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Large numbers

Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. Googology is the study of nomenclature and properties of large numbers.

In the everyday world
Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, or a 1 followed by nine zeros: 1 000 000 000. The reciprocal, 1.0 × 10−9, means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort, and hazard of counting a long series of zeros to see how large the number is. In addition to scientific (powers of 10) notation, the following examples include (short scale) systematic nomenclature of large numbers.

Examples of large numbers describing everyday real-world objects include:
 * The number of cells in the human body (estimated at 3.72 × 1013), or 37.2 trillion
 * The number of bits on a computer hard disk (, typically about 1013, 1–2 TB), or 10 trillion
 * The number of neuronal connections in the human brain (estimated at 1014), or 100 trillion
 * The Avogadro constant is the number of “elementary entities” (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 – approximately $6.022$, or 602.2 sextillion.
 * The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3 ± 3.6) × 1037, or 53±36 undecillion
 * The mass of Earth consists of about 4 × 1051, or 4 sexdecillion, nucleons
 * The estimated number of atoms in the observable universe (1080), or 100 quinvigintillion
 * The lower bound on the game-tree complexity of chess, also known as the “Shannon number” (estimated at around 10120), or 1 novemtrigintillion
 * Note that this value of the Shannon number is for Standard Chess. It has even larger values for larger-board chess variants such as Grant Acedrex, Tai Shogi, and Taikyoku Shogi.

Astronomical
Other large numbers, as regards length and time, are found in astronomy and cosmology. For example, the current Big Bang model suggests that the universe is 13.8 billion years (4.355 × 1017 seconds) old, and that the observable universe is 93 billion light years across (8.8 × 1026 metres), and contains about 5 × 1022 stars, organized into around 125 billion (1.25 × 1011) galaxies, according to Hubble Space Telescope observations. There are about 1080 atoms in the observable universe, by rough estimation.

According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is

 * $$10^{10^{10^{10^{10^{1.1}}}}} \mbox{ years}$$

which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses. This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.

Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations on a set of fixed objects, grows very rapidly with the number of objects. Stirling's formula gives a precise asymptotic expression for this rate of growth.

Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms.

Gödel numbers, and similar numbers used to represent bit-strings in algorithmic information theory, are very large, even for mathematical statements of reasonable length. However, some pathological numbers are even larger than the Gödel numbers of typical mathematical propositions.

Logician Harvey Friedman has done work related to very large numbers, such as with Kruskal's tree theorem and the Robertson–Seymour theorem.

"Billions and billions"
To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b". Sagan never did, however, say "billions and billions". The public's association of the phrase and Sagan came from a Tonight Show skit. Parodying Sagan's effect, Johnny Carson quipped "billions and billions". The phrase has, however, now become a humorous fictitious number—the Sagan. Cf., Sagan Unit.

Examples

 * googol = $$10^{100}$$
 * centillion = $$10^{303}$$ or $$10^{600}$$, depending on number naming system
 * millinillion = $$10^{3003}$$ or $$10^{6000}$$, depending on number naming system
 * The largest known Smith number = (101031−1) × (104594 + 3 + 1)1476
 * The largest known Mersenne prime = $$2^{82,589,933}-1$$
 * googolplex = $$10^{\text{googol}}=10^{10^{100}}$$
 * Skewes's numbers: the first is approximately $$10^{10^{10^{34}}}$$, the second $$10^{10^{10^{964}}}$$
 * Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation.
 * Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number.
 * Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007.

Standardized system of writing
A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.

To compare numbers in scientific notation, say 5×104 and 2×105, compare the exponents first, in this case 5 > 4, so 2×105 > 5×104. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×104 > 2×104 because 5 > 2.

Tetration with base 10 gives the sequence $$10 \uparrow \uparrow n=10 \to n \to 2=(10\uparrow)^n 1$$, the power towers of numbers 10, where $$(10\uparrow)^n$$ denotes a functional power of the function $$f(n)=10^n$$ (the function also expressed by the suffix "-plex" as in googolplex, see the googol family).

These are very round numbers, each representing an order of magnitude in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.

More precisely, numbers in between can be expressed in the form $$(10\uparrow)^n a$$, i.e., with a power tower of 10s, and a number at the top, possibly in scientific notation, e.g. $$10^{10^{10^{10^{10^{4.829}}}}} = (10\uparrow)^5 4.829$$, a number between $$10\uparrow\uparrow 5$$ and $$10\uparrow\uparrow 6$$ (note that $$10 \uparrow\uparrow n < (10\uparrow)^n a < 10 \uparrow\uparrow (n+1)$$ if $$ 1 < a < 10$$). (See also extension of tetration to real heights.)

Thus googolplex is $$10^{10^{100}} = (10\uparrow)^2 100 = (10\uparrow)^3 2$$

Another example:
 * $$2 \uparrow\uparrow\uparrow 4 =

\begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\ \qquad\quad\ \ \ 65,536\mbox{ copies of }2 \end{matrix} \approx (10\uparrow)^{65,531}(6 \times 10^{19,728}) \approx (10\uparrow)^{65,533} 4.3 $$ (between $$10\uparrow\uparrow 65,533$$ and $$10\uparrow\uparrow 65,534$$)

Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (n) one has to take the $$log_{10}$$ to get a number between 1 and 10. Thus, the number is between $$10\uparrow\uparrow n$$ and $$10\uparrow\uparrow (n+1)$$. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 1010, or the next, between 0 and 1.

Note that
 * $$10^{(10\uparrow)^{n}x}=(10\uparrow)^{n}10^x$$

I.e., if a number x is too large for a representation $$(10\uparrow)^{n}x$$ the power tower can be made one higher, replacing x by log10x, or find x from the lower-tower representation of the log10 of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).

If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g. $$10\uparrow\uparrow(7.21\times 10^8)$$) can be used. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.

Examples:
 * $$10\uparrow\uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}$$ (between $$10\uparrow\uparrow\uparrow 2$$ and $$10\uparrow\uparrow\uparrow 3$$)
 * $$10\uparrow\uparrow 10\uparrow\uparrow (10\uparrow)^{497}(9.73\times 10^{32})=(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})$$ (between $$10\uparrow\uparrow\uparrow 4$$ and $$10\uparrow\uparrow\uparrow 5$$)

Similarly to the above, if the exponent of $$(10\uparrow)$$ is not exactly given then giving a value at the right does not make sense, and instead of using the power notation of $$(10\uparrow)$$, it is possible to add $$1$$ to the exponent of $$(10\uparrow\uparrow)$$, to obtain e.g. $$(10\uparrow\uparrow)^{3} (2.8\times 10^{12})$$.

If the exponent of $$(10\uparrow \uparrow)$$ is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and instead of using the power notation of $$(10\uparrow \uparrow)$$ it is possible use the triple arrow operator, e.g. $$10\uparrow\uparrow\uparrow(7.3\times 10^{6})$$.

If the right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g. $$10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})$$ (between $$10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 4$$ and $$10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 5$$). This can be done recursively, so it is possible to have a power of the triple arrow operator.

Then it is possible to proceed with operators with higher numbers of arrows, written $$\uparrow^n$$.

Compare this notation with the hyper operator and the Conway chained arrow notation:
 * $$a\uparrow^n b$$ = ( a → b → n ) = hyper(a, n + 2, b)

An advantage of the first is that when considered as function of b, there is a natural notation for powers of this function (just like when writing out the n arrows): $$(a\uparrow^n)^k b$$. For example:

 * $$(10\uparrow^2)^3 b$$ = ( 10 → ( 10 → ( 10 → b → 2 ) → 2 ) → 2 )

and only in special cases the long nested chain notation is reduced; for $$b = 1$$ obtains:
 * $$10\uparrow^3 3 = (10\uparrow^2)^3 1$$ = ( 10 → 3 → 3 )

Since the b can also be very large, in general it can be written instead a number with a sequence of powers $$(10 \uparrow^n)^{k_n}$$ with decreasing values of n (with exactly given integer exponents $${k_n}$$) with at the end a number in ordinary scientific notation. Whenever a $${k_n}$$ is too large to be given exactly, the value of $${k_{n+1}}$$ is increased by 1 and everything to the right of $$({n+1})^{k_{n+1}}$$ is rewritten.

For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, $$10 \uparrow (10 \uparrow \uparrow)^5 a=(10 \uparrow \uparrow)^6 a$$, and $$10 \uparrow (10 \uparrow \uparrow \uparrow 3)=10 \uparrow \uparrow (10 \uparrow \uparrow 10 + 1)\approx 10 \uparrow \uparrow \uparrow 3$$. Thus is obtained the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10x are "almost equal" (for arithmetic of large numbers see also below).

If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it act, instead it is possible to simply use a standard value at the right, say 10, and the expression reduces to $$10 \uparrow^n 10=(10 \to 10 \to n)$$ with an approximate n. For such numbers the advantage of using the upward arrow notation no longer applies, so the chain notation can be used instead.

The above can be applied recursively for this n, so the notation $$\uparrow^n$$ is obtained in the superscript of the first arrow, etc., or a nested chain notation, e.g.:

 * (10 → 10 → (10 → 10 → $$3 \times 10^5$$) ) = $$10 \uparrow ^{10 \uparrow ^{3 \times 10^5} 10} 10 $$

If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function $$f(n)=10 \uparrow^{n} 10$$ = (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form $$f^m(n)$$ where m is given exactly and n is an integer which may or may not be given exactly (for example: $$f^2(3 \times 10^5)$$). If n is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form fm(1) = (10→10→m→2). For example, $$(10 \to 10 \to 3\to 2) = 10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 $$

Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus $$ G < 3\rightarrow 3\rightarrow 65\rightarrow 2 <(10 \to 10 \to 65\to 2)=f^{65}(1)$$, but also $$ G < f^{64}(4)<f^{65}(1)$$.

If m in $$f^m(n)$$ is too large to give exactly, it is possible to use a fixed n, e.g. n = 1, and apply the above recursively to m, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of f this gives multiple levels of f. Introducing a function $$g(n)=f^{n}(1)$$ these levels become functional powers of g, allowing us to write a number in the form $$g^m(n)$$ where m is given exactly and n is an integer which may or may not be given exactly. For example, if (10→10→m→3) = gm(1). If n is large any of the above can be used for expressing it. Similarly a function h, etc. can be introduced. If many such functions are required, they can be numbered instead of using a new letter every time, e.g. as a subscript, such that there are numbers of the form $$f_k^m(n)$$ where k and m are given exactly and n is an integer which may or may not be given exactly. Using k=1 for the f above, k=2 for g, etc., obtains (10→10→n→k) = $$f_k(n)=f_{k-1}^n(1)$$. If n is large any of the above can be used to express it. Thus is obtained a nesting of forms $${f_k}^{m_k}$$ where going inward the k decreases, and with as inner argument a sequence of powers $$(10 \uparrow^n)^{p_n}$$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

When k is too large to be given exactly, the number concerned can be expressed as $${f_n}(10)$$=(10→10→10→n) with an approximate n. Note that the process of going from the sequence $$10^{n}$$=(10→n) to the sequence $$10 \uparrow^n 10$$=(10→10→n) is very similar to going from the latter to the sequence $${f_n}(10)$$=(10→10→10→n): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions $${f_{qk}}^{m_{qk}}$$, nested in lexicographical order with q the most significant number, but with decreasing order for q and for k; as inner argument yields a sequence of powers $$(10 \uparrow^n)^{p_n}$$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

For a number too large to write down in the Conway chained arrow notation it size can be described by the length of that chain, for example only using elements 10 in the chain; in other words, one could specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number same techniques can be applied again.

Examples
Numbers expressible in decimal notation: Numbers expressible in scientific notation:
 * 22 = 4
 * 22 2 = 2 ↑↑ 3 = 16
 * 33 = 27
 * 44 = 256
 * 55 = 3,125
 * 66 = 46,656
 * $$2^{2^{2^{2}}}$$ = 2 ↑↑ 4 = 2↑↑↑3 = 65,536
 * 77 = 823,543
 * 106 = 1,000,000 = 1 million
 * 88 = 16,777,216
 * 99 = 387,420,489
 * 109 = 1,000,000,000 = 1 billion
 * 1010 = 10,000,000,000
 * 1012 = 1,000,000,000,000 = 1 trillion
 * 33 3 = 3 ↑↑ 3 = 7,625,597,484,987 ≈ 7.63 × 1012
 * 1015 = 1,000,000,000,000,000 = 1 million billion = 1 quadrillion
 * 1018 = 1,000,000,000,000,000,000 = 1 billion billion = 1 quintilion
 * Approximate number of atoms in the observable universe = 1080 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
 * googol = 10100 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
 * 44 4 = 4 ↑↑ 3 = 2512 ≈ 1.34 × 10154 ≈ (10 ↑)2 2.2
 * Approximate number of Planck volumes composing the volume of the observable universe = 8.5 × 10184
 * 55 5 = 5 ↑↑ 3 = 53125 ≈ 1.91 × 102184 ≈ (10 ↑)2 3.3
 * $$2^{2^{2^{2^2}}} = 2 \uparrow \uparrow 5 = 2^{65,536} \approx 2.0 \times 10^{19,728} \approx (10 \uparrow)^2 4.3$$
 * 66 6 = 6 ↑↑ 3 ≈ 2.66 × 1036,305 ≈ (10 ↑)2 4.6
 * 77 7 = 7 ↑↑ 3 ≈ 3.76 × 10695,974 ≈ (10 ↑)2 5.8
 * 88 8 = 8 ↑↑ 3 ≈ 6.01 × 1015,151,335 ≈ (10 ↑)2 7.2
 * $$M_{82,589,933} \approx 1.49 \times 10^{24,862,047} \approx 10^{10^{7.3955}} = (10 \uparrow)^2 \ 7.3955$$, the 51st and the largest known Mersenne prime.
 * 99 9 = 9 ↑↑ 3 ≈ 4.28 × 10369,693,099 ≈ (10 ↑)2 8.6
 * 1010 10 =10 ↑↑ 3 = 1010,000,000,000 = (10 ↑)3 1
 * $$3^{3^{3^{3}}} = 3 \uparrow \uparrow 4 \approx 1.26 \times 10^{3,638,334,640,024} \approx (10 \uparrow)^3 1.10$$

Numbers expressible in (10 ↑)n k notation:
 * googolplex = $$10^{10^{100}} = (10 \uparrow)^3 2$$
 * $$2^{2^{2^{2^{2^2}}}} = 2 \uparrow \uparrow 6 = 2^{2^{65,536}} \approx 2^{(10 \uparrow)^2 4.3} \approx 10^{(10 \uparrow)^2 4.3} = (10 \uparrow)^3 4.3$$
 * $$10^{10^{10^{10}}}=10 \uparrow \uparrow 4=(10 \uparrow)^4 1$$
 * $$3^{3^{3^{3^3}}} = 3 \uparrow \uparrow 5 \approx 3^{10^{3.6 \times 10^{12}}} \approx (10 \uparrow)^4 1.10$$
 * $$2^{2^{2^{2^{2^{2^2}}}}} = 2 \uparrow \uparrow 7 \approx (10 \uparrow)^4 4.3$$
 * 10 ↑↑ 5 = (10 ↑)5 1
 * 3 ↑↑ 6 ≈ (10 ↑)5 1.10
 * 2 ↑↑ 8 ≈ (10 ↑)5 4.3
 * 10 ↑↑ 6 = (10 ↑)6 1
 * 10 ↑↑↑ 2 = 10 ↑↑ 10 = (10 ↑)10 1
 * 2 ↑↑↑↑ 3 = 2 ↑↑↑ 4 = 2 ↑↑ 65,536 ≈ (10 ↑)65,533 4.3 is between 10 ↑↑ 65,533 and 10 ↑↑ 65,534

Bigger numbers:
 * 3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑ 3) ≈ 3 ↑↑ 7.6 × 1012 ≈ 10 ↑↑ 7.6 × 1012 is between (10 ↑↑)2 2 and (10 ↑↑)2 3
 * $$10\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow)^3 1$$ = ( 10 → 3 → 3 )
 * $$(10\uparrow\uparrow)^2 11$$
 * $$(10\uparrow\uparrow)^2 10^{\,\!10^{10^{3.81\times 10^{17}}}}$$
 * $$10\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow)^4 1$$ = ( 10 → 4 → 3 )
 * $$(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})$$
 * $$10\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow)^5 1$$ = ( 10 → 5 → 3 )
 * $$10\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow)^6 1$$ = ( 10 → 6 → 3 )
 * $$10\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow)^7 1$$ = ( 10 → 7 → 3 )
 * $$10\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow)^8 1$$ = ( 10 → 8 → 3 )
 * $$10\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow)^9 1$$ = ( 10 → 9 → 3 )
 * $$10 \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow)^{10} 1$$ = ( 10 → 2 → 4 ) = ( 10 → 10 → 3 )
 * The first term in the definition of Graham's number, g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) ≈ 3 ↑↑↑ (10 ↑↑ 7.6 × 1012) ≈ 10 ↑↑↑ (10 ↑↑ 7.6 × 1012) is between (10 ↑↑↑)2 2 and (10 ↑↑↑)2 3 (See Graham's number)
 * $$10\uparrow\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow\uparrow)^3 1$$ = (10 → 3 → 4)
 * $$4 \uparrow \uparrow \uparrow \uparrow 4$$ = ( 4 → 4 → 4 ) $$\approx (10 \uparrow \uparrow \uparrow)^2 (10 \uparrow \uparrow)^3 154$$
 * $$10\uparrow\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow\uparrow)^4 1$$ = ( 10 → 4 → 4 )
 * $$10\uparrow\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow\uparrow)^5 1$$ = ( 10 → 5 → 4 )
 * $$10\uparrow\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow\uparrow)^6 1$$ = ( 10 → 6 → 4 )
 * $$10\uparrow\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow\uparrow)^7 1=$$ = ( 10 → 7 → 4 )
 * $$10\uparrow\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow\uparrow)^8 1=$$ = ( 10 → 8 → 4 )
 * $$10\uparrow\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow\uparrow)^9 1=$$ = ( 10 → 9 → 4 )
 * $$10 \uparrow \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow\uparrow)^{10} 1$$ = ( 10 → 2 → 5 ) = ( 10 → 10 → 4 )
 * ( 2 → 3 → 2 → 2 ) = ( 2 → 3 → 8 )
 * ( 3 → 2 → 2 → 2 ) = ( 3 → 2 → 9 ) = ( 3 → 3 → 8 )
 * ( 10 → 10 → 10 ) = ( 10 → 2 → 11 )
 * ( 10 → 2 → 2 → 2 ) = ( 10 → 2 → 100 )
 * ( 10 → 10 → 2 → 2 ) = ( 10 → 2 → $$10^{10}$$ ) = $$10 \uparrow ^{10^{10}} 10 $$
 * The second term in the definition of Graham's number, g2 = 3 ↑g1 3 > 10 ↑g1 – 1 10.
 * ( 10 → 10 → 3 → 2 ) = (10 → 10 → (10 → 10 → $$10^{10}$$) ) = $$10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 $$
 * g3 = (3 → 3 → g2) > (10 → 10 → g2 – 1) > (10 → 10 → 3 → 2)
 * g4 = (3 → 3 → g3) > (10 → 10 → g3 – 1) > (10 → 10 → 4 → 2)
 * g9 = (3 → 3 → g8) is between (10 → 10 → 9 → 2) and (10 → 10 → 10 → 2)
 * ( 10 → 10 → 10 → 2 )
 * g10 = (3 → 3 → g9) is between (10 → 10 → 10 → 2) and (10 → 10 → 11 → 2)
 * g63 = (3 → 3 → g62) is between (10 → 10 → 63 → 2) and (10 → 10 → 64 → 2)
 * ( 10 → 10 → 64 → 2 )
 * Graham's number, g64
 * ( 10 → 10 → 65 → 2 )
 * ( 10 → 10 → 10 → 3 )
 * ( 10 → 10 → 10 → 4 )
 * ( 10 → 10 → 10 → 10 )
 * ( 10 → 10 → 10 → 10 → 10 )
 * ( 10 → 10 → 10 → 10 → 10 → 10 )
 * ( 10 → 10 → 10 → 10 → 10 → 10 → 10 → ... → 10 → 10 → 10 → 10 → 10 → 10 → 10 → 10 ) where there are ( 10 → 10 → 10 ) "10"s
 * ( 10 → 10 → 10 → 10 → 10 → 10 )
 * ( 10 → 10 → 10 → 10 → 10 → 10 → 10 → ... → 10 → 10 → 10 → 10 → 10 → 10 → 10 → 10 ) where there are ( 10 → 10 → 10 ) "10"s

Other notations
Some notations for extremely large numbers:
 * Knuth's up-arrow notation/hyperoperators/Ackermann function, including tetration
 * Conway chained arrow notation
 * Steinhaus-Moser notation; apart from the method of construction of large numbers, this also involves a graphical notation with polygons. Alternative notations, like a more conventional function notation, can also be used with the same functions.
 * Fast-growing hierarchy

These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument.

A function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.

Comparison of base values
The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.

$$100^{12}=10^{24}$$, with base 10 the exponent is doubled.

$$100^{100^{12}}=10^{2*10^{24}}$$, ditto.

$$100^{100^{100^{12}}} \approx 10^{10^{2*10^{24}+0.30103}}$$, the highest exponent is very little more than doubled (increased by log102).

 * $$100\uparrow\uparrow 2=10^ {200} $$
 * $$100\uparrow\uparrow 3=10^ {2 \times 10^ {200}}$$
 * $$100\uparrow\uparrow 4=(10\uparrow)^2 (2 \times 10^ {200}+0.3)=(10\uparrow)^2 (2\times 10^ {200})=(10\uparrow)^3 200.3=(10\uparrow)^4 2.3$$
 * $$100\uparrow\uparrow n=(10\uparrow)^{n-2} (2 \times 10^ {200})=(10\uparrow)^{n-1} 200.3=(10\uparrow)^{n}2.3<10\uparrow\uparrow (n+1)$$ (thus if n is large it seems fair to say that $$100\uparrow\uparrow n$$ is "approximately equal to" $$10\uparrow\uparrow n$$)
 * $$100\uparrow\uparrow\uparrow 2=(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow)^{100} 2.3$$
 * $$100\uparrow\uparrow\uparrow 3=10\uparrow\uparrow(10\uparrow)^{98} (2 \times 10^ {200})=10\uparrow\uparrow(10\uparrow)^{100} 2.3$$
 * $$100\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{100} 2.3<10\uparrow\uparrow\uparrow (n+1)$$ (compare $$10\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{10}1<10\uparrow\uparrow\uparrow (n+1)$$; thus if n is large it seems fair to say that $$100\uparrow\uparrow\uparrow n$$ is "approximately equal to" $$10\uparrow\uparrow\uparrow n$$)
 * $$100\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3$$ (compare $$10\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1$$)
 * $$100\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3$$ (compare $$10\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1$$)
 * $$100\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3$$ (compare $$10\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1$$; if n is large this is "approximately" equal)

Accuracy
For a number $$10^n$$, one unit change in n changes the result by a factor 10. In a number like $$10^{\,\!6.2 \times 10^3}$$, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor $$10^{50}$$ too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).

For very large numbers
In the case of an approximation of an extremely large number, the relative error may be large, yet there may still be a sense in which one wants to consider the numbers as "close in magnitude". For example, consider

 * $$10^{10}$$ and $$10^9$$

The relative error is

 * $$1 - \frac{10^9}{10^{10}} = 1 - \frac{1}{10} = 90\%$$

a large relative error. However, one can also consider the relative error in the logarithms; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%.

The point is that exponential functions magnify relative errors greatly – if a and b have a small relative error,

 * $$10^a$$ and $$10^b$$

the relative error is larger, and

 * $$10^{10^a}$$ and $$10^{10^b}$$

will have an even larger relative error. The question then becomes: on which level of iterated logarithms do to compare two numbers? There is a sense in which one may want to consider

 * $$10^{10^{10}}$$ and $$10^{10^9}$$

to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:

 * $$\log_{10}(\log_{10}(10^{10^{10}})) = 10$$ and $$\log_{10}(\log_{10}(10^{10^9})) = 9$$

Such comparisons of iterated logarithms are common, e.g., in analytic number theory.

Classes
One solution to the problem of comparing large numbers is to define classes of numbers, such as the system devised by Robert Munafo, which is based on different "levels" of perception of an average person. Class 0 – numbers between zero and six – is defined to contain numbers that are easily subitized, that is, numbers that show up very frequently in daily life and are almost instantly comparable. Class 1 – numbers between six and 1,000,000=10$6$ – is defined to contain numbers whose decimal expressions are easily subitized, that is, numbers who are easily comparable not by cardinality, but "at a glance" given the decimal expansion.

Each class after these are defined in terms of iterating this base-10 exponentiation, to simulate the effect of another "iteration" of human indistinguishibility. For example, class 5 is defined to include numbers between 10$1010106$ and 10$101010106$, which are numbers where X becomes humanly indistinguishable from X$2$ (taking iterated logarithms of such X yields indistinguishibility firstly between log(X) and 2log(X), secondly between log(log(X)) and 1+log(log(X)), and finally an extremely long decimal expansion whose length can't be subitized).

Approximate arithmetic
There are some general rules relating to the usual arithmetic operations performed on very large numbers:

Hence:
 * The sum and the product of two very large numbers are both "approximately" equal to the larger one.
 * $$(10^a)^{\,\!10^b}=10^{a 10^b}=10^{10^{b+\log _{10} a}}$$
 * A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large $$n$$ there is $$n^n\approx 10^n$$ (see e.g. the computation of mega) and also $$2^n\approx 10^n$$. Thus $$2\uparrow\uparrow 65536 \approx 10\uparrow\uparrow 65533$$, see table.

Systematically creating ever-faster-increasing sequences
Given a strictly increasing integer sequence/function $$f_0(n)$$ (n≥1), it is possible to produce a faster-growing sequence $$f_1(n) = f_0^n(n)$$ (where the superscript n denotes the nth functional power). This can be repeated any number of times by letting $$f_k(n) = f_{k-1}^n(n)$$, each sequence growing much faster than the one before it. Thus it is possible to define $$f_\omega(n) = f_n(n)$$, which grows much faster than any $$f_k$$ for finite k (here ω is the first infinite ordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.

For example, starting with f0(n) = n + 1:

$$
 * f1(n) = f0n(n) = n + n = 2n
 * f2(n) = f1n(n) = 2nn > (2 ↑) n for n ≥ 2 (using Knuth up-arrow notation)
 * f3(n) = f2n(n) > (2 ↑)n n ≥ 2 ↑2 n for n ≥ 2
 * fk+1(n) > 2 ↑k n for n ≥ 2, k < ω
 * fω(n) = fn(n) > 2 ↑n – 1 n > 2 ↑n − 2 (n + 3) − 3 = A(n, n) for n ≥ 2, where A is the Ackermann function (of which fω is a unary version)
 * fω+1(64) > fω64(6) > Graham's number (= g64 in the sequence defined by g0 = 4, gk+1 = 3 ↑gk 3)
 * This follows by noting fω(n) > 2 ↑n – 1 n > 3 ↑n – 2 3 + 2, and hence fω(gk + 2) > gk+1 + 2
 * fω(n) > 2 ↑n – 1 n = (2 → n → n-1)  =  (2 → n → n-1 → 1) (using Conway chained arrow notation)
 * fω+1(n) = fωn(n) > (2 → n → n-1 → 2) (because if gk(n) = X → n → k then X → n → k+1 = gkn(1))
 * fω+k(n) > (2 → n → n-1 → k+1) > (n → n → k)
 * fω2(n) = fω+n(n) > (n → n → n) = (n → n → n→ 1)
 * fω2+k(n) > (n → n → n → k)
 * fω3(n) > (n → n → n → n)
 * fωk(n) > (n → n → ... → n → n) (Chain of k+1 n's)
 * fω2(n) = fωn(n) > (n → n → ... → n → n) (Chain of n+1 n's)

In some noncomputable sequences
The busy beaver function Σ is an example of a function which grows faster than any computable function. Its value for even relatively small input is huge. The values of Σ(n) for n = 1, 2, 3, 4, 5 are 1, 4, 6, 13, 4098. Σ(6) is not known but is at least 10↑↑15.

Infinite numbers
Although all the numbers discussed above are very large, they are all still decidedly finite. Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite set of natural numbers, and aleph-one is the next greatest cardinal number. $$\mathfrak{c}$$ is the cardinality of the reals. The proposition that $$\mathfrak{c} = \aleph_1$$ is known as the continuum hypothesis.

title: Large numbers


text:
Feature

Feature may refer to:

Computing

 * Feature recognition, could be a hole, pocket, or notch
 * Feature (computer vision), could be an edge, corner or blob
 * Feature (machine learning), in statistics: individual measurable properties of the phenomena being observed
 * Software feature, a distinguishing characteristic of a software program

Science and analysis

 * Feature data, in geographic information systems, comprise information about an entity with a geographic location
 * Features, in audio signal processing, an aim to capture specific aspects of audio signals in a numeric way
 * Feature (archaeology), any dug, built, or dumped evidence of human activity

Media

 * Feature film, a film with a running time long enough to be considered the principal or sole film to fill a program
 * Feature length, the standardized length of such films
 * Feature story, a piece of non-fiction writing about news
 * Radio documentary (feature), a radio program devoted to covering a particular topic in some depth, usually with a mixture of commentary and sound pictures
 * A feature as a guest appearance

Music

 * Feature (band), a British punk trio.
 * The Features, an American rock band

Linguistics

 * Feature (linguistics), a property of a class of linguistic items which describes individual members of that class
 * Distinctive feature, the most basic unit of structure that can be analyzed by phonetics and phonology

Other uses

 * The Feature, a film collaboration between filmmakers Michel Auder and Andrew Neel
 * The Feature (originally named Give Me Something to Read), a standalone website that features a few high-quality, long-form, nonfiction articles every day from Instapaper's most frequently saved articles

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Feature


text:
Emperor Kōmei

Osahito (22 July 1831 – 30 January 1867), posthumously honored as Emperor Kōmei, was the 121st emperor of Japan, according to the traditional order of succession. Kōmei's reign spanned the years from 1846 through 1867, corresponding to the final years of the Edo period.

During his reign there was much internal turmoil as a result of Japan's first major contact with the United States, which occurred under Commodore Perry in 1853 and 1854, and the subsequent forced re-opening of Japan to western nations, ending a 220-year period of national seclusion. Emperor Kōmei did not care much for anything foreign, and he opposed opening Japan to Western powers. His reign would continue to be dominated by insurrection and partisan conflicts eventually culminating in the collapse of the Tokugawa shogunate shortly after his death and the Meiji Restoration in the beginning of the reign of his son and successor Emperor Meiji.

Early life

Before Kōmei's accession to the Chrysanthemum Throne, his personal name (imina) was Osahito (統仁) and his title was Hiro-no-miya (煕宮). Osahito was born on 22 July 1831 and was the fourth son of Emperor Ninkō and his consort Ōgimachi Naoko (正親町雅子). Osahito's Imperial Family lived with him in the Dairi of the Heian Palace.

Reign
Prince Osahito was enthroned as Emperor on 10 March 1846 upon the death of his father. The succession was considered to have been received by the new monarch; and shortly thereafter, Emperor Kōmei is said to have acceded the throne. The years of Kōmei's reign correspond with a period in which Tokugawa Ieyoshi, Tokugawa Iesada, Tokugawa Iemochi, and Tokugawa Yoshinobu were leaders at the pinnacle of the Tokugawa shogunate. On 8 July 1853 US Commodore Matthew Perry arrived with his "Black Ships" to force trade with Japan. The Tokugawa shogunate, which had controlled military and civil affairs in Japan's feudal provinces for some three centuries, proved unable to meet the new challenge of open trade with the West. At the time, Emperor Kōmei still retained only symbolic power at his court in Kyoto. As the shogunate, divided by internal disputes, gradually surrendered sovereignty to the foreign powers under threat of military force, Emperor Kōmei began to assert himself and regain many of the powers his ancestors had conceded to the Tokugawa clan at the close of the Sengoku ("warring states") period. The Emperor's younger sister, Imperial princess Kazu-no-Miya Chikako (和宮親子内親王), was married to the Tokugawa shōgun Tokugawa Iemochi as part of the Movement to Unite Court and Bakufu. Both the Emperor and his sister were against the marriage, even though he realized the gains to be had from such familial connections with the true ruler of Japan. Emperor Kōmei did not care much for anything foreign, and he opposed opening Japan to Western powers, even as the shogun continued to accept foreign demands.

Consultation with the shogunate
On 22 January 1858, Daigaku-no-kami Hayashi Akira headed the bakufu delegation which sought advice from Emperor Kōmei in deciding how to deal with newly assertive foreign powers. This would have been the first time the Emperor's counsel was actively sought since the establishment of the Tokugawa shogunate. The most easily identified consequence of this transitional overture would be the increased numbers of messengers streaming back and forth between Edo and Kyoto during the next decade. Concerning these difficult Imperial audiences in Kyoto, it is somewhat remarkable that the shogun and his bakufu were represented by Hayashi Akira, a 19th-century neo-Confucian scholar/bureaucrat who might have been somewhat surprised to find himself at a crucial nexus of managing political change—moving arguably "by the book" through uncharted waters with well-settled theories and history as the only reliable guide. Hayashi Akira was dispatched from Edo to Kyoto in October 1858 to explain the terms of the Treaty of Amity and Commerce (日米修好通商条約), also known as the Harris Treaty. Hayashi's twofold task was to both explain the terms to a sceptical Emperor and gain the sovereign's assent to it. Kōmei did ultimately acquiesce in February 1859 when he came to understand that there was no alternative.

The pilgrimage of the 14th shogun Tokugawa Iemochi to Kyoto in 1863 was a defining moment not only in 19th century relations between the military bakufu and the Imperial Court, but also in what history would come to call the Meiji Restoration. The reception by Emperor Kōmei of the shogun in the Kyoto palace can be seen as a moment at which the political realm was thoroughly redefined, becoming a transitional imperial realm. This impression was enforced by the ensuing pilgrimage by Emperor Kōmei to the Kamo shrine, with the shogun in tow. This public demonstration showed that a new order had now emerged in the realm.

After reluctantly accepting the Harris Treaty, Japan quickly signed similar treaties, called the Ansei Treaties (also known as the Ansei Five-Power Treaties, with Russia, France, Great Britain, and the Netherlands). The treaties stipulated that the citizens of those foreign nations would be allowed to reside and trade at will in the cities of Edo, Nagasaki, Niigata, Kobe and Yokohama. Those five cities were to be opened to foreign trade with the four Western nations in the treaties. In addition, the treaties stipulated that a system of extraterritoriality would provide for the subjugation of foreign residents to the laws of their own consular courts instead of the Japanese legal system.

The "Order to Expel Barbarians"
Emperor Kōmei was infuriated with nearly every development during his reign as Emperor, and during his lifetime he never saw any foreigners nor did he know much about them. Unequal trade treaties with the Western powers, such as the Treaty of Kanagawa and the Harris Treaty were signed without Imperial sanction and in spite of the Emperor's refusal to approve them. He twice expressed his will to resign from his position in protest. During his reign he started to gain more power as the Tokugawa shogunate declined, though this was limited to consultation and other forms of deference according to protocol. Emperor Kōmei generally agreed with anti-Western sentiments, and, breaking with centuries of imperial tradition, began to take an active role in matters of state. As opportunities arose, he fulminated against the treaties and attempted to interfere in the shogunal succession. His efforts culminated in 1863 with his "Order to expel barbarians". Although the Shogunate had no intention of enforcing the order, it nevertheless inspired attacks against the Shogunate itself and against foreigners in Japan: the most famous incident was the killing of British trader Charles Lennox Richardson, for which the Tokugawa government paid an indemnity of 100,000 pound sterling. Other incidents included the bombardments of Shimonoseki and Kagoshima, and the destruction of Japanese warships, coastal guns, and assorted military infrastructure throughout the country. These incidents showed that Japan could not match the military might of the Western powers at the time, and that military confrontation could not prove to be a diplomatic solution.

Illness and death
In January 1867 the Emperor was diagnosed with smallpox, which caused surprise because Kōmei had allegedly never been ill before. On 30 January 1867 he suffered a fatal violent bout of vomiting and diarrhea, and had purple spots on his face. Emperor Kōmei's death was distinctly convenient for the anti-bakufu forces whom Kōmei had consistently opposed. It was rumored at the time that he was assassinated either by radicals from Choshu, or radical officials in the court. British diplomat Sir Ernest Satow wrote, "it is impossible to deny that [the Emperor Kōmei's] disappearance from the political scene, leaving as his successor a boy of fifteen or sixteen [actually fourteen], was most opportune". However, there is no evidence of this and it is generally believed that he was simply one more victim of what was a worldwide pandemic at the time. Nevertheless, by the time of Emperor Kōmei's death the government was faced with bankruptcy and near collapse. Japan was also surrounded by colonial powers, who stood poised to gain considerable influence with substantial investments in Japanese trade. Kōmei's son, Imperial Prince Mutsuhito, was crowned as Emperor Meiji on 12 September 1868 and these issues were put to rest under the Meiji Restoration.

After Kōmei's death in 1867, his kami was enshrined in the Imperial mausoleum, Nochi no Tsukinowa no Higashi no misasagi (後月輪東山陵), which is at Sennyū-ji in Higashiyama-ku, Kyoto. Also enshrined in this mausoleum complex are Kōmei's immediate predecessors since Emperor Go-Mizunoo: Meishō, Go-Kōmyō, Go-Sai, Reigen, Higashiyama, Nakamikado, Sakuramachi, Momozono, Go-Sakuramachi, Go-Momozono, Kōkaku and Ninkō. Empress Dowager Eishō is also entombed at this Imperial mausoleum complex.

Emperor Kōmei was the last Emperor to be given a posthumous name chosen after his death. Beginning with the reign of his son, Emperor Meiji, posthumous names were chosen in advance, being the same as their reign names.

Kugyō
Kugyō (公卿) is a collective term for the very few most powerful men attached to the court of the Emperor of Japan in pre-Meiji eras. Even during those years in which the court's actual influence outside the palace walls was minimal, the hierarchic organization persisted.

In general, this elite group included only three to four men at a time. These were hereditary courtiers whose experience and background would have brought them to the pinnacle of a life's career. During Kōmei's reign, this apex of the Daijō-kan included:
 * Kampaku, Takatsukasa Masamichi, 1823–1856
 * Kampaku, Kujō Hisatada, 1856–1862
 * Kampaku, Konoe Tadahiro, 1862–1863
 * Kampaku, Takatsukasa Sukehiro, 1863
 * Kampaku, Nijō Nariyuki, 1863–1866
 * Sadaijin
 * Udaijin
 * Naidaijin
 * Dainagon

Eras of Kōmei's reign

Emperor Kōmei was the last Japanese Emperor who had more than one era name (nengō) during a single ruling term. Beginning with his successor, Emperor Meiji, a single era name (identical to the Emperor's eventual posthumous name) was selected and did not change until his death. There were seven nengō during Kōmei's reign.

 * Kōka (1844–1848)
 * Kaei (1848–1854)
 * Ansei (1854–1860)
 * Man'en (1860–1861)
 * Bunkyū (1861–1864)
 * Genji (1864–1865)
 * Keiō (1865–1868)

Genealogy
The family included six children, four daughters and two sons; but the future Emperor Meiji was the only one to survive to adulthood. Kōmei's principal consort was Asako Kujō (九条夙子). After Kōmei's death in 1867, Asako was given the title Empress Dowager Eishō (英照皇太后) by Emperor Meiji.

title: Emperor Kōmei


text:
Christopher of Bavaria

Christopher of Bavaria (Christoph von Pfalz-Neumarkt; Danish and Norwegian: Christoffer af/av Bayern; Swedish Kristofer av Bayern; 26 February 1416 – 5/6 January 1448), was King of Denmark (1440–48, as Christopher III), Sweden (1441–48) and Norway (1442–48) during the era of the Kalmar Union.

Coming to power
He was the son of John, Count Palatine of Neumarkt (1383–1443) and Catherine of Pomerania (c. 1390–1426). Catherine was the daughter of Wartislaw VII, Duke of Pomerania in Pomerania-Stolp, and sister of the Scandinavian king, Eric of Pomerania. Count Palatine John was a son of King Rupert of Germany (1352–1410). Christopher was probably born at Neumarkt in der Oberpfalz in Upper Palatinate, in Bavaria, Germany. In 1445, Christopher married Dorothea of Brandenburg (1430 – 25 November 1495) in Copenhagen.

Eric of Pomerania was deposed as king of Denmark and Sweden in 1439. Eric's nephew, Christopher, who was rather unfamiliar with Scandinavian conditions, was elected by the Danish State Council as the successor to his uncle, first as regent from 1439, and then proclaimed King of Denmark at the Viborg Assembly (Danish landsting) on 9 April 1440. He was meant to be a puppet, as evidenced by the saying: "Had the Council demanded the stars of heaven from him, he would have ordered it." However he succeeded in maintaining some personal control. As a whole his rule, according to the politics of the nobility and his succession, might be called the start of the long period of balance between royal power and nobility which lasted until 1660. He was later elected king of Sweden in 1441, and Norway in June 1442.

For himself Christopher used the otherwise unknown title of arch king (archirex), because in his opinion he ruled an empire, not simply three different countries, and thus ranked immediately under the European emperor.''

Peasant rebellions
At the start of his reign, he put down peasant rebellions in Funen and Jutland. Once the rebellion on Funen was suppressed, he turned his attention the uprising in Jutland. North Jutland, especially Vendsyssel, was so restive that a peasant army of 25,000 led by Henrik Tagesen Reventlow (executed 1441) posed a serious threat to Christopher's continued reign. Before the king could act, Jutland's noble families raised their own army and marched west of Aalborg to meet Reventlow's forces.

The peasants had created a gigantic wagon fortress three layers deep to protect themselves from the mounted knights they knew would come against them. They also placed tree branches across the bog in front of the camp and then cast earth on top to make it look like solid ground. The overconfident army of nobles led by Eske Jensen Brock appeared at St Jorgen's Hill (St. Jørgensbjerg) on 3 May 1441. The knights charged the camp, and were quickly mired down in the bog. The peasants moved in for the kill. Brock was killed in the Battle of St Jorgen's Hill (Slaget ved Skt. Jørgensbjerg) and dismembered and the pieces sent to the towns in the area as a warning. The peasants then raided Aalborghus (the area's most important manor) forcing the noble Niels Guldenstierne to flee.

The treatment of the captives after the battle strengthened Christopher's determination to put down the peasants. With his own army Christopher rode north to the rebel camp at Husby Hole near St Jorgen's Hill in northern Jutland. Because the rebels outnumbered his troops, Christopher sent word that anyone who left the camp and went home would not be punished for rebellion. The men from the island of Mors and Thisted left, for which they were called cowards and traitors ever after. Christopher ordered the attack on the rebel camp on 8 June 1441 and despite fighting ferociously the rebels could not overcome the heavily armed knights. Thousands of rebels were killed, those who survived were fined heavily. The more severe consequence was that rebels lost their free status and became serfs on the farms where they worked. The king made it a capital crime for peasants to carry weapons longer than a short knife. The subjugation of Denmark's once free peasants was complete.

Coronation, relations with Swedes
In May 1442 Christopher traveled to Lödöse to meet with the nobles from all three kingdoms. He was elected King of Norway there and then went to Oslo where he was crowned on 2 July 1442. The next year he was proclaimed King of Denmark at the Urnehoved Assembly near Ribe. When his residence at Roskilde burned down, Christopher moved to Copenhagen and made it the capital of Denmark.

The Swedish nobles were not happy to relinquish any power and thus didn't like him, claiming he was too German for them and that he allowed his uncle (ex-King Eric) to plunder shipping from his castle on Gotland without any attempt to stop him. They blamed a series of bad harvests on him. People were so hungry they mixed ground tree bark with the little flour they could find. Christopher was contemptuously nicknamed the "Bark King" in Sweden. On the other hand, he tried to support the cities and their merchants as far as the limits of nobility and Hanseatic cities allowed. During his reign Copenhagen was made permanently the capital of Denmark (municipal charter of 1443).

He carried on an ineffective policy of war and negotiations against Eric in Gotland which did little to help the dissatisfaction within both Sweden and the Hanseatic League. The Kalmar Union Treaty was changed so that the aristocracy had most of the policy-making powers, and the king lost many of the powers monarchs had acquired since Viking times. The results of this policy of balance were still not reached when he suddenly died as the last descendant of Valdemar IV of Denmark.

Death
In 1448, Christopher died suddenly at Helsingborg at age 31. King Christopher was buried in Roskilde Cathedral. His widow, Queen Dorothea, married the new king of Denmark, Christian I.

Other sources

 * Dansk Biografisk Leksikon, vol. 7, Copenhagen 1980.
 * Politikens Danmarkshistorie, vol. 4 by Erik Kjersgaard, Copenhagen 1962.
 * Politikens bog om Danske Monarker by Benito Scocozza, Copenhagen 1998.

title: Christopher of Bavaria


text:
Talk:Stainless steel

Today's stailness steel 304 18gauge seat
Today's stailness steel 18gauge 304 seat rate 103.214.60.231 (talk) 05:27, 22 July 2023 (UTC)

Questionable Statement about Thermal "Conduction"?
I'm not a native english Speaker myself, but I stumbled over the following passage in de Section "Thermal Conduction" (which I believe should be "Thermal Conductivity"?):

"Typically, stainless steel has a thermal conductivity ranging from 15 to 20 W/mK (watts per meter Kelvin). Due to this, it keeps more energy that stabilizes the surrounding temperature."

I cannot figure out what this sentence tries to convey! Thermal Conductivity is about the ability conduct energy from the hotter part of an object to the cooler part - not about keeping energy - that's what I would rather associate with Thermal Capacitance !

Unless there is some hidden aspect that I fail to grasp, the sentence "due to this, it keeps ..." does not make any sense in this form, and I would therefore suggest that it should be either clarified or removed.

Interestingly, both sections "Hardness" and "Thermal Conduction" seem to have been introduced at the same time in March 10, 2023, and are sourced from the same obscure Website "https://blog.thepipingmart.com/", which does not seem to be scientific. 2A01:C23:799E:1900:5C09:9A68:64B2:D760 (talk) 22:41, 14 August 2023 (UTC)

 * It was just someone adding text as an excuse to spam their website. I removed it. - MrOllie (talk) 17:16, 21 September 2023 (UTC)

Reducing acids
Stating that acids can be either reducing or oxidizing is incompetent! An example of acid, which may act as a reducing agent, is HI. Formic acid can be considered reducing as well. Hydrochloric and dilute sulfuric acid are non-oxidizing, but ARE NOT reducing! Stating that acids can be either oxidizing or non-oxidizing is fine, because a reducing acid is also non-oxidizing. But many non-oxidizing acids ARE NOT reducing.

See Reducing agent. Gradatmit (talk) 20:30, 21 September 2023 (UTC)
 * The article has had that wording for years. Per WP:BRD it should stay that way until (if) consensus is reached to change it. And the personal attack is not appropriate. Meters (talk) 21:00, 21 September 2023 (UTC)
 * And where is the personal attack!? I did NOT say that you are incompetent! I said that the statement is incompetent! 94.25.177.135 (talk) 21:27, 21 September 2023 (UTC)
 * I'm not going to quibble about. Please don't make comments like that.
 * One could just as well say that acids can be reducing and non-reducing. If acids are categorized as "oxidizing" and "non-oxidizing" then let's add a source to that effect. Similarly if they are categorized as "oxidizing" and "reducing" as claimed then let's add a source for that. If that was just an editor's terminology the we should reword it properly. user:Geimoe added this more than 5 years ago and it has not been contested since, but that editor has not [typo corrected 25 September 2023] been active for several years. Meters (talk) 23:55, 21 September 2023 (UTC)
 * The section in question discusses oxidative corrosion, which is a chemical process. The concepts of oxidation/reduction are taught in high-school chemistry classes. The same applies to basic reactions of mineral acids with metals and other materials. Sources are necessary to support statements about research/technical material, not high-school-level concepts.
 * In basic terms, a substance is considered a reducer/oxidizer if it can be readily oxidized/reduced by common oxidizers/reducers. In a reaction of common metals (and other reducers) with oxidizing acids, protons are not normally reduced to hydrogen, but the element forming the acid anion is reduced instead. For example, concentrated sulfuric acid reacts with oxidizable materials and may yield a variety of reduced forms of sulfate, such as sulfur dioxide: H2SVIO4->SIVO2. When diluted, the basic process H+->H2 normally occurs, if possible. The only element, which can be oxidized in sulfuric acid is oxygen, so someone understanding elementary chemistry would not call sulfuric acid "reducing" regardless of its concentration. The chlorine ion in hydrochloric acid can be oxidized a bit easier, but considering its oxidation potential, again it cannot be called "reducing" either. HI is an example of an acid, which can be considered reducing.
 * Of course, "neutral" acids can be called as both non-oxidizing and non-reducing. Only in the context in question, oxidative corrosion is discussed, which means talking about reducing/non-reducing acids is pointless. Gradatmit (talk) 10:50, 22 September 2023 (UTC)
 * I was not suggesting that the terminology "reducing/non-reducing" would be of any use here, just that with the current lack of justification in the article one can't tell whether it should be oxidizing/reducing, oxidizing/non-oxidizing or reducing/non-reducing. And the section does seem to
 * Not all of our readers have a high school chemistry education, and not all of those who have a high school education in chemistry will remember that education. For that matter, I'm not aware of any policy that says that only technical matters beyond high school level require sourcing. Certainly WP:NOTCITE doesn't say that. This is Wikipedia, not a technical research paper. There seems to be a large difference between what "the average adult recognizes as true" (Wikipedia's standard for not needing sourcing) and what you (a PhD in chemistry according to your user page) think everyone should know about chemistry. You are trying to change something that has been in the article (unchallenged) for more than 5 years. Again, if the norm is to refer to acids as "oxidizing" and "non-oxidizing" then just provide a source to that effect and we'll fix it, but right now all we have is your say so.
 * And your contention that this section is only discussing oxidative corrosion seems at odds with the statement "Increasing chromium and molybdenum content provides increased resistance to reducing acids". Meters (talk) 20:07, 25 September 2023 (UTC)

Availability section is oddly specific to the Netherlands and reads a bit like an advertisement
Should it really be linking to specific businesses like that? Also it seems odd that "maintenance" is a subheading of "availability"; those seem like unrelated topics. 2600:1700:9DA1:1D2F:E47D:8E4E:B8B3:2A54 (talk) 09:21, 15 October 2023 (UTC)
 * Agreed. Removing it. Meters (talk) 09:24, 15 October 2023 (UTC)
 * That was added just a few days ago. Meters (talk) 09:26, 15 October 2023 (UTC)

Whatever 30x alloy Tesla is using?
Tesla claims to have made some modifications to existing 30x stainless steel alloys, with R&D done by their own (apparently, in concert with SpaceX) materials engineering/metalurgy shop, in order to make a new alloy of stainless steel. Apparently, they have recently started calling this particular alloy "HFS" (per this and other sources) and are using it extensively in their new pickup truck EV. They previously just called the alloy 30X.

They don't make the steel. They buy large coils from regular/standard suppliers of stainless steel alloys. I read somewhere it may be similar, or identical?, to the cold-rolled alloy they are using for the new SpaceX huge launch vehicle: Starship. This the the world's largest and most-powerful launch vehicle ever now, larger than the Saturn V, and has made a couple of test flights. But am unsure if the particular alloy for that rocket is confirmed in reliable sources.

Given the number of consumer orders and projected production volume for the new stainless-steel-body Cybertruck, it would appear that this new alloy is an economically-significant, and clearly a notable alloy.

When might we find serious (laboratory? or academic?) analysis of just what this new alloy is? Composition? Properties? etc.? Would be good to add that particular grade/type/formulation to improve the article. N2e (talk) 18:19, 12 December 2023 (UTC)

title: Talk:Stainless steel


text:
Talk:Telegraph key

Suggested external link: http://www.ae4rv.com/tn/education/bug.htm —Preceding unsigned comment added by 216.248.55.132 (talk) 17:54, 22 April 2008 (UTC)

This is a great article! The Anome 12:36 Feb 24, 2003 (UTC)

""Glass arm" or "Telegrapher's paralysis" may be reduced or eliminated by increasing the side play of the straight key by loosening the adjustable trunnion screws." - Citation needed? I think they mean by not using a straight key and using a bug or a keyer instead, using a side to side movement. —Preceding unsigned comment added by 216.248.55.132 (talk) 13:15, 3 May 2011 (UTC)

The Fist
I've broken fist out into its own section, because it's interesting. But I have to admit, maybe it doesn't really belong here, except as a passing mention. Fist is more to do with the operator than with the equipment (although the equipment is a factor). Can anyone suggest a better place to put it? I don't think it deserves its own article, but I'm not sure where it does belong either. Regards, Ben Aveling 07:58, 7 November 2006 (UTC)
 * It's relevant, as types of keys influence one's "fist", but not as a separate section. I have arranged the format in terms of intro, then key classifications. - LuckyLouie 22:28, 1 April 2007 (UTC)
 * The overall article's purpose seems to be about the telegraph key, however there is space for appropriate copy regarding "fist." With due respect LuckyLouie, it seems the cited source is discussing how fist can be a result of the key, so I edited it back down into its own section with a focus on how the key affects one's personal characteristics. - Davandron | Talk 03:32, 2 April 2007 (UTC)
 * Good stuff, I like what you did. -- LuckyLouie 04:19, 2 April 2007 (UTC)

Removed section on "fist"
I've removed the section on fist, as was a very thinly-modified copy-and-paste copyvio from the referenced source (http://www.zerobeat.net/tasrt/c09a.htm). Copying, then adding a pointer to the original, is still copying. Tlesher (talk) 03:50, 16 July 2008 (UTC)

title: Talk:Telegraph key


text:
Louis III of France

Louis III (863/65 – 5 August 882) was King of West Francia (precursor to France) from 879 until his death in 882. He succeeded his father Louis the Stammerer, and ruled over West Francia in tandem with his brother Carloman II. Louis controlled the northern part of West Francia (Neustria), including the capital of Paris, while Carloman controlled the southern portion (Aquitania). Louis ruled from March 880 to 5 August 882, when he died and left the rest of West Francia to his brother. His short reign was profoundly influenced by his military success, including his defeating Vikings in August 881.

Early life
Louis was the eldest son of Louis the Stammerer and his first wife, Ansgarde of Burgundy. He was born while his father was King of Aquitaine and his grandfather Charles the Bald was ruling West Francia. Some doubts were raised about his legitimacy, mostly due to the fact that his parents had married secretly and Ansgarde was later repudiated at Charles' insistence.

When Charles died in 877, followed by Louis the Stammerer on 10 April 879, some Frankish nobles advocated electing Louis as the sole king, but another party favoured each brother ruling a separate part of the kingdom. In September 879, Louis and his brother, Carloman, were crowned at Ferrières Abbey.

Military success
In March 880 at Amiens, Louis and his brother divided their father's kingdom, with Louis receiving the northern and western part, called Neustria. The other brother received the southern and eastern part, and Duke Boso, one of Charles the Bald's most trusted lieutenants, renounced his allegiance to both brothers and was elected King of Provence.

In the summer of 880, Louis and Carloman marched against Boso and captured Mâcon and the northern part of Provence. They then proceeded to unite their forces with those of their cousin, Charles the Fat, then ruling East Francia and Kingdom of Italy, and unsuccessfully besieged Vienne from August to November 880.

In 881, Louis III and Carloman achieved a victory against the Vikings, whose invasions had been ongoing since his grandfather's reign, at the Battle of Saucourt-en-Vimeu. The king and his victory were so acclaimed that within a year of the battle, an anonymous poet celebrated it and the king for both his prowess and piety in a short poem in Old High German, the Ludwigslied.

Death and legacy
Louis III, at the time of his great successes, was very young, at a mere 16 to 17 years of age. His colourful victories against the Vikings entertained the public, and he was loved by most of the Francian people. In Neustria, he was widely celebrated, and he was the most popular person in the realm.

Louis III died on 5 August 882, aged around 17, at Saint-Denis in the centre of his realm. He was chasing after a girl, who was retreating to her father's house, when he hit his head on the lintel of a low door and later died. Because Louis III had no children, his brother Carloman II became the sole king of West Francia, and the young king was buried in the royal mausoleum of the Basilica of Saint-Denis.

title: Louis III of France


text:
Louis III

Louis III may refer to:

 * Louis the Younger, sometimes III of Germany (835–882)
 * Louis III of France (865–882)
 * Louis the Blind, Louis III, Holy Roman Emperor, (c. 880–928)
 * Louis the Child, sometimes III of Germany (893–911)
 * Louis III, Count of Chiny (died 1189)
 * Ludwig III of Thuringia (ruled 1172–1190), see Hermann I, Landgrave of Thuringia
 * Louis III of Châtillon (died 1391)
 * Louis III of Naples (1403–1434)
 * Louis III, Elector Palatine (ruled 1410–1436)
 * Louis III, Landgrave of Hesse, nicknamed "the Frank" (1438–1471)
 * Louis III, Count of Löwenstein (1530–1611)
 * Louis III, Duke of Württemberg (1554–1593)
 * Louis III, Cardinal of Guise (1575–1621)
 * Louis III, Prince of Condé (1668–1710)
 * Ludwig III, Grand Duke of Hesse and by Rhine (1806–1877)
 * Ludwig III of Bavaria (1845–1921)

title: Louis III


text:
Pseudoscorpion

Pseudoscorpions, also known as false scorpions or book scorpions, are small, scorpion-like arachnids belonging to the order Pseudoscorpiones, also known as Pseudoscorpionida or Chelonethida.

Pseudoscorpions are generally beneficial to humans since they prey on clothes moth larvae, carpet beetle larvae, booklice, ants, mites, and small flies. They are tiny, and are rarely noticed due to their small size, despite being common in many environments. When people do see pseudoscorpions, especially indoors, they are often mistaken for ticks or small spiders. Pseudoscorpions often carry out phoresis, a form of commensalism in which one organism uses another for the purpose of transport.

Characteristics
Pseudoscorpions belong to the class Arachnida. They are small arachnids with a flat, pear-shaped body, and pincer-like pedipalps that resemble those of scorpions. They usually range from 2 to 8 mm in length. The largest known species is Garypus titanius of Ascension Island at up to 12 mm. Range is generally smaller at an average of 3 mm.

A pseudoscorpion has eight legs with five to seven segments each; the number of fused segments is used to distinguish families and genera. They have two very long pedipalps with palpal chelae (pincers), which strongly resemble the pincers found on a scorpion.

The pedipalps generally consist of an immobile "hand" and mobile "finger", the latter controlled by an adductor muscle. Members of the clade Iocheirata, which contains the majority of pseudoscorpions, are venomous, with a venom gland and duct usually located in the mobile finger; the venom is used to immobilize the pseudoscorpion's prey. During digestion, pseudoscorpions exude a mildly corrosive fluid over the prey, then ingest the liquefied remains.

The abdomen, referred to as the opisthosoma, is made up of twelve segments, each protected by sclerotized plates (called tergites above and sternites below). The abdomen is short and rounded at the rear, rather than extending into a segmented tail and stinger like true scorpions. The color of the body can be yellowish-tan to dark-brown, with the paired claws often a contrasting color. They may have two, four or no eyes.

Pseudoscorpions spin silk from a gland in their jaws to make disk-shaped cocoons for mating, molting, or waiting out cold weather, but they do not have book lungs like true scorpions and the Tetrapulmonata. Instead, they breathe exclusively through tracheae, which open laterally through two pairs of spiracles on the posterior margins of the sternites of abdominal segments 3 and 4.

Behavior

The male produces a spermatophore which is attached to the substrate and is picked up by the female. Members of the Cheliferoidea (Atemnidae, Cheliferidae, Chernetidae and Withiidae) have an elaborate mating dance, which ends with the male navigating the female over his spermatophore. In Cheliferidae, the male also uses his forelegs to open the female genital operculum, and after she has mounted the packet of sperm, assisting the spermatophore's entry by pushing it into her genital opening. Females in species that possess a spermatheca (sperm storing organ) can store the sperm for a longer period of time before fertilizing the eggs, but species without the organ fertilize the eggs shortly after mating. The female carries the fertilized eggs in a brood pouch attached to her abdomen.

Between 2 and 50 young are hatched in a single brood, with more than one brood per year possible. The young go through three molts called the protonymph, deutonymph and tritonymph. The developing embryo and the protonymph, which remain attached to the mother, is nourished by a ‘milk’ produced by her ovary. Many species molt in a small, silken igloo that protects them from enemies during this vulnerable period.

After reaching adulthood they no longer molt, and will live for 2–3 years. They are active in the warm months of the year, overwintering in silken cocoons when the weather grows cold. Smaller species live in debris and humus. Some species are arboreal, while others are phagophiles, eating parasites in an example of cleaning symbiosis. Some species are phoretic, others may sometimes be found feeding on mites under the wing covers of certain beetles.

Distribution

More than 3,300 species of pseudoscorpions are recorded in more than 430 genera, with more being discovered on a regular basis. They range worldwide, even in temperate to cold regions such as Northern Ontario and above the timberline in Wyoming's Rocky Mountains in the United States and the Jenolan Caves of Australia, but have their most dense and diverse populations in the tropics and subtropics, where they spread even to island territories such as the Canary Islands, where around 25 endemic species have been found. There are also two endemic species on the Maltese Islands. Species have been found under tree bark, in leaf and pine litter, in soil, in tree hollows, under stones, in caves such as the Movile Cave, at the seashore in the intertidal zone, and within fractured rocks.

Chelifer cancroides is the species most commonly found in homes, where it is often observed in rooms with dusty books. There, the tiny animals (2.5 - 4.5 mm) can find their food such as booklice and house dust mites. They enter homes by riding insects (phoresy) larger than themselves, or are brought in with firewood.

Evolution
The oldest known fossil pseudoscorpion, Dracochela deprehendor is known from cuticle fragments of nymphs found in the Panther Mountain Formation near Gilboa in New York, dating to the mid-Devonian, around 383 million years ago. It has all of the traits of a modern pseudoscorpion, indicating that the order evolved very early in the history of land animals. Its morphology suggests that it is more primitive than any living pseudoscorpion. As with most other arachnid orders, the pseudoscorpions have changed very little since they first appeared, retaining almost all the features of their original form. After the Devonian fossils, almost no other fossils of pseudoscorpions are known for over 250 million years until Cretaceous fossils in amber, all belonging to modern families, suggesting that the major diversification of pseudoscorpions had already taken place by this time. The only fossil from this time gap is Archaeofeaella from the Triassic of Ukraine, approximately 227 million years ago, which is suggested to be an early relative of the family Feaellidae.

Historical references
Pseudoscorpions were first described by Aristotle, who probably found them among scrolls in a library where they would have been feeding on booklice. Robert Hooke referred to a "Land-Crab" in his 1665 work Micrographia. Another reference in the 1780s, when George Adams wrote of "a lobster-insect, spied by some labouring men who were drinking their porter, and borne away by an ingenious gentleman, who brought it to my lodging."

Classification
The following taxon numbers are calculated as of the end of 2023.

 * Atemnidae Kishida, 1929 (21 genera, 194 species)
 * Bochicidae Chamberlin, 1930 (12 genera, 44 species)
 * Cheiridiidae Hansen, 1894 (9 genera, 81 species)
 * Cheliferidae Risso, 1827 (64 genera, 312 species)
 * Chernetidae Menge, 1855 (120 genera, 728 species)
 * Chthoniidae Daday, 1888 (54 genera, 909 species)
 * Feaellidae Ellingsen, 1906 (8 genus, 37 species)
 * Garypidae Simon, 1879 (11 genera, 110 species)
 * Garypinidae Daday, 1888 (21 genera, 94 species)
 * Geogarypidae Chamberlin, 1930 (2 genera, 81 species)
 * Gymnobisiidae Beier, 1947 (4 genera, 17 species)
 * Hyidae Chamberlin, 1930 (2 genera, 41 species)
 * Ideoroncidae Chamberlin, 1930 (15 genera, 86 species)
 * Larcidae Harvey, 1992 (1 genus, 15 species)
 * Menthidae Chamberlin, 1930 (5 genera, 12 species)
 * Neobisiidae Chamberlin, 1930 (34 genera, 748 species)
 * Olpiidae Banks, 1895 (24 genera, 211 species)
 * Parahyidae Harvey, 1992 (1 genus, 1 species)
 * Pseudochiridiidae Chamberlin, 1923 (2 genera, 13 species)
 * Pseudogarypidae Chamberlin, 1923 (2 genera, 12 species)
 * Pseudotyrannochthoniidae Beier, 1932 (6 genera, 80 species)
 * Sternophoridae Chamberlin, 1923 (3 genera, 21 species)
 * Syarinidae Chamberlin, 1930 (18 genera, 125 species)
 * Withiidae Chamberlin, 1931 (37 genera, 170 species)
 * †Dracochelidae Schawaller, Shear & Bonamo, 1991 (1 genus, 1 species)

Cladogram
After Benavides et al, 2019, with historic taxonomic groups from Harvey (1992).

title: Pseudoscorpion