/Bird's array notation | {45,2} | ||
| Fast-growing hierarchy | f2(7) | f2(8) | |
| Hardy hierarchy | Hω(1024) | Hω(1024) | |
| Middle-growing hierarchy | m(ω,11) | m(ω,11) | |
| Hyper-E notation | E3.3113 | ||
| Hyper-E notation (non-10 base) | \(E[45]2\) | ||
| Hyperfactorial array notation | 6! | 7! | |
| X-Sequence Hyper-Exponential Notation | 45{1}2 | ||
| Steinhaus-Moser Notation | 4[3] | 5[3] | |
| PlantStar's Debut Notation | [1] | [2] | |
| H* function | H(0.1) | H(0.2) | |
| Bashicu matrix system with respect to version 4 | (0)[45] | (0)[46] | |
| m(n) map | m(1)(4) | m(1)(5) | |
| s(n) map | \(s(1)(\lambda x . x+1)(1023)\) | \(s(1)(\lambda x . x+1)(1024)\) | |
Sources[]
- ↑ Pointless Gigantic List of Numbers - Part 1 (0 ~ 1,000,000) - Pointless Large Number Stuff
- ↑ http://michaelhalm.tripod.com/andre_joyce_s_coined_words.htm
- ↑ Hayden's Big Numbers - Meet yourself in 105 degrees Celsius. Retrieved 2023-01-18.
- ↑ OEIS A002808 - Composite numbers
- ↑ Wolfram Alpha Is 2048 composite?
- ↑ Wolfram Alpha 2048's factors
- ↑ OEIS A005843 - Even numbers
- ↑ Wolfram Alpha Is 2048 even?
- ↑ Wolfram Alpha Unhappy Numbers
- ↑ OEIS A031177 - Unhappy Numbers
- ↑ OEIS A005100 - Deficient numbers
- ↑ https://www.washingtonpost.com/news/arts-and-entertainment/wp/2014/04/23/everything-you-ever-wanted-to-know-about-2048-the-internets-latest-impossible-hit-game/
- ↑ https://cupola.gettysburg.edu/cgi/viewcontent.cgi?article=1025&context=csfac#:~:text=Beyond%20this%20primary%20goal%2C%20however,would%20be%20improbable%20to%20achieve.