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71 (number)

71 (number)

71 is the natural number following 70 and preceding 72.


Cardinal	seventy-one
Ordinal	71st (seventy-first)
Factorization	prime
Roman numeral	LXXI
Binary	1000111
Hexadecimal	47

In mathematics

It is the 20th prime number. The next is 73, with which it composes a twin prime. It is also a permutable prime with 17. If we add up the primes less than 71 (2 through 67), we get 568, which is divisible by 71, 8 times. 71 is the largest supersingular prime. Also, 71² = 7! + 1, making it part of the last known pair of Brown numbers, as (71, 7). It is an Eisenstein prime with no imaginary part and real part of the form

3n - 1

. 71 is a centered heptagonal number.

In science

- The atomic number of lutetium, a lanthanide In astronomy, : Messier object M71, a magnitude 8.5 globular cluster in the constellation Sagitta : The New General Catalogue [http://www.ngcic.org/ object] NGC 71, a magnitude 13.2 peculiar spiral galaxy in the constellation Andromeda :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -702 October 8 and ended on 777 March 14. The duration of Saros series 71 was 1478.4 years, and it contained 83 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -472 June 4 and ended on 808 July 11. The duration of Saros series 71 was 1280.1 years, and it contained 72 lunar eclipses.

In other fields

Seventy-one is also:
- The designation of USA Interstate 71, which runs from Kentucky to Ohio.
- The registry of the U.S. Navy's nuclear aircraft carrier USS Theodore Roosevelt (CVN-71), named after U.S. President Theodore Roosevelt.
- The number of the French department Saône-et-Loire.
- The year AD 71, 71 BC, or 1971. Category:Integers ko:71 ja:71

Natural number

In mathematics, a natural number is either a positive integer (, , , , ...) or a non-negative integer (, , , , , ...). The former definition is generally used in number theory, while the latter is preferred in set theory. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics.

History of natural numbers and the status of zero

The natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful place-value system based essentially on the numerals for 1 and 10. The ancient Egyptians had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. A much later advance in abstraction was the development of the idea of zero as a number with its own numeral. A zero digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.¹ The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage did not spread beyond Mesoamerica. The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628 AD. Nevertheless, zero was used as a number by all medieval computists (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it. Instead, the Latin word for "nothing," nullae, was employed. The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. In the nineteenth century, a set-theoretical definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. This convention is followed by set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers. The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers.

Notation

Mathematicians use N or

\mathbb

(an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "
- " is added in the latter case: : N₀ = ; N^- = . (Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R₊ = [0,∞) and Z₊ = , at least in European literature. The notation "
- ", however, is quite standard for nonzero or rather invertible elements.) Less frequently, W or

\mathbb

is used for the set of "whole numbers", which are sometimes identified with the natural numbers as defined here, sometimes with the integers. Set theorists often denote the set of all natural numbers by ω. When this notation is used, zero is explicitly included as a natural number.

Formal definitions

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano postulates state conditions that any successful definition must satisfy. Certain constructions show that, given set theory, models of the Peano postulates must exist.

Peano axioms

- There is a natural number 0.
- Every natural number a has a natural number successor, denoted by S(a).
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.) It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one).

Constructions based on set theory

The standard construction

A standard construction in set theory is to define the natural numbers as follows: :We set 0 := :and define S(a) = a U for all a. :The set of natural numbers is then defined to be the intersection of all sets containing 0 which are closed under the successor function. :Assuming the axiom of infinity, this definition can be shown to satisfy the Peano axioms. :Each natural number is then equal to the set of natural numbers less than it, so that :
- 0 = :
- 1 = = :
- 2 = = = :
- 3 = = = :and so on. When you see a natural number used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) iff n is a subset of m. :Also, with this definition, different possible interpretations of notations like R^{n (n-tuples vs. mappings of n into R) coincide.

Other constructions

Although the standard construction is useful, it is not the only possible construction. For example:
:one could define 0 =
:and S(a) = ,
:producing
:: 0 =
:: 1 = =
:: 2 = = , etc.
Or we could even define 0 =
:and S(a) = a U
:producing
:: 0 =
:: 1 = =
:: 2 = , etc.

For the rest of this article, we follow the standard construction described above.

Properties

One can recursively define an addition on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers.

If we define S(0) := 1, then S(b) = S(b + 0) = b + S(0) = b + 1; i.e. the successor of b is simply b + 1.

Analogously, given that addition has been defined, a multiplication × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (N, ×) into a commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Addition and multiplication are compatible, which is expressed in the distribution law:
a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.

If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that a + 1 = S(a) and a × 1 = a.

For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operations.

Furthermore, one defines a total order on the natural numbers by writing a ≤ b if and only if there exists another natural number c with a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 we can find natural numbers q and r such that

:a = bq + r and r < b

The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

Generalizations

Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in a ordered sequence and cardinal numbers are used to specify the size of a given set.

For finite sequences or finite sets, both of these properties are embodied in the natural numbers.

Other generalizations are discussed in the article on numbers.

Footnote

¹ "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place." [http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html]

Category:Elementary mathematics
Category:Integers
Category:Number theory
Category:Numbers
Category:Set theory

ko:자연수

ja:自然数

th:จำนวนธรรมชาติ

72 (number)
72 is the natural number following 71 and preceding 73.

Cardinal seventy-two
Ordinal 72nd (seventy-second)
Factorization $2^3 \cdot 3^2$
Divisors 2, 3, 4, 6, 8, 9,
12, 18, 24, 36
Roman numeral LXXII
Binary 1001000
Hexadecimal 48

In mathematics
Seventy-two is the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). 72 is a Harshad number and a pronic number.

In normal space, the exterior angles of an equilateral pentagon measure 72 degrees each.

There are 17 solutions to the equation φ(x) = 72, more than any integer below 72, making it a highly totient number.

In science

- The atomic number of hafnium

In astronomy,

: Messier object M72, a magnitude 10.0 globular cluster in the constellation Aquarius

: The New General Catalogue [http://www.ngcic.org/ object] NGC 72, a magnitude 13.5 barred spiral galaxy in the constellation Andromeda

:The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -727 August 16 and ended on 752 January 21. The duration of Saros series 72 was 1478.1 years, and it contained 83 solar eclipses.

:The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -407 June 5 and ended on 891 July 25. The duration of Saros series 72 was 1298.1 years, and it contained 73 lunar eclipses.

In other fields
Seventy-two is also:

- The conventional number of scholars translating the Septuagint, according to the legendary account in the "Letter of Aristeas".

- The conventional number of disciples sent forth by Jesus in Luke 10 in some manuscripts (seventy in others).

- The number of names of God, according to Kabbalah (see names of God in Judaism).

- The number of hours in three days.

- The total number of books in the Holy Bible in the Catholic version if the Book of Lamentations is considered part of the Book of Jeremiah.

- In dpi, the standard resolution of an Apple Macintosh screen.

- The number of the French department Sarthe.

- The designation of USA Interstate 72, a freeway that runs from Missouri to Illinois.

- The registry of the U.S. Navy's nuclear aircraft carrier USS Abraham Lincoln (CVN-72), named after U.S. President Abraham Lincoln.

- The designation of the Soviet T-72 tank.

- In the movie The Red Violin, the lot number of the Bussotti red violin at the DuVal auction house.

- The year AD 72, 72 BC, or 1972.

- The Rule of 72 in finance.
Category:Integers

ko:72

ja:72

Ordinal number
Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc., whereas a cardinal number says "how many there are": one, two, three, four, etc. (See How to name numbers.)

An ordinal scale defines a total preorder of objects; the scale values themselves have a total order; names may be used like "bad", "medium", "good"; if numbers are used they are only relevant up to strictly monotonically increasing transformations (order isomorphism). See also level of measurement.
----
In mathematics, ordinal numbers are an extension of the natural numbers to accommodate infinite sequences, introduced by Georg Cantor in 1897. It is this generalization which will be explained below.

Introduction

A natural number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two. The notion of size leads to cardinal numbers, which were also discovered by Cantor, while the position is generalized by the ordinal numbers described here.

In set theory, the natural numbers are commonly constructed as sets, such that each natural number is the set of all smaller natural numbers:

:0 = (empty set)
:1 = =
:2 = =
:3 = =
:4 = =

etc.

Viewed this way, every natural number is a well-ordered set: the set 4, for instance, has the elements 0, 1, 2, 3 which are of course ordered as 0 < 1 < 2 < 3. A natural number is smaller than another if and only if it is an element of the other.

We don't want to distinguish between two well-ordered sets if they only differ in the "notation for their elements", or more formally: if we can pair off the elements of the first set with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism (or a strictly increasing function) and the two well-ordered sets are said to be order-isomorphic.

Under this convention, one can show that every finite well-ordered set is order-isomorphic to one (and only one) natural number. This provides the motivation for the generalization to infinite numbers.

Modern definition and first properties

We want to construct ordinal numbers as special well-ordered sets in such a way that every well-ordered set is order-isomorphic to one and only one ordinal number.
The following definition improves on Cantor's approach and was first given by John von Neumann:

:A set S is an ordinal if and only if S is totally ordered with respect to set containment and every element of S is also a subset of S.

(Here, "set containment" is another name for the subset relationship.)
Such a set S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every nonempty set S has an element a which is disjoint from S.

Note that the natural numbers are ordinals by this definition. For instance,
2 is an element of 4 = , and 2 is equal to and so it is a subset of .

It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals.

Furthermore, the elements of every ordinal are ordinals themselves. Whenever you have two ordinals S and T, S is an element of T if and only if S is a proper subset of T, and moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. And in fact, much more is true: Every set of ordinals is well-ordered. This important result generalizes the fact that every set of natural numbers is well-ordered and it allows us to use transfinite induction liberally with ordinals.

Another consequence is that every ordinal S is a set having as elements precisely the ordinals smaller than S. This statement completely determines the set-theoretic structure of every ordinal in terms of other ordinals. It's used to prove many other useful results about ordinals. One example of these is an important characterization of the order relation between ordinals: every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set.
Another example is the fact that the collection of all ordinals is not a set. Indeed, since every ordinal contains only other ordinals, it follows that every member of the collection of all ordinals is also its subset. Thus, if that collection were a set, it would have to be an ordinal itself by definition; then it would be its own member, which contradicts the axiom of regularity. (See also the Burali-Forti paradox).

An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its subsets has a greatest element.

Other definitions

There are other modern formulations of the definition of ordinal. Each of these is essentially equivalent to the definition given above. One of these definitions is the following. A class S is transitive if, whenever x is an element of y and y is an element of S, then x is an element of S. An ordinal is then defined to be a class S which is transitive, and such that every member of S is also transitive.

Arithmetic of ordinals

To define the sum S + T of two ordinal numbers S and T, one proceeds as follows: first the elements of T are relabeled so that S and T become disjoint, then the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S∪T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This way, a new well-ordered set is formed, and this well-ordered set is order-isomorphic to a unique ordinal, which is called S + T. This addition is associative and generalizes the addition of natural numbers.

The first transfinite ordinal is ω, the set of all natural numbers.
Let's try to visualize the ordinal ω+ω: two copies
of the natural numbers ordered in the normal fashion and the second copy completely to the right of the first. If we write the second copy as then ω+ω looks like
:0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...
This is different from ω because in ω only 0 does not have a direct predecessor while in ω+ω the two elements 0 and 0' don't have direct predecessors.
Here's 3 + ω:
:0 < 1 < 2 < 0' < 1' < 2' < ...
and after relabeling, this just looks like ω itself: we have 3 + ω = ω. But ω + 3 is not equal to ω since the former has a largest element and the latter doesn't. So our addition is not commutative.

You should now be able to see that (ω + 4) + ω = ω + (4 + ω) = ω + ω for example.

To multiply the two ordinals S and T you write down the well-ordered set T and replace each of its elements with a different copy of the well-ordered set S. This results in a well-ordered set, which defines a unique ordinal; we call it ST. Again, this operation is associative and generalizes the multiplication of natural numbers.

Here's ω2:
:0₀ < 1₀ < 2₀ < 3₀ < ... < 0₁ < 1₁ < 2₁ < 3₁ < ...
and we see: ω2 = ω + ω. But 2ω looks like this:
:0₀ < 1₀ < 0₁ < 1₁ < 0₂ < 1₂ < 0₃ < 1₃ < ...
and after relabeling, this looks just like ω and so we get 2ω = ω.
Multiplication of ordinals is not commutative.

Distributivity partially holds for ordinal arithmetic: R(S+T) = RS + RT. However, the other distributive law (T+U)R = TR + UR is not generally true: (1+1)ω is equal to 2ω = ω while 1ω + 1ω equals ω+ω. Therefore, the ordinal numbers do not form a ring.

A ring-like structure such as this, with only the left distributive law, is called a (left) nearring: however, ordinals are not quite a nearring either because they do not admit additive inverses (negation). Since a ring without negation is sometimes referred to as a rig, the ordinals may be said to form a left nearrig: a nearring without negation.

One can now go on to define exponentiation of ordinal numbers.
For finite exponents, this should be obvious, for instance $\omega^2 = \omega \omega$ using the operation of ordinal multiplication. But instead this can be visualized as a set of ordered pairs of natural numbers, ordered by a variant of lexicographical order which puts the least significant position first:

:(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...

And similarly for any finite n, $\omega^n$ can be visualized as the set of n-tuples of natural numbers.

Then for $\omega^\omega$ , we might try to visualize the set of infinite sequences of natural numbers. However, if we try to use any variant of lexicographical order on this set, we find it is not well-ordered. We must add the restriction that only a finite number of elements of the sequence are different from zero. Now our ordering works, and it looks like the ordering of natural numbers written in decimal notation, except with digit positions reversed, and with arbitrary natural numbers instead of just the digits 0-9:

:(0,0,0,...) < (1,0,0,0,...) < (2,0,0,0,...) < ... <
:(0,1,0,0,0,...) < (1,1,0,0,0,...) < (2,1,0,0,0,...) < ... <
:(0,2,0,0,0,...) < (1,2,0,0,0,...) < (2,2,0,0,0,...)
:: < ... <
:(0,0,1,0,0,0,...) < (1,0,1,0,0,0,...) < (2,0,1,0,0,0,...)
:: < ...

So in general, raise an ordinal S to the power of an ordinal T, we write down copies of the well-ordered set T, and then replace each element with some element of S, with the restriction that all but a finite number of elements of the sequence must be the first element of S. We find $1^\omega = 1$ , $2^\omega = \omega$ , $2^ = \omega 2 = \omega + \omega$ . We also have the expected exponentiation laws $(S^T)^U = S^$ and $S^ = S^T S^U$ .

Cantor normal form

Ordinal numbers present an extremely rich arithmetic. Every ordinal number $\alpha>0$ can be uniquely written as $\omega^ c_1 + \omega^c_2 + \ldots + \omega^c_k$ , where $k, c_1, c_2, \ldots, c_k$ are positive integers, and $\beta_1 > \beta_2 > \ldots > \beta_k$ are ordinal numbers (possibly $\beta_k=0$ ). This decomposition of $\alpha$ is called the Cantor normal form of $\alpha$ , and can be considered the positional base-ω numeral system. The highest exponent $\beta_1$ is called the degree of $\alpha$ , and satisfies $\beta_1\le\alpha$ . In case $\beta_1<\alpha$ , we have a finite representation of α with integers only (attached to a skeleton of ωs, additions, multiplications, and exponentiations).

There are ordinal numbers which cannot be reached from ω with a finite number of the arithmetical operations addition, multiplication and exponentiation. The smallest such is denoted by ε₀. This ordinal is very important in many induction proofs, because for many purposes, transfinite induction is only required up to ε₀. Note that $\epsilon_0 = \omega^$ , so that $\epsilon_0 = \omega^$ .

$\epsilon_0$ is still countable. There exist uncountable ordinals. The smallest uncountable ordinal is equal to the set of all countable ordinals, and is usually denoted by ω₁.

Topology and limit ordinals

The ordinals also carry an interesting order topology by virtue of being totally ordered. In this topology, the sequence 0, 1, 2, 3, 4, ... has limit ω and the sequence ω, ω^ω, ω^(ω^ω), ... has limit ε₀. Ordinals which don't have an immediate predecessor can always be written as a limit of a net of other ordinals (but not necessarily as the limit of a sequence, i.e. as a limit of countably many smaller ordinals) and are called limit ordinals; the other ordinals are the successor ordinals.

The topological spaces ω₁ and its successor ω₁+1 are frequently used as text-book examples of non-countable topological spaces.
For example, in the topological space ω₁+1, the element ω₁ is in the closure of the subset ω₁ even though no sequence of elements in ω₁ has the element ω₁ as its limit. The space ω₁ is first-countable, but not second-countable, and ω₁+1 has neither of these two properties.

Some special ordinals can be used to measure the size or cardinality of sets. These are the cardinal numbers.

See also

- successor ordinal

- limit ordinal

- cardinal number

- Serial number

- Nominal number

References

- Conway, J. H. and Guy, R. K. "Cantor's Ordinal Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 266-267 and 274, 1996.

- Sierpinski, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Pantswowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.

Category:Ordinal numbers
Category:Numbers

ko:순서수

ja:順序数

simple:Ordinal number

Roman numeral
The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. The system used in antiquity was slightly modified in the Middle Ages to produce the system we use today.
It is based on certain letters which are given values as numerals:
:I or i for one,
:V or v for five,
:X or x for ten,
:L or l for fifty,
:C or c for one hundred (centum),
:D or d for five hundred,
:M or m for one thousand (mille).

For larger numbers (five thousand and above), a bar is placed above a base numeral to indicate multiplication by 1000.

: for five thousand

: for ten thousand

: for fifty thousand

: for one hundred thousand

: for five hundred thousand

: for one million

Roman numerals are commonly used today in numbered lists (in outline format), clockfaces, pages preceding the main body of a book, chord triads in music analysis, the numbering of movie sequels, and the numbering of some sport events, like the Super Bowls or Olympic Games.

For arithmetics involving Roman numerals, see Roman arithmetic and Roman abacus.

Origins

Although the Roman numerals are now written with letters of the Roman alphabet, they were originally separate symbols. The Etruscans, for example, used I Λ X ⋔ 8 ⊕ for I V X L C M.

They appear to derive from notches on tally sticks, such as those used by Italian and Dalmatian shepherds into the 19^th century. Thus, the I descends from a notch scored across the stick. Every fifth notch was double cut (⋀, ⋁, ⋋, ⋌, etc.), and every tenth was cross cut (X), much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies, IIIIΛIII, but this could be written ΛIII (or VIII), as the Λ implies the four prior notches. Likewise, number four on the stick was the I-notch that could be felt just before the cut of the V, so it could be written as either IIII or IV. Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were later transfered to writing, the marks were easily identified with the existing Roman letters I, V, X.

(A folk etymology has it that the V represented a hand, and that the X was made by placing two Vs on top of each other, one inverted.)

The tenth V or X along the stick received an extra stroke. Thus 50 was written variously as N, И, K, Ψ, ⋔, etc., but perhaps most often as a chicken-track shape like a superimposed V and I. This had flattened to ⊥ (an inverted T) by the time of Augustus, and soon thereafter became identified with the graphically similar letter L. Likewise, 100 was variously Ж, ⋉, ⋈, H, or as any of the symbols for 50 above plus an extra stroke. The form Ж (that is, a superimposed X and I) came to predominate, was written variously as >I< or ƆIC, was then shortened to Ɔ or C, with C finally winning out because, as a letter, it stood for centum (Latin for 'hundred').

The hundredth V or X was marked with a box or circle. Thus 500 was like a Ɔ superposed on a ⋌ or ⊢ (that is, like a Þ with a cross bar), becoming a struck-through D or a Ð by the time of Augustus, under the graphic influence of the letter D. It was later identified as the letter D. Meanwhile, 1000 was a circled X: Ⓧ, ⊗, ⊕, and by Augustinian times was partially identified with the Greek letter Φ. It then evolved along several independent routes. Some variants, such as Ψ and CD (more accurately a reversed D adjacent to a regular D), were historical dead ends (although folk etymology later identified D for 500 as half of Φ for 1000 because of the CD variant), while two variants of ↀ survive to this day. One, CIƆ, lead to the convention of using parentheses to indicate multiplication by 1000 (later extended to double parentheses as in ↁ, ↂ, etc.); in the other, ↀ became ∞ and ⋈, eventually changing to M under the influence of the word mille ('thousand').

Zero
In general, the number zero did not have its own Roman numeral, but the concept of zero as a number was well known by all medieval computists (responsible for calculating the date of Easter). They included zero (via the Latin word nulla meaning nothing) as one of nineteen epacts, or the age of the moon on March 22. The first three epacts were nullae, xi, and xxii (written in minuscule or lower case). The first known computist to use zero was Dionysius Exiguus in 525, but the concept of zero was no doubt well known earlier. Only one instance of a Roman numeral for zero is known. About 725, Bede or one of his colleagues used the letter N, the initial of nullae, in a table of epacts, all written in Roman numerals.

A notation for the value zero is quite distinct from the role of the digit zero in a positional notation system. The lack of a zero digit prevented Roman numerals from developing into a positional notation, and led to their gradual replacement by Arabic numerals in the early second millennium.

IIII or IV?
The notation of Roman numerals has varied through the centuries. Originally, it was common to use IIII to represent "four", because IV represented the god Jove (and later YHWH). The subtractive notation (which uses IV instead of IIII) has become universally used only in modern times. For example, Forme of Cury, a manuscript from 1390, uses IX for "nine", but IIII for "four". Another document in the same manuscript, from 1381, uses IV and IX. A third document in the same manuscript uses both IIII and IV, and IX. Constructions such as IIX for "eight" have also been discovered. In many cases, there seems to have been a certain reluctance in the use of the less intuitive subtractive notation. Its use increased the complexity of performing Roman arithmetic, without conveying the benefits of a full positional notation system.

Calendars and clocks
Clock faces that are labelled using Roman numerals conventionally show IIII for 4 o'clock and IX for 9 o'clock, using the subtractive principle in one case and not in the other. There are several suggested explanations for this, several of which may be true:

- The four-character form IIII creates a visual symmetry with the VIII on the other side, which IV would not.

- The number of symbols on the clock totals twenty I's, four V's, and four X's, so clock makers need only a single mold with five I's, a V, and an X in order to make the correct number of numerals for the clocks, cast four times for each clock:
:: V IIII IX
:: VI II IIX
:: VII III X
:: VIII I IX
:IIX and one of the IX's can be rearranged or inverted to form XI and XII. The alternative uses seventeen I's, five V's, and four X's, possibly requiring the clock maker to have several different molds.

- IIII was the preferred way for the ancient Romans to write 4, since they to a large extent avoided subtraction.

- It has been suggested that since IV is the first two letters of IVPITER, the main god of the Romans, it was not appropriate to use.

- The I symbol would be the only symbol in the first 4 hours of the clock, the V symbol would only appear in the next 4 hours, and the X symbol only in the last 4 hours. This would add to the clock's radial symmetry.

- IV is difficult to read upside down and on an angle, particularly at that location on the clock.

- Louis XIV, king of France, preferred IIII over IV, ordered his clockmakers to produce clocks with IIII and not IV, and thus it has remained.

XCIX or IC?
Rules regarding Roman numerals often state that a symbol representing 10^x may not precede any symbol larger than 10^x+1. For example, C cannot be preceded by I or V, only by X (or, of course, by a symbol representing a value larger than C). Thus, one should represent the number "ninety-nine" as XCIX, not as the "shortcut" IC. However, these rules are not universally followed.

This 'problem' manifested in questions as to why 1999 was not written simply IMM or MIM.

Year in Roman numerals
In seventeenth century Europe, using Roman numerals for the year of publication for books was standard; there were many other places it was used as well. Publishers attempted to make the number easier to read by those more accustomed to Arabic positional numerals. On British title pages, there were often spaces between the groups of digits: M DCC LXI is one example. This may have come from the French, who separated the groups of digits with periods, as: M.DCC.LXI. or M. DCC. LXI. Notice the period at the end of the sequence; many foreign countries did this for Roman numerals in general, but not necessarily Britain. (Periods were also common on each side of numerals in running text, as in "commonet .iij. viros illos".)

These practices faded from general use before the start of the twentieth century, though the cornerstones of major buildings still occasionally use them. Roman numerals are today still used on building faces for dates: 2005 can be represented as MMV.

The film industry has used them perhaps since its inception to denote the year a film was made, so that it could be redistributed later, either locally or to a foreign country, without making it immediately clear to viewers what the actual date was. This became more useful when films were broadcast on television to partially conceal the age of films. From this came the policy of the broadcasting industry, including the BBC, to use them to denote the year in which a television program was made (the Australian Broadcasting Corporation has largely stopped this practice but still occasionally lapses).

Other modern usage by English-speaking peoples
Roman numerals remained in common use until about the 14th century, when they were replaced by Arabic numerals (thought to have been introduced to Europe from al-Andalus, by way of Arab traders and arithmetic treatises, around the 11th century). The use of Roman numerals today is mostly restricted to ordinal numbers, such as volumes or chapters in a book or the numbers identifying monarchs or popes (e.g. Elizabeth II, Benedict XVI, etc.).

Sometimes the numerals are written using lower-case letters (thus: i, ii, iii, iv, etc.), particularly if numbering paragraphs or sections within chapters, or for the pagination of the front matter of a book.

Undergraduate degrees at British universities are generally graded using I, IIi, IIii, III for first, upper second (often pronounced "two one"), lower second (often pronounced "two two") and third class respectively.

Modern English usage also employs Roman numerals in many books (especially anthologies), movies (e.g., Star Wars), sporting events (e.g., the Super Bowl), and historic events (e.g., World War I, World War II ). The common unifying theme seems to be stories or events that are episodic or annual in nature, with the use of classical numbering suggesting importance or timelessness.

In chemistry, Roman numerals were used to denote the group in the periodic table of the elements. But there was not international agreement as to whether the group of metals which dissolve in water should be called Group IA or IB, for example, so although references may use them, the international norm has recently switched to Arabic numerals.

In music theory a scale degrees or diatonic functions are often identified by Roman numerals (as in chord symbols) as follows:

Modern non-English speaking usage
The above uses are customary for English-speaking countries. Although many of them are also maintained in other countries, those countries have additional uses for Roman numerals which are unknown in English-speaking regions.

The French, the Portuguese, and the Spanish use capital Roman numerals to denote centuries. For example, 'XVIII' refers to the eighteenth century, so as to avoid confusion between the '18^th century' and the '1800s'. (The Italians take the opposite approach, basing names of centuries on the digits of the years; quattrocento for example is the Italian name for the fifteenth century.) Some scholars in English-speaking countries have adopted the French method, among them Lyon Sprague de Camp.

In Germany, Poland, and Russia, mixed Roman numerals are used to record dates. Just as an old clock recorded the hour by Roman numerals while the minutes were measured in Arabic numerals, the month is written in Roman numerals while the day is in Arabic numerals: 14-VI-1789 is June the fourteenth, 1789. This is how dates are inscribed on the walls of the Kremlin, for example. This method has the advantage that days and months are not confused in rapid note-taking, and that any range of days or months can be expressed without confusion. For instance, V-VIII is May to August, while 1-V-31-VIII is May first to August thirty-first.

In Eastern Europe, especially the Baltic nations, Roman numerals are used to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses. Monday is represented by I, which is the initial day of the week. Sunday is represented by VII, which is the final day of the week. The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. The following example hours-of-operation table would be for a business whose hours of operation are 9:30AM to 5:30PM on Mondays, Wednesdays, and Thursdays; 9:30AM to 7:00PM on Tuesdays and Fridays; and 9:30AM to 1:00PM on Saturdays; and which is closed on Sundays.

Since the French use capital Roman numerals to refer to the quarters of the year ('III' is the third quarter), and this has become the norm in some European standards organisation, the mixed Roman-Arabic method of recording the date has switched to lowercase Roman numerals in many circles, as '4-viii-1961'. (ISO has since specified that dates should be given in all Arabic numerals, in ISO 8601 formats.)

In Romania, Roman numerals are used for floor numbering.

Alternate forms
In the Middle Ages, Latin writers used a horizontal line above a particular numeral to represent one thousand times that numeral, and additional vertical lines on both sides of the numeral to denote one hundred times the number, as in these examples:

: for one thousand
: for five thousand
: || for one hundred thousand
: || for five hundred thousand

The same overline was also used with a different meaning, to clarify that the characters were numerals. Sometimes both underline and overline were used, e. g. , and in certain font-faces, particularly Times New Roman, the capital letters when used without spaces simulates the appearance of the under/over bar, e.g. MCMLXVII, which is often exagerated when written by hand.

Sometimes 500, usually D, was written as I followed by an apostrophus, resembling a backwards C (Ɔ), while 1,000, usually M, was written as CIƆ. This is believed to be a system of encasing numbers to denote thousands (imagine the Cs as parentheses). This system has its origins from Etruscan numeral usage. The D and M symbols to represent 500 and 1,000 were most likely derived from IƆ and CIƆ, respectively.

An extra Ɔ denoted 500, and multiple extra Ɔs are used to denote 5,000, 50,000, etc. For example:

Sometimes CIƆ was reduced to an lemniscate symbol ( $\infty$ ) for denoting 1,000. John Wallis is often credited for introducing this symbol to represent infinity, and one conjecture is that he based it off of this usage, since 1,000 was hyperbolically used to represent very large numbers.

In medieval times, before the letter j emerged as a distinct letter, a series of letters i in Roman numerals was commonly ended with a flourish; hence they actually looked like ij, iij, iiij, etc. This proved useful in preventing fraud, as it was impossible, for example, to add another i to vij to get viij. This practice is now merely an antiquarian's note; it is never used. (It did, however, lead to the Dutch diphthong IJ.)

Table of Roman numerals
The "modern" Roman numerals, post-Victorian era, are shown below:

An accurate way to write large numbers in Roman numerals is to handle first the thousands, then hundreds, then tens, then units.

Example: the number 1988.

One thousand is M, nine hundred is CM, eighty is LXXX, eight is VIII.

Put it together: MCMLXXXVIII (ⅯⅭⅯⅬⅩⅩⅩⅤⅠⅠⅠ).

Unicode has a number of characters specifically designated as Roman numerals, as part of the Number Forms range from U+2160 to U+2183. For example, MCMLXXXVIII could alternatively be written as
ⅯⅭⅯⅬⅩⅩⅩⅧ. This range includes both upper- and lowercase numerals, as well as pre-combined glyphs for numbers up to 12 (Ⅻ or XII), mainly intended for the clock faces for compatibility with non–West-European encodings. The pre-combined glyphs should only be used to represent the individual numbers where the use of individual glyphs is not wanted, and not to replace compounded numbers. Similarly precombined glyphs for 5000 and 10000 exist.

The Unicode characters are present only for compatibility with other character standards which provide these characters; for ordinary uses, the regular Latin letters are preferred. Displaying these characters requires a user agent that can handle Unicode and a font that contains appropriate glyphs for them.

Games
After the Renaissance, the Roman system could also be used to write chronograms. It was common to put in the first page of a book some phrase, so that when adding the I, V, X, L, C, D, M present in the phrase, the reader would obtain a number, usually the year of publication. The phrase was often (but not always) in Latin, as chronograms can be rendered in any language that utilises the Roman alphabet.

References

-

External links

- [http://ostermiller.org/calc/roman.html Roman Numeral Conversion Calculator and Self Test]

- [http://www.ubr.com/clocks/faq/iiii.html FAQ: Roman IIII vs. IV on Clock Dials]

- [http://www.straightdope.com/classics/a2_153 Why do clocks with Roman numerals use "IIII" instead of "IV"?] (from The Straight Dope)

- [http://web.archive.org/web/20041119032330/http://www.wilkiecollins.demon.co.uk/roman/front.htm Roman Numerals listing from Archive.org]

- [http://alanwood.net/unicode/number_forms.html Roman numbers in Unicode]

Category:Ancient Rome

ko:로마 숫자

ja:ローマ数字

th:เลขโรมัน

Hexadecimal
In mathematics and computer science, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. The current hexadecimal system was first introduced to the computing world in 1963 by IBM. An earlier version, using the digits 0–9 and u–z, was used by the Bendix G-15 computer, introduced in 1956.

For example, the decimal numeral 79 whose binary representation is 01001111 can be written as 4F in hexadecimal (4 = 0100, F = 1111).

It is a useful system in computers because there is an easy mapping from four bits to a single hex digit. A byte can be represented as two consecutive hexadecimal digits.

It was IBM that decided on the prefix of "hexa" rather than the proper Latin prefix of "sexa". The word "hexadecimal" is strange in that hexa is derived from the Greek έξι (hexi) for "six" and decimal is derived from the Latin for "ten". It may have been derived from the Latin root, but Greek deka is so similar to the Latin decem that some would not consider this nomenclature inconsistent. An older term was the pure Latin "sexidecimal", but that was changed because some people thought it too risqué, and it also had an alternative meaning of "base 60". However, the word "sexagesimal" (base 60) retains the prefix. The earlier Bendix documentation used the term "sexadecimal".

Since several years an alternate, unambiguous set of hexadecimal digits is proposed. (Cf. Hexadecimal time)

Representing hexadecimal

Some hexadecimal numbers are indistinguishable from a decimal number (to both humans and computers). Therefore, some convention is usually used to flag them.

In typeset text, the indication is often a subscripted suffix such as 5A3₁₆, 5A3_SIXTEEN, or 5A3_HEX.

In computer programming languages (which are nearly always plain text without such typographical distinctions as subscript and superscript) a wide variety of ways of marking hexadecimal numbers have appeared. These are also seen even in typeset text especially if that text relates to a programming language.

Some of the more common textual representations:

- Ada and VHDL enclose hexadecimal numerals in based "numeric quotes", e.g. "16#5A3#". (Note: Ada accepts this notation for all bases from 2 through 16 and for both integer and real types.)

- C and languages with a similar syntax (such as C++, C# and Java) prefix hexadecimal numerals with "0x", e.g. "0x5A3". The leading "0" is used so that the parser can simply recognize a number, and the "x" stands for hexadecimal (cf. 0 for Octal). The "x" in "0x" can be either in upper or lower case but is almost always seen written in lower case.

- In HTML, hexadecimal character references also use the x: ֣ should give the same as ֣ – with your browser ֣ and ֣ respectively (Hebrew accent munah).

- Some assemblers indicate hex by an appended "h" (if the numeral starts with a letter, then also with a preceding 0, to indicate that it is a number), e.g., "0A3Ch", "5A3h".

- Postscript indicates hex by a prefix "16#".

- Common Lisp use the prefixes "#x" and "#16r".

- Pascal, other assemblers (AT&T, Motorola), and some versions of BASIC use a prefixed "$", e.g. "$5A3".

- The Smalltalk programming language uses the prefix "16r". Note Smalltalk accepts the format "r" where radix is a number base from 2 upwards (i.e. 2r1110 is 10r14 or 16rE), with the practical limitation being within the ASCII character set range 0-9 and A-Z used to represent the digits. Some versions of Smalltalk allow fractional digits following a period character, ".", enabling hexadecimal (and other bases of) floating point numbers to be represented.

- Some versions of BASIC, notably Microsoft's variants including QBasic and Visual Basic), prefix hexadecimal numerals with "&H", e.g. "&H5A3"; others such as BBC BASIC just used "&" (used for octal in Microsoft's BASIC!).

- Notations such as X'5A3' are sometimes seen; PL/I uses such notation.

- Donald Knuth introduced the use of different fonts to represent radices in his book The TeXbook. In his notation, hexadecimal numbers are represented in a typewriter type, e.g. 5A3

Donald Knuth]
There is no single agreed-upon standard, so all the above conventions are in use, sometimes even in the same paper. However, as they are quite unambiguous, little difficulty arises from this.

The most commonly used (or encountered) notations are the ones with a prefix "0x" or a subscript-base 16, for hex numbers. For example, both 0x2BAD and 2BAD₁₆ represent the decimal number 11181 (or 11181₁₀).

The choice of the letters A through F to represent the additional digits was not universal in the early history of computers. During the 1950's, some installations favored using the digits 0 through 5 with a macron to indicate the values 10-15. Users of Bendix computers used the letters U through Z.

Uses

A common use of hexadecimal numerals is found in HTML and CSS. They use hexadecimal notation (hex triplets) to specify colours on web pages; there is just the # symbol, not a separate symbol for "hexadecimal". Twenty-four-bit color is represented in the format #RRGGBB, where RR specifies the value of the Red component of the color, GG the Green component and BB the Blue component. For example, a shade of red that is 238,9,63 in decimal is coded as #EE093F. This syntax is borrowed from the X Window System.

In URLs, special characters can be coded hexadecimally, with a percent sign used to introduce each byte; e.g., http://en.wikipedia.org/wiki/Main%20Page

The canonical written form of numeric IPv6 addresses represents each group of 16 bits as a separate hexadecimal number, to ease reading and transcription of the 128-bit addresses.

Fractions
As with other numeral systems, the hexadecimal system can be used in forming vulgar fractions, although recurring digits are common:

Because the radix 16 is a square (4²), hexadecimal fractions have an odd period much more often than decimal ones. Recurring decimals occur when the denominator in lowest terms has a prime factor not found in the radix. In the case of hexadecimal numbers, all fractions with denominators that are not a power of two will result in a recurring decimal.

Humor
Hexadecimal is sometimes used in programmer jokes because certain words can be formed using only hexadecimal digits. Some of these words are "dead", "beef", "babe", and with appropriate substitutions "c0ffee". [http://www.thinkgeek.com/tshirts/frustrations/6596/ This] is an example of such a joke. Since these are quickly recognisable by programmers, debugging setups sometimes initialise memory to them to help programmers see when something has not been initialised.

Another example is the magic number in FAT Mach-O files, which is "CAFEBABE".

A Knuth reward check is one hexadecimal dollar, or $2.56.

0xdeadbeef is often put into uninitialized memory.

Mapping to binary
When working with computers we often need to deal with binary data. It is much easier to handle in hexadecimal or octal than in binary (just think of lots of '0's and '1's) and whilst we are more familiar with the base 10 system, it is much easier to map binary to hexadecimal or octal than to decimal since each hexadecimal or octal digit maps to a whole number of bits (4 for hexadecimal 3 for octal). Hexadecimal's big advantage over octal is that exactly 2 digits represent a byte. This means that with hexadecimal, you can easily see from the value of a word what the value of the individual bytes will be; conversely, if you have the values of the bytes, you can easily assemble them to get the value of a word.

Consider converting 1111₂ to base 10. Since each position in a binary (base 2) number can only be either a 1 or 0, its value may be easily determined by its position from the right:

- 0001₂ = 1₁₀

- 0010₂ = 2₁₀

- 0100₂ = 4₁₀

- 1000₂ = 8₁₀
Therefore:

1111₂
= 8₁₀ + 4₁₀ + 2₁₀ + 1₁₀

= 15₁₀

This is a very simple example which still requires the addition of 4 numbers; whereas, with some practice, 1111₂ can be mapped directly to F₁₆ in one step (see table in Representing hexadecimal). When the binary number is very much greater, conversion to decimal becomes very much more tedious; however, when mapping to hexadecimal, it is simple to divide the binary number up in blocks of 4 positions and map each block of 4 bits to a single position hexadecimal number. For example a tedious conversion to decimal:

01011110101101010010₂
= 262144₁₀ + 65536₁₀ + 32768₁₀ + 16384₁₀ + 8192₁₀ + 2048₁₀ + 512₁₀ + 256₁₀ + 64₁₀ + 16₁₀ + 2₁₀

= 387922₁₀

Compared to the conversion to hexadecimal:

01011110101101010010₂
=
0101
1110
1011
0101
0010₂

=
5
E
B
5
2₁₆

=
5EB52₁₆

Conversion from hexadecimal back to binary is just as direct.

Octal is similarly easy to map to binary, only in blocks of 3 bits instead of 4.

Converting from Other bases

as with most bases there is a simple algorithm for converting a number of any base you can do integer division and remainders to a hexadecimal number.

Let d be the decimal number to convert, and the series h_ih_i-1...h₂h₁ be the hexadecimal digits representing the number.

1. h₁ := d mod 16

2. d := (d-h₁) / 16

3. if d 0 (return series h_i)

else go to 1

The following is a JavaScript implementation of the above algorithm probablly written by someone who didn't realize the computer was already working binary which makes far more direct conversion methods possible.

function toHex(d)

function toChar(n)

See also

- Base32

- Base64

- Numeral system — a list of other base systems

- Hexadecimal time

- Hexspeak

- Nibble — one hexadecimal digit can exactly represent one "nibble"

External links

- [http://www.intuitor.com/hex/ Intuitor Hex Headquarters] - A site dedicated to changing the traditional base 10 (decimal) standard to hexadecimal.

- [http://www.insidereality.net/site/content/math/base_conversion.php Simple Conversion Methods]

- [http://leetkey.mozdev.org Leet Key], a Firefox extension that supports ASCII/Hex conversions and typing

- [http://acms.synonet.com/bendix/intro/bitsof.pdf Bits of Meaning (pdf)] - Introduction to Computer Arithmetic for Bendix G-15 computer

- [http://www.pcnineoneone.com/howto/hex1.html Hexadecimal basics]

Calculators

- [http://www.iboost.com/tools/number.htm Hex/Unsigned Decimal/Binary Converter] (integer only)

- [http://www.statman.info/conversions/hexadecimal.html Hex/Unsigned Decimal Converter]

- [http://www.edepot.com/win95.html Virtual Calc 2000 - Arbitrary Precision Calculator] will do floating-point hexadecimal arithmetic

- [http://www.paulschou.com/tools/xlate/ Online HEX, Binary, Base64, etc... Encoder/Decoder]

Category:Computer arithmetic
Category:Numeration
16

ko:십육진법

ja:十六進記数法

th:เลขฐานสิบหก

73 (number)
73 is the natural number following 72 and preceding 74.

Cardinal seventy-three
Ordinal 73rd (seventy-third)
Factorization $prime$
Roman numeral LXXIII
Binary 1001001
Hexadecimal 49

In mathematics
Seventy-three is the 21st prime number. The previous is seventy-one, with which it comprises a twin prime. It is also a permutable prime with thirty-seven. 73 is a star number.

Every positive integer is the sum of at most 73 sixth powers (see Waring's problem).

In science

- The atomic number of tantalium

In astronomy,

: Messier object M73, a magnitude 9.0 apparent open cluster in the constellation Aquarius

: The New General Catalogue [http://www.ngcic.org/ object] NGC 73, a barred spiral galaxy in the constellation Cetus

:The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -716 July 16 and ended on 582 September 3. The duration of Saros series 73 was 1298.1 years, and it contained 73 solar eclipses.

:The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -378 May 16 and ended on 902 June 23. The duration of Saros series 73 was 1280.1 years, and it contained 72 lunar eclipses.

In other fields
Seventy-three is also:

- The total number of books in the Holy Bible in the Catholic version if the Book of Lamentations is counted as a book separate from the Book of Jeremiah.

- The designation of USA Interstate 73, a freeway in North Carolina.

- The registry of the U.S. Navy's nuclear aircraft carrier USS George Washington (CVN-73), named after U.S. President George Washington.

- In baseball, the single-season home run record set by Barry Bonds in 2001.

- The year AD 73, 73 BC, or 1973.

- Ham radio operators often use the number 73 as a way of saying aloha. This is because in morse code it forms a palindrome --... ...-- which has a very distinctive sound.

Category:Integers

ko:73

ja:73

Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. (Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin.)

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture. A strong form of the twin prime conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

Using his celebrated sieve method, Viggo Brun shows that the number of twin primes less than x is << x/(log x)². This result implies that the sum of the reciprocals of all twin primes converges (see Brun's constant). This is in stark contrast to the sum of the reciprocals of all primes, which diverges.
He also shows that every even number can be represented in infinitely many ways as a difference of two numbers both having at most 9 prime factors. Chen Jingrun's well known theorem states that for any m even, there are infinitely many primes that differs by m from a number having at most two prime factors.
(Before Brun attacked the twin prime problem, Jean Merlin had also attempted to solve this problem using the sieve method. He was killed in World War I.)

Every twin prime pair greater than 3 is of the form (6n - 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8.

It has been proven that the pair m, m + 2 is a twin prime if and only if

: $4((m-1)! + 1) = -m \mod (m(m+2))$

As of 2005, the largest known twin prime is 16869987339975 · 2¹⁷¹⁹⁶⁰ ± 1; it was found in 2005 by the Hungarians Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza and Antal Járai. It has 51779 digits [http://primes.utm.edu/bios/code.php?code=x24].

An empirical analysis of all prime pairs up to 4.35 · 10¹⁵ shows that the number of such pairs less than x is x·f(x)/(log x)² where f(x) is about 1.7 for small x and decreases to about 1.3 as x tends to infinity.
The limiting value of f(x) is conjectured to equal the twin prime constant
: $2 \prod_ (1 - \frac) = 1.3203236\ldots;$
this conjecture would imply the twin prime conjecture, but remains unresolved.

The first 35 twin prime pairs

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463), (521, 523), (569, 571), (599, 601), (617, 619), (641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883)

Only four pairs of these twin primes are irregular primes. The lower member of a pair is always a Chen prime.

See also

- Prime number

- Factorial

- Modular arithmetic

- Gödel's incompleteness theorem

External links

- [http://www.utm.edu/research/primes/lists/top20/twin.html Chris Caldwell: Twin Primes]

- [http://numbers.computation.free.fr/Constants/Primes/twin.html Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant]

- [http://www.rankyouragent.com/primes/primes.htm Martin Winer: Randomness and Prime Twin Proof]

Category:Prime numbers

ja:双子素数

Permutable prime
A permutable prime is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number. In base 10, the first few permutable primes are (with the permutations listed in parentheses):

2, 3, 5, 7, 11, 13(31), 17(71), 37(73), 79(97), 113(131, 311), 199(919, 991), 337(373, 733)

Any repunit prime can automatically be assumed to be a permutable prime as well. In base 2, only repunits can be permutable primes, because any 0 permuted to the one's place results in an even number; unless we consider 1 a prime number and 10 permutable with 01. The generalization can safely be made that for any number system based on an even number (such as decimal and sexagesimal), permutable primes can only have digits that are individually odd, for any even digit permuted to the one's place results in a number divisible by two.

External link
[http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A003459 Sloane's A003459]

Category:Prime numbers

Supersingular prime
In mathematics, a supersingular prime is a certain kind of prime number.

Formally, let H denote the upper half-plane. For a natural number n, let Γ₀(n) denote the modular group Γ₀, and let w_n be the Fricke involution defined by the block matrix 0, −1], [n, 0. Furthermore, let the modular curve X₀(n) be the compactification (with added cusps) of

:Y₀(n) = Γ₀(n)\H,

and for any prime p, define

:X₀⁺(p) = X₀(p) / w_p.

Then p is supersingular means by definition that the genus of X₀⁺(p) is zero.

It is also possible to define supersingular primes in a number-theoretic way using supersingular elliptic curves defined over the algebraic closure of the finite field GF(p) that have their j-invariant in GF(p). [details, anyone?]

As is turns out, there are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 . It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.

Note the set of supersingular primes is a subset of the set of the Chen primes.

Category:Prime numbers
Category:Moonshine theory

Centered heptagonal number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for n is given by the formula

: $\over2$ .

This can also be calculated by multiplying the triangular number for (n - 1) by 7, then adding 1.

The first few centered heptagonal numbers are

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953

Centered heptagonal numbers alternate parity in the pattern odd-even-even-odd.

centered heptagonal prime

A centered heptagonal prime is a centered heptagonal number that is prime. The first few centered heptagonal primes are

43, 71, 197, 463, 547, 953, 1471, 1933, ....

See also regular heptagonal number.

Category:Figurate numbers

Atomic number
The atomic number (Z) is a term used in chemistry and physics to represent the number of protons found in the nucleus of an atom. In an atom of neutral charge, the number of electrons also equals the atomic number.

The atomic number originally meant the number of an element's place in the periodic table. When Mendeleev arranged the known chemical elements grouped by their similarities in chemistry, it was noticeable that placing them in strict order of atomic mass resulted in some mismatches. Iodine and tellurium, if listed by atomic mass, appeared to be in the wrong order, and would fit better if their places in the table were swapped. Placing them in the order which fit chemical properties most closely, their number in the table was their atomic number. This number appeared to be approximately proportional to the mass of the atom, but, as the discrepancy showed, reflected some other property than mass.

The anomalies in this sequence were finally explained after research by Henry Gwyn Jeffreys Moseley in 1913. Moseley discovered a strict relationship between the x-ray diffraction spectra of elements, and their correct location in the periodic table. It was later shown that the atomic number corresponds to the electric charge of the nucleus — in other words the number of protons. It is the charge which gives elements their chemical properties, rather than the atomic mass.

The atomic number is closely related to the mass number (although they should not be confused) which is the number of protons and neutrons in the nucleus of an atom. The mass number often comes after the name of the element, e.g. carbon-14 (used in carbon dating).

See also

- Periodic table

- List of elements by number

- Effective atomic number

Category:Chemical properties
Category:Nuclear physics

als:Ordnungszahl

ko:원자 번호

ja:原子番号

simple:Atomic number

th:เลขอะตอม

Lutetium

Lutetium is a chemical element in the periodic table that has the symbol Lu and atomic number 71. A metallic element of the rare earth group, lutetium usually occurs in association with yttrium and is sometimes used in metal alloys and as a catalyst in various processes. A strict correlation between periodic table blocks and chemical series for neutral atoms would describe lutetium as a transition metal, but it is commonly considered a lanthanide.

Notable characteristics and applications
Lutetium is a silvery white corrosion-resistant trivalent metal that is relatively stable in air and is the heaviest and hardest of the rare earth elements. Lutetium has the highest spin quantum number of the elements, at 7.

This element is very expensive to obtain in useful quantities and therefore it has very few commercial uses. However, stable lutetium can be used as catalysts in petroleum cracking in refineries and can also be used in alkylation, hydrogenation, and polymerization applications.

History
Lutetium (Latin Lutetia meaning Paris) was independently discovered in 1907 by French scientist Georges Urbain and Austrian mineralogist Baron Carl Auer von Welsbach. Both men found lutetium as an impurity in the mineral ytterbia which was thought by Swiss chemist Jean Charles Galissard de Marignac (and most others) to consist entirely of the element ytterbium.

The separation of lutetium from Marignac's ytterbium was first described by Urbain and the naming honor therefore went to him. He chose the names neoytterbium (new ytterbium) and lutecium for the new element but neoytterbium was eventually reverted back to ytterbium and in 1949 the spelling of element 71 was changed to lutetium.

Welsbach proposed the names cassiopium for element 71 (after the constellation Cassiopeia) and albebaranium for the new name of ytterbium but these naming proposals where rejected (although many German scientists still call element 71 cassiopium).

Occurrence
Found with almost all other rare-earth metals but never by itself, lutetium is very difficult to separate from other elements and is the least abundant of all naturally occurring elements. Consequently, it is also one of the most expensive metals, costing about six times as much per gram as gold.

The principal commercially viable ore of lutetium is the rare earth phosphate mineral monazite: (Ce, La, etc.)PO₄ which contains 0.003% of the element. Pure lutetium metal has only relatively recently been isolated and is very difficult to prepare (thus it is one of the most rare and expensive of the rare earth metals). It is separated from other rare earth elements by ion exchange (reduction of anhydrous LuCl₃ or LuF₃ by either an alkali metal or alkaline earth metal).

Isotopes
Naturally occurring lutetium is composed of 1 stable isotope Lu-175 (97.41% natural abundance). 33 radioisotopes have been characterized, with the most stable being Lu-176 with a half-life of 3.78 × 10¹⁰ years (2.59% natural abundance), Lu-174 with a half-life of 3.31 years, and Lu-173 with a half-life of 1.37 years. All of the remaining radioactive isotopes have half-lifes that are less than 9 days, and the majority of these have half lifes that are less than a half an hour. This element also has 18 meta states, with the most stable being Lu-177m (t_½ 160.4 days), Lu-174m (t_½ 142 days) and Lu-178m (t_½ 23.1 minutes).

The isotopes of lutetium range in atomic weight from 149.973 (Lu-150) to 183.961 (Lu-184). The primary decay mode before the most abundant stable isotope, Lu-175, is electron capture (with some alpha and positron emission), and the primary mode after is beta emission. The primary decay products before Lu-175 are element 70 (ytterbium) isotopes and the primary products after are element 72 (hafnium) isotopes.

Compounds
Fluoride: LuF₃, Chloride: LuCl₃, Bromide: LuBr₃, Iodide: LuI₃, Oxide: Lu₂O₃, Sulfide: Lu₂S₃, Telluride: Lu₂Te₃, Nitride: LuN

Precautions
Like other rare-earth metals lutetium is regarded as having a low toxicity rating but it and especially its compounds should be handled with care nonetheless. Metal dust of this element is a fire and explosion hazard. Lutetium plays no biological role in the human body but is thought to help stimulate metabolism.

References

- Guide to the Elements - Revised Edition, Albert Stwertka, (Oxford University Press; 1998) ISBN 0-19-508083-1

- [http://periodic.lanl.gov/elements/71.html Los Alamos National Laboratory's Chemistry Division: Periodic Table - Lutetium]

External links

- [http://www.webelements.com/webelements/elements/text/Lu/index.html WebElements.com - Lutetium] (also used as a reference)

- [http://education.jlab.org/itselemental/ele071.html It's Elemental - Lutetium]

Category:Chemical elements
Category:Transition metals

ja:ルテチウム

th:ลูทีเตียม

Astronomy
:This article is about the science branch. For information about the magazine, see Astronomy (magazine).

Astronomy (magazine) as they circled the Moon in 1969. Located near the center of the far side of Earth's Moon, its diameter is about 58 miles (93 km).]]

Astronomy (Greek: αστρονομία = άστρον + νόμος, astronomia = astron + nomos, literally, "law of the stars") is the science of celestial objects and phenomena that originate outside the Earth's atmosphere, such as stars, planets, comets, galaxies, and the cosmic background radiation. It is concerned with the formation and development of the universe, the evolution and physical and chemical properties of celestial objects and the calculation of their motions. Astronomical observations are not only relevant for astronomy as such, but provide essential information for the verification of fundamental theories in physics, such as general relativity theory. Complementary to observational astronomy, theoretical astrophysics seeks to explain astronomical phenomena.

Astronomy is one of the oldest sciences, with a scientific methodology existing at the time of Ancient Greece and advanced observation techniques possibly much earlier (see archaeoastronomy). Historically, amateurs have contributed to many important astronomical discoveries, and astronomy is one of the few sciences where amateurs can still play an active role, especially in the discovery and observation of transient phenomena.

Astronomy is not to be confused with astrology, which assumes that people's destiny and human affairs in general are correlated to the apparent positions of astronomical objects in the sky -- although the two fields share a common origin, they are quite different; astronomers embrace the scientific method, while astrologers do not.
In other words, there is no proof that the stars predict our future, but there is proof
that our planet is 93 million miles from the sun.
Divisions

In ancient Greece and other early civilizations, astronomy consisted largely of astrometry, measuring positions of stars and planets in the sky. Later, the work of Kepler and Newton, whose work led to the development of celestial mechanics, mathematically predicting the motions of celestial bodies interacting under gravity, and solar system objects in particular. Much of the effort in these two areas, once done largely by hand, is highly automated nowadays, to the extent that they are rarely considered as independent disciplines anymore. Motions and positions of objects are now more easily determined, and modern astronomy is more concerned with observing and understanding the actual physical nature of celestial objects.

Since the twentieth century, the field of professional astronomy has split into observational astronomy and theoretical astrophysics. Although most astronomers incorporate elements of both into their research, because of the different skills involved, most professional astronomers tend to specialize in one or the other. Observational astronomy is concerned mostly with acquiring data, which involves building and maintaining instruments and processing the results; this branch is at times referred to as "astrometry" or simply as "astronomy". Theoretical astrophysics is concerned mainly with ascertaining the observational implications of different models, and involves working with computer or analytic models.

The fields of study can also be categorized in other ways. Categorization by the region of space under study (for example, Galactic astronomy, Planetary Sciences); by subject, such as star formation or cosmology; or by the method used for obtaining information.

By subject or problem addressed

theoretical astrophysics. Photographed by Mars Global Surveyor, the long dark streak is formed by a moving swirling column of Martian atmosphere (with similarities to a terrestrial tornado). The dust devil itself (the black spot) is climbing the crater wall. The streaks on the right are sand dunes on the crater floor.]]

- Astrometry: the study of the position of objects in the sky and their changes of position. Defines the system of coordinates used and the kinematics of objects in our galaxy.

- Astrophysics: the study of physics of the universe, includ}

71 (number)

In mathematics

In science

In other fields

Natural number

History of natural numbers and the status of zero

Notation

Formal definitions

Peano axioms

Constructions based on set theory

The standard construction

Other constructions

Properties

Generalizations

Footnote

72 (number)

In mathematics

In science

In other fields

Ordinal number

Introduction

Modern definition and first properties

Other definitions

Arithmetic of ordinals

Cantor normal form

Topology and limit ordinals

See also

References

Roman numeral

Origins

Zero

IIII or IV?

Calendars and clocks

XCIX or IC?

Year in Roman numerals

Other modern usage by English-speaking peoples

Modern non-English speaking usage

Alternate forms

Table of Roman numerals

Games

References

External links

Hexadecimal

Representing hexadecimal

Uses

Fractions

Humor

Mapping to binary

Converting from Other bases

0 (return series hi) else go to 1 The following is a JavaScript implementation of the above algorithm probablly written by someone who didn't realize the computer was already working binary which makes far more direct conversion methods possible. function toHex(d) function toChar(n)

See also

External links

Calculators

73 (number)

In mathematics

In science

In other fields

Twin prime

The first 35 twin prime pairs

See also

External links

Permutable prime

External link

Supersingular prime

Centered heptagonal number

centered heptagonal prime

Atomic number

See also

Lutetium

Notable characteristics and applications

History

Occurrence

Isotopes

Compounds

Precautions

References

External links

Astronomy

Divisions

By subject or problem addressed

0 (return series h_i)
else go to 1 The following is a JavaScript implementation of the above algorithm probablly written by someone who didn't realize the computer was already working binary which makes far more direct conversion methods possible.
function toHex(d) function toChar(n)