The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom. It is identical to the charge number of the nucleus; the atomic number uniquely identifies a chemical element. In an uncharged atom, the atomic number is equal to the number of electrons; the sum of the atomic number Z and the number of neutrons, N, gives the mass number A of an atom. Since protons and neutrons have the same mass and the mass defect of nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units, is within 1% of the whole number A. Atoms with the same atomic number Z but different neutron numbers N, hence different atomic masses, are known as isotopes. A little more than three-quarters of occurring elements exist as a mixture of isotopes, the average isotopic mass of an isotopic mixture for an element in a defined environment on Earth, determines the element's standard atomic weight, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century.
The conventional symbol Z comes from the German word Zahl meaning number, before the modern synthesis of ideas from chemistry and physics denoted an element's numerical place in the periodic table, whose order is but not consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was the nuclear charge and a physical characteristic of atoms, did the word Atomzahl come into common use in this context. Loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, so they can be numbered in order. Dmitri Mendeleev claimed. However, in consideration of the elements' observed chemical properties, he changed the order and placed tellurium ahead of iodine; this placement is consistent with the modern practice of ordering the elements by proton number, Z, but that number was not known or suspected at the time. A simple numbering based on periodic table position was never satisfactory, however.
Besides the case of iodine and tellurium several other pairs of elements were known to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not obvious, led to inconsistency and uncertainty in the periodic numbering of elements at least from lutetium onward. In 1911, Ernest Rutherford gave a model of the atom in which a central core held most of the atom's mass and a positive charge which, in units of the electron's charge, was to be equal to half of the atom's atomic weight, expressed in numbers of hydrogen atoms; this central charge would thus be half the atomic weight. In spite of Rutherford's estimation that gold had a central charge of about 100, a month after Rutherford's paper appeared, Antonius van den Broek first formally suggested that the central charge and number of electrons in an atom was equal to its place in the periodic table.
This proved to be the case. The experimental position improved after research by Henry Moseley in 1913. Moseley, after discussions with Bohr, at the same lab, decided to test Van den Broek's and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fitted the Bohr theory's postulation that the frequency of the spectral lines be proportional to the square of Z. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube; the square root of the frequency of these photons increased from one target to the next in an arithmetic progression. This led to the conclusion that the atomic number does correspond to the calculated electric charge of the nucleus, i.e. the element number Z. Among other things, Moseley demonstrated that the lanthanide series must have 15 members—no fewer and no more—which was far from obvious from the chemistry at that time.
After Moseley's death in 1915, the atomic numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, corresponding to atomic numbers 43, 61, 72, 75, 85, 87 and 91. From 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had been discovered, so that the periodic table was complete with no gaps as far as curium. In 1915 the rea
The Messier objects are a set of 110 astronomical objects cataloged by the French astronomer Charles Messier in his Catalogue des Nébuleuses et des Amas d'Étoiles. Because Messier was interested in finding only comets, he created a list of non-comet objects that frustrated his hunt for them; the compilation of this list, in collaboration with his assistant Pierre Méchain, is known as the Messier catalogue. This catalogue of objects is one of the most famous lists of astronomical objects, many Messier objects are still referenced by their Messier number; the catalogue includes some astronomical objects that can be observed from Earth's Northern Hemisphere such as deep-sky objects, a characteristic which makes the Messier objects popular targets for amateur astronomers. A preliminary version first appeared in the Memoirs of the French Academy of Sciences in 1771, the last item was added in 1966 by Kenneth Glyn Jones, based on Messier's observations; the first version of Messier's catalogue contained 45 objects and was published in 1774 in the journal of the French Academy of Sciences in Paris.
In addition to his own discoveries, this version included objects observed by other astronomers, with only 17 of the 45 objects being Messier's. By 1780 the catalogue had increased to 80 objects; the final version of the catalogue containing 103 objects was published in 1781 in the Connaissance des Temps for the year 1784. However, due to what was thought for a long time to be the incorrect addition of Messier 102, the total number remained 102. Other astronomers, using side notes in Messier's texts filled out the list up to 110 objects; the catalogue consists of a diverse range of astronomical objects, ranging from star clusters and nebulae to galaxies. For example, Messier 1 is a supernova remnant, known as the Crab Nebula, the great spiral Andromeda Galaxy is M31. Many further inclusions followed in the next century when the first addition came from Nicolas Camille Flammarion in 1921, who added Messier 104 after finding Messier's side note in his 1781 edition exemplar of the catalogue. M105 to M107 were added by Helen Sawyer Hogg in 1947, M108 and M109 by Owen Gingerich in 1960, M110 by Kenneth Glyn Jones in 1967.
The first edition of 1771 covered 45 objects numbered M1 to M45. The total list published by Messier in 1781 contained 103 objects, but the list was expanded through successive additions by other astronomers, motivated by notes in Messier's and Méchain's texts indicating that at least one of them knew of the additional objects; the first such addition came from Nicolas Camille Flammarion in 1921, who added Messier 104 after finding a note Messier made in a copy of the 1781 edition of the catalog. M105 to M107 were added by Helen Sawyer Hogg in 1947, M108 and M109 by Owen Gingerich in 1960, M110 by Kenneth Glyn Jones in 1967. M102 was observed by Méchain. Méchain concluded that this object was a re-observation of M101, though some sources suggest that the object Méchain observed was the galaxy NGC 5866 and identify that as M102. Messier's final catalogue was included in the Connaissance des Temps for 1784, the French official yearly publication of astronomical ephemerides; these objects are still known by their "Messier number" from this list.
Messier did his astronomical work at the Hôtel de Cluny, in Paris, France. The list he compiled contains only objects found in the sky area he could observe: from the north celestial pole to a celestial latitude of about −35.7°. He did not observe or list objects visible only from farther south, such as the Large and Small Magellanic Clouds; the Messier catalogue comprises nearly all the most spectacular examples of the five types of deep-sky object – diffuse nebulae, planetary nebulae, open clusters, globular clusters, galaxies – visible from European latitudes. Furthermore all of the Messier objects are among the closest to Earth in their respective classes, which makes them studied with professional class instruments that today can resolve small and visually spectacular details in them. A summary of the astrophysics of each Messier object can be found in the Concise Catalog of Deep-sky Objects. Since these objects could be observed visually with the small-aperture refracting telescope used by Messier to study the sky, they are among the brightest and thus most attractive astronomical objects observable from Earth, are popular targets for visual study and astrophotography available to modern amateur astronomers using larger aperture equipment.
In early spring, astronomers sometimes gather for "Messier marathons", when all of the objects can be viewed over a single night. Lists of astronomical objects List of Messier objects Caldwell catalogue Deep-sky object Herschel 400 Catalogue New General Catalogue SEDS Messier Database Charles Messier Charles Messier's Catalog of Nebulae and Star Clusters History of the Messier Catalog Interactive Messier Catalog Greenhawk Observatory Listing of Copyright-free Images of all Messier Objects CCD Images of Messier Objects 12 Dimensional String Messier Gallery The Messier Catalogue Merrifield, Mike. "Messier Objects". Deep Sky Videos. Brady Haran. Messier Objects at Constellation Guide
40 is the natural number following 39 and preceding 41. Though the word is related to "four", the spelling "forty" replaced "fourty" in the course of the 17th century and is now the standard form. Forty is a composite number, an octagonal number, as the sum of the first four pentagonal numbers, it is a pentagonal pyramidal number. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number. Given 40, the Mertens function returns 0. 40 is the smallest number n with 9 solutions to the equation φ = n. Forty is the number of n-queens problem solutions for n = 7. 40 is a repdigit in base 3 and a Harshad number in base 10. The atomic number of zirconium. Negative forty is the unique temperature, it is referred to as either "minus forty" or "forty below". Messier object M40, a magnitude 9.0 double star in the constellation Ursa Major The New General Catalogue object NGC 40, a magnitude 12.4 planetary nebula in the constellation Cepheus The number 40 is found in many traditions without any universal explanation for its use.
In Jewish, Christian and other Middle Eastern traditions it is taken to represent a large, approximate number, similar to "umpteen". Enki or Enkil is a god in Sumerian mythology known as Ea in Akkadian and Babylonian mythology, he was patron god of the city of Eridu, but the influence of his cult spread throughout Mesopotamia and to the Canaanites and Hurrians. He was the deity of crafts, he was associated with the southern band of constellations called stars of Ea, but with the constellation AŠ-IKU, the Field. Beginning around the second millennium BCE, he was sometimes referred to in writing by the numeric ideogram for "40," referred to as his "sacred number."A large number of myths about Enki have been collected from many sites, stretching from Southern Iraq to the Levantine coast. He figures in the earliest extant cuneiform inscriptions throughout the region and was prominent from the third millennium down to Hellenistic times; the exact meaning of his name is uncertain: the common translation is "Lord of the Earth": the Sumerian en is translated as a title equivalent to "lord".
The name Ea is Hurrian in origin while others claim that it is of Semitic origin and may be a derivation from the West-Semitic root *hyy meaning "life" in this case used for "spring", "running water." In Sumerian E-A means "the house of water", it has been suggested that this was the name for the shrine to the God at Eridu. In the Hebrew Bible, forty is used for time periods, forty days or forty years, which separate "two distinct epochs". Rain fell for "forty days and forty nights" during the Flood. Spies were sent by Moses to explore the land of Canaan for "forty days"; the Hebrew people lived in the lands outside of the promised land for "forty years". This period of years represents the time. Several Jewish leaders and kings are said to have ruled for "forty years", that is, a generation. Examples include Eli, Saul and Solomon. Goliath challenged the Israelites twice a day for forty days. Moses spent three consecutive periods of "forty days and forty nights" on Mount Sinai:He went up on the seventh day of Sivan, after God gave the Torah to the Jewish people, in order to learn the Torah from God, came down on the seventeenth day of Tammuz, when he saw the Jews worshiping the Golden Calf and broke the tablets.
He went up on the eighteenth day of Tammuz to beg forgiveness for the people's sin and came down without God's atonement on the twenty-ninth day of Av. He went up on the first day of Elul and came down on the tenth day of Tishrei, the first Yom Kippur, with God's atonement. A mikvah consists of 40 se'ah of water The prophet Elijah had to walk 40 days and 40 nights before arriving to mount Horeb. 40 lashes is one of the punishments meted out by the Sanhedrin, though in actual practice only 39 lashes were administered. Alludes to the same with ties to the prophecy in The Book of Daniel. "For forty years—one year for each of the forty days you explored the land—you will suffer for your sins and know what it is like to have me against you." One of the prerequisites for a man to study Kabbalah is. "The registering of these men was carried on cruelly, assiduously, from the rising of the sun to its going down, was not brought to an end in forty days". Christianity uses forty to designate important time periods.
Before his temptation, Jesus fasted "forty days and forty nights" in the Judean desert. Forty days was the period from the resurrection of Jesus to the ascension of Jesus. According to Stephen, Moses' life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, his return to lead his people out. In modern Christian practice, Lent consists of t
Holmium is a chemical element with symbol Ho and atomic number 67. Part of the lanthanide series, holmium is a rare-earth element. Holmium was discovered by Swedish chemist Per Theodor Cleve, its oxide was first isolated from rare-earth ores in 1878. The element's name comes from the Latin name for the city of Stockholm. Elemental holmium is a soft and malleable silvery-white metal, it is too reactive to be found uncombined in nature, but when isolated, is stable in dry air at room temperature. However, it reacts with water and corrodes and burns in air when heated. Holmium is found in the minerals monazite and gadolinite and is commercially extracted from monazite using ion-exchange techniques, its compounds in nature and in nearly all of its laboratory chemistry are trivalently oxidized, containing Ho ions. Trivalent holmium ions have fluorescent properties similar to many other rare-earth ions, thus are used in the same way as some other rare earths in certain laser and glass-colorant applications.
Holmium has the highest magnetic permeability of any element and therefore is used for the polepieces of the strongest static magnets. Because holmium absorbs neutrons, it is used as a burnable poison in nuclear reactors. Holmium is a soft and malleable element, corrosion-resistant and stable in dry air at standard temperature and pressure. In moist air and at higher temperatures, however, it oxidizes, forming a yellowish oxide. In pure form, holmium possesses a metallic, bright silvery luster. Holmium oxide has some dramatic color changes depending on the lighting conditions. In daylight, it has a tannish yellow color. Under trichromatic light, it is fiery orange-red indistinguishable from the appearance of erbium oxide under the same lighting conditions; the perceived color change is related to the sharp absorption bands of holmium interacting with a subset of the sharp emission bands of the trivalent ions of europium and terbium, acting as phosphors. Holmium has the highest magnetic moment of any occurring element and possesses other unusual magnetic properties.
When combined with yttrium, it forms magnetic compounds. Holmium is paramagnetic at ambient conditions, but is ferromagnetic at temperatures below 19 K. Holmium metal tarnishes in air and burns to form holmium oxide: 4 Ho + 3 O2 → 2 Ho2O3Holmium is quite electropositive and is trivalent, it reacts with cold water and quite with hot water to form holmium hydroxide: 2 Ho + 6 H2O → 2 Ho3 + 3 H2 Holmium metal reacts with all the halogens: 2 Ho + 3 F2 → 2 HoF3 2 Ho + 3 Cl2 → 2 HoCl3 2 Ho + 3 Br2 → 2 HoBr3 2 Ho + 3 I2 → 2 HoI3 Holmium dissolves in dilute sulfuric acid to form solutions containing the yellow Ho ions, which exist as a 3+ complexes: 2 Ho + 3 H2SO4 → 2 Ho3+ + 3 SO2−4 + 3 H2 Holmium's most common oxidation state is +3. Holmium in solution is in the form of Ho3+ surrounded by nine molecules of water. Holmium dissolves in acids. Natural holmium contains one stable isotope, holmium-165; some synthetic radioactive isotopes are known. All other radioisotopes have ground-state half-lives not greater than 1.117 days, most have half-lives under 3 hours.
However, the metastable 166m1Ho has a half-life of around 1200 years because of its high spin. This fact, combined with a high excitation energy resulting in a rich spectrum of decay gamma rays produced when the metastable state de-excites, makes this isotope useful in nuclear physics experiments as a means for calibrating energy responses and intrinsic efficiencies of gamma ray spectrometers. Holmium was discovered by Jacques-Louis Soret and Marc Delafontaine in 1878 who noticed the aberrant spectrographic absorption bands of the then-unknown element; the following year, Per Teodor Cleve independently discovered the element while he was working on erbia earth. Using the method developed by Carl Gustaf Mosander, Cleve first removed all of the known contaminants from erbia; the result of that effort was one brown and one green. He named the green one thulia. Holmia was found to be the holmium oxide, thulia was thulium oxide. In Henry Moseley's classic paper on atomic numbers, holmium was assigned an atomic number of 66.
Evidently, the holmium preparation he had been given to investigate had been grossly impure, dominated by neighboring dysprosium. He would have seen x-ray emission lines for both elements, but assumed that the dominant ones belonged to holmium, instead of the dysprosium impurity. Like all other rare earths, holmium is not found as a free element, it does occur combined with other elements in gadolinite and other rare-earth minerals. No holmium-dominant mineral has yet been found; the main mining areas are China, United States, India, Sri Lanka, Australia with reserves of holmium estimated as 400,000 tonnes. Holmium makes up 1.4 parts per million of the Earth's crust by mass. This makes it the 56th most abundant element in the Earth's crust. Holmium makes up 1 part per million of the soils, 400 parts per quadrillion of seawater, none of Earth's atmosphere. Holmium is rare for a lanthanide, it makes up 500 parts per trillion of the universe by mass. It is commercially extracted by ion exch
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. 100 is the square of 10. The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages. 100 is the sum of the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g. 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. 100 is the sum of the cubes of the first four integers. This is related by Nicomachus's theorem to the fact that 100 equals the square of the sum of the first four integers: 100 = 102 = 2.26 + 62 = 100, thus 100 is a Leyland number.100 is an 18-gonal number. It is divisible by 25, the number of primes below it, it can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors.
100 is a Harshad number in base 10, in base 4, in that base it is a self-descriptive number. There are 100 prime numbers whose digits are in ascending order. 100 is the smallest number. One hundred is the atomic number of fermium, an actinide and the first of the heavy metals that cannot be created through neutron bombardment. On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level; the Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is used to define the boundary between Earth's atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of the Jewish New Year. A religious Jew is expected to utter at least 100 blessings daily. In the Hindu book of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas; the United States Senate has 100 Senators. Most of the world's currencies are divided into 100 subunits; the 100 Euro banknotes feature a picture of a Rococo gateway on the obverse and a Baroque bridge on the reverse.
The U. S. hundred-dollar bill has Benjamin Franklin's portrait. S. bill in print. American savings bonds of $100 have Thomas Jefferson's portrait, while American $100 treasury bonds have Andrew Jackson's portrait. One hundred is also: The number of years in a century; the number of pounds in an American short hundredweight. In Greece, India and Nepal, 100 is the police telephone number. In Belgium, 100 is the firefighter telephone number. In United Kingdom, 100 is the operator telephone number; the HTTP status code indicating that the client should continue with its request. The 100 The age at which a person becomes a centenarian; the number of yards in an American football field. The number of runs required for a cricket batsman to score a significant milestone; the number of points required for a snooker player to score a century break, a significant milestone. The record number of points scored in one NBA game by a single player, set by Wilt Chamberlain of the Philadelphia Warriors on March 2, 1962.
1 vs. 100 AFI's 100 Years... Hundred Hundred Hundred Days Hundred Years' War List of highways numbered 100 Top 100 Greatest 100 Wells, D; the Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group.: 133 Chisholm, Hugh, ed.. "Hundred". Encyclopædia Britannica. Cambridge University Press. On the Number 100
The apparent magnitude of an astronomical object is a number, a measure of its brightness as seen by an observer on Earth. The magnitude scale is logarithmic. A difference of 1 in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The brighter an object appears, the lower its magnitude value, with the brightest astronomical objects having negative apparent magnitudes: for example Sirius at −1.46. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry. Apparent magnitudes are used to quantify the brightness of sources at ultraviolet and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or simply as V, as in "mV = 15" or "V = 15" to describe a 15th-magnitude object; the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes.
The brightest stars in the night sky were said to be of first magnitude, whereas the faintest were of sixth magnitude, the limit of human visual perception. Each grade of magnitude was considered twice the brightness of the following grade, although that ratio was subjective as no photodetectors existed; this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest and is believed to have originated with Hipparchus. In 1856, Norman Robert Pogson formalized the system by defining a first magnitude star as a star, 100 times as bright as a sixth-magnitude star, thereby establishing the logarithmic scale still in use today; this implies that a star of magnitude m is about 2.512 times as bright as a star of magnitude m + 1. This figure, the fifth root of 100, became known as Pogson's Ratio; the zero point of Pogson's scale was defined by assigning Polaris a magnitude of 2. Astronomers discovered that Polaris is variable, so they switched to Vega as the standard reference star, assigning the brightness of Vega as the definition of zero magnitude at any specified wavelength.
Apart from small corrections, the brightness of Vega still serves as the definition of zero magnitude for visible and near infrared wavelengths, where its spectral energy distribution approximates that of a black body for a temperature of 11000 K. However, with the advent of infrared astronomy it was revealed that Vega's radiation includes an Infrared excess due to a circumstellar disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures. However, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the magnitude scale was extrapolated to all wavelengths on the basis of the black-body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, as a function of wavelength, can be computed. Small deviations are specified between systems using measurement apparatuses developed independently so that data obtained by different astronomers can be properly compared, but of greater practical importance is the definition of magnitude not at a single wavelength but applying to the response of standard spectral filters used in photometry over various wavelength bands.
With the modern magnitude systems, brightness over a wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30; the brightness of Vega is exceeded by four stars in the night sky at visible wavelengths as well as the bright planets Venus and Jupiter, these must be described by negative magnitudes. For example, the brightest star of the celestial sphere, has an apparent magnitude of −1.4 in the visible. Negative magnitudes for other bright astronomical objects can be found in the table below. Astronomers have developed other photometric zeropoint systems as alternatives to the Vega system; the most used is the AB magnitude system, in which photometric zeropoints are based on a hypothetical reference spectrum having constant flux per unit frequency interval, rather than using a stellar spectrum or blackbody curve as the reference. The AB magnitude zeropoint is defined such that an object's AB and Vega-based magnitudes will be equal in the V filter band.
As the amount of light received by a telescope is reduced by transmission through the Earth's atmosphere, any measurement of apparent magnitude is corrected for what it would have been as seen from above the atmosphere. The dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of 100. Therefore, the apparent magnitude m, in the spectral band x, would be given by m x = − 5 log 100 , more expressed in terms of common logarithms as m x
In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.
That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.
But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.
For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.
An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to