Talk:Power of two

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Powers of two whose exponents are powers of two

Because data (specifically integers) and the addresses of data are stored using the same hardware, and the data is stored in one or more octets ( $23$ ), double exponentials of two are common. The first 24 of them are:


$n$	$2 n$	$2 2 n$ (sequence A001146 in the OEIS)
0	1	2
1	2	4
2	4	16
3	8	256
4	16	65,536
5	32	4,294,967,296
6	64	18,446,744,073,709,551,616 (20 digits)
7	128	340,282,366,920,938,463,463,374,607,431,768,211,456 (39 digits)
8	256	115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936 (78 digits)
9	512	13,407,807,929,942,597,099,574,02...1,946,569,946,433,649,006,084,096 (155 digits)
10	1,024	179,769,313,486,231,590,772,930,5...6,304,835,356,329,624,224,137,216 (309 digits)
11	2,048	32,317,006,071,311,007,300,714,87...8,193,555,853,611,059,596,230,656 (617 digits)
12	4,096	1,044,388,881,413,152,506,691,752,...0,243,804,708,340,403,154,190,336 (1,234 digits)
13	8,192	1,090,748,135,619,415,929,462,984,...1,997,186,505,665,475,715,792,896 (2,467 digits)
14	16,384	1,189,731,495,357,231,765,085,759,...2,460,447,027,290,669,964,066,816 (4,933 digits)
15	32,768	1,415,461,031,044,954,789,001,553,...7,541,122,668,104,633,712,377,856 (9,865 digits)
16	65,536	2,003,529,930,406,846,464,979,072,...2,339,445,587,895,905,719,156,736 (19,729 digits)
17	131,072	4,014,132,182,036,063,039,166,060,...1,850,665,812,318,570,934,173,696 (39,457 digits)
18	262,144	16,113,257,174,857,604,736,195,72...0,753,862,605,349,934,298,300,416 (78,914 digits)
19	524,288	259,637,056,783,100,077,612,659,6...1,369,814,364,528,226,185,773,056 (157,827 digits)
20	1,048,576	67,411,401,254,990,734,022,690,65...2,009,289,119,068,940,335,579,136 (315,653 digits)
21	2,097,152	4,544,297,019,161,366,309,996,159,...5,826,312,131,885,036,518,506,496 (631,306 digits)
22	4,194,304	20,650,635,398,358,879,243,991,19...6,933,296,051,236,698,394,198,016 (1,262,612 digits)
23	8,388,608	426,448,742,355,952,787,243,272,8...7,419,485,551,374,411,818,336,256 (2,525,223 digits)

Also see tetration and lower hyperoperations.

Last digits for powers of two whose exponents are powers of two

All of these numbers end in 6. Starting with 16 the last two digits are periodic with period 4, with the cycle 16–56–36–96–, and starting with 16 the last three digits are periodic with period 20. These patterns are generally true of any power, with respect to any base. The pattern continues where each pattern has starting point $2 k$ , and the period is the multiplicative order of 2 modulo $5 k$ , which is $φ (5 k) = 4 \times 5 k -1$ (see Multiplicative group of integers modulo n).^{[citation needed]}

Facts about powers of two whose exponents are powers of two

In a connection with nimbers, these numbers are often called Fermat 2-powers.

The numbers $2^{2^{n}}$ form an irrationality sequence: for every sequence $x_{i}$ of positive integers, the series

\sum _{i=0}^{\infty }{\frac {1}{2^{2^{i}}x_{i}}}={\frac {1}{2x_{0}}}+{\frac {1}{4x_{1}}}+{\frac {1}{16x_{2}}}+\cdots

converges to an irrational number. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known.^[1]

References

^ Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, p. 346, ISBN 0-387-20860-7, Zbl 1058.11001, archived from the original on 2016-04-28

Powers of two whose exponents are powers of two in computer science

Several of these numbers represent the number of values representable using common computer data types. For example, a 32-bit word consisting of 4 bytes can represent $232$ distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to $232 - 1$ , or as the range of signed numbers between $-2 31$ and $231 - 1$ . For more about representing signed numbers see two's complement. Faster328 (talk) 07:09, 13 April 2023 (UTC)[reply]

I improved the table, seperated sections, and added colour. Faster328 (talk) 07:12, 13 April 2023 (UTC)[reply]

Trim the table. Numbers-Mathworld (talk) 07:17, 13 April 2023 (UTC)[reply]

You can use multiply and calculator, not use table and have infinite integer.

124.122.238.117 (talk) — Preceding undated comment added 13:54, 12 September 2023 (UTC)[reply]

Powers of 1024

The section #Powers of 1024 has the following:

It takes approximately 17 powers of 1024 to reach 50% deviation and approximately 29 powers of 1024 to reach 100% deviation of the same powers of 1000.^{[citation needed]}

Does this really need a citation? The calculation is very easily done:

\log _{1024/1000}1.5\approx 17.1

\log _{1024/1000}2\approx 29.2

Kuulopuhe (talk) 01:24, 17 December 2023 (UTC)[reply]

I added this calculation in a footnote and took out the cn template. –jacobolus (t) 01:40, 17 December 2023 (UTC)[reply]

Thank you! Kuulopuhe (talk) 00:43, 18 December 2023 (UTC)[reply]

Number of unique states in one unit of data

I recently added a note to 2^1024 that it was the range of possible states in one kibibit (kilobit), as well as a similar note for 2^8192 and one kibibyte (kilobyte). Those edits were reverted by user:David Eppstein with the following edit summary:

“gibidigibidigibidi. Nobody uses fixed-precision arithmetic with this many bits, and if they did the max value would be half what you state it is, minus one”

My edit was not talking about the max value, it said “range of possible values”. Even if it were to be misunderstood as talking about max value, there is no reason to assume that it is referring to a signed integer. I assert that it is in fact correct that one kibibit can have 2^1024 possible unique values, and one kibibyte can have 2^8192.

Regarding the comment that “Nobody uses fixed-precision arithmetic with this many bits”, it is certainly true that there are more efficient ways to store numbers. However, needless to say numbers aren't the only type of data that can be represented with binary data. Having a sense of scale for how many different states are possible at a common data size is useful.

As for the “gibidigibidigibidi”, this seems to be referring to the user's dislike of the aesthetics of the standard names for binary prefixes, which is not relevant. Pinball larry (talk) 22:32, 2 March 2024 (UTC)[reply]

The range of possible values is an interval, not a number. Perhaps you mean "the number of distinct values"? Also, I dispute that "these are the standard names". A standards body has declared them to be the names, but very few others actually use them. —David Eppstein (talk) 22:56, 2 March 2024 (UTC)[reply]

Negative integer exponents

Is there anything in sources about negative exponents, it numbers 1/2, 1/4, 1/8...? Or shal the lede speak about nonnegative integer exponents? Is there a ref for the def? (I mean, to establish that usually nonnegative exp is meant (that is my impression, but {{citation needed}} for either way :-). - Altenmann >talk 20:41, 29 March 2024 (UTC)[reply]

Sure, there are plenty of sources using the phrase "negative power of two". I think it would be fine to add a section about this if you feel motivated. You could start by looking at binary logarithm. –jacobolus (t) 21:34, 29 March 2024 (UTC)[reply]

If I said simply that a number is a power of two, I think I would likely mean a non-negative exponent unless otherwise qualified. If I meant a negative-integer power I would probably say something like "inverse power of two" or "negative power of two". I definitely would not count

2^{\log _{2}3}

as a power of two. But as you say, adding clarifications about this requires sourcing. —David Eppstein (talk) 21:44, 29 March 2024 (UTC)[reply]

Sure, the numbers 1, 2, 4, 8, ... are clearly the primary focus of this article. But if we wanted to add a section called "Negative powers of two" about 1/2, 1/4, 1/8, ..., and briefly mention it in the lead section, I think it would be plausibly in scope, not particularly distracting, and potentially helpful to readers. I certainly agree with you that the number 3 doesn't seem like a "power of two". –jacobolus (t) 22:13, 29 March 2024 (UTC)[reply]

plenty of sources -- fooled by google. Only a handgful unique hits. Anyway, I found one nontrivial.- Altenmann >talk 21:50, 29 March 2024 (UTC)[reply]

P.S. thre is also something to write about computer representations, like exact representation of decimal fractions such as 1/625, but I dont "feel motivated" enough (Must be bad weather :-). - Altenmann >talk 22:02, 29 March 2024 (UTC)[reply]

I did a search in google scholar, which found nearly 400 examples of "negative power of two". –jacobolus (t) 22:10, 29 March 2024 (UTC)[reply]

The part about exact computer representation really belongs under related article dyadic rational, where it is already mentioned in section "In computing". —David Eppstein (talk) 23:47, 29 March 2024 (UTC)[reply]

The current section § Powers of two in music theory is poorly placed/organized in my opinion, and should not be an independent section. I think this article should (near the bottom) have short sections about (a) negative powers of two; (b) fractional powers of two (possibly linking to binary logarithm), which can talk about ISO 216 paper sizes, f-numbers, equally tempered semitones, etc.; (c) dyadic rationals, including music time signatures. These sections can each briefly explain their relationship to powers of two. –jacobolus (t) 03:21, 30 March 2024 (UTC)[reply]

Non-negative and Zero

The first ten powers of 2 for non-negative values of n are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ..

I think the use of the term "non-negative" , should be hypertext ,that , when clicked on sends you to the Wikipedia page titled "Sign(mathematics)" and then to the section called , "Terminology for signs" , where an explanation of the term is given. EuclidIncarnated (talk) 17:18, 26 April 2024 (UTC)[reply]

I partially rewrote the lead section. does that help? –jacobolus (t) 00:13, 27 April 2024 (UTC)[reply]

Slightly , although the hyperlink takes you to the Negative number Wikipedia page whereas there exists a Wikipedia page with a direct explanation of the term "non-negative". Instead we can put "non-negative" in hypertext and when clicked take us to the "Sign(mathematics)" Wikipedia page and the part named "Terminology for signs".

I shall make the edit now but I wanted to run it by the community first to point it out in case the edit was reverted. EuclidIncarnated (talk) 14:50, 27 April 2024 (UTC)[reply]

I think negative number is a more useful/relevant wikilink than Sign (mathematics) § Terminology for signs to attach to the word "negative". –jacobolus (t) 15:10, 27 April 2024 (UTC)[reply]

The negative number page talks of non-negative whole numbers being referred to as natural integers. Whereas the page Sign (mathematics) § Terminology for signs gives a direct explanation of "non-negative", which is not constricted to that of solely whole numbers. Therefore I think a general definition is more suitable than that which talks about only whole numbers. EuclidIncarnated (talk) 23:58, 27 April 2024 (UTC)[reply]

[1] Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, p. 346, ISBN 0-387-20860-7, Zbl 1058.11001, archived from the original on 2016-04-28

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