The octonions, constructed by John T. Graves in 1843, were the first example of a ring that is not associative.

This algebra was discovered by John T. Graves in 1843, and is called the octonions or the " Cayley numbers ".

In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley – Dickson construction to the octonions.

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold.

This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions.

This quasigroup coincides with the multiplicative structure defined by the unit octonions e < sub > 1 </ sub >, e < sub > 2 </ sub >, ..., e < sub > 7 </ sub > ( omitting 1 ) if the signs of the octonion products are ignored.

Aut ( O < sub > s </ sub >) is the automorphism group of the algebra O < sub > s </ sub > of split octonions, a closed subgroup of SO ( 3, 4 ), and P is the intersection of G with the stabilizer of the isotropic line spanned by the first standard basis vector in R < sup > 7 </ sup > viewed as the purely imaginary split octonions ( orthogonal complement of the unit element in O < sub > s </ sub >).

Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the reals, complexes, quaternions, and octonions, respectively, and using the multiplication operations from these algebras.

* John T. Graves discovers the octonions.

In the 1840s William Rowan Hamilton had developed his quaternions and John T. Graves and Arthur Cayley the octonions.

Addition of octonions is also associative, but multiplication of octonions is non-associative.

Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

The latter carries the structure of a Lie quasigroup ( a nonassociative group ), which can be identified with the set of unit octonions.

Every associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions.

Like ( Cayley – Dickson ) octonions, multiplication of Cayley – Dickson sedenions is neither commutative nor associative.

But in contrast to the octonions, the sedenions do not even have the property of being alternative.

There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions, double the number of the quaternions from which they are an extension.

Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.

The octonions can be thought of as octets ( or 8-tuples ) of real numbers.

In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature.

Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.

Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space over the real numbers.

The multiplication of octonions is even stranger than that of quaternions.

For the reason of this non-associativity, octonions have no matrix representation.

The algebra immediately following the octonions is called the sedenions.

http :// math. ucr. edu / home / baez / octonions / node15. html.

* More generally, the conjugation involution in any Cayley – Dickson algebra such as the complex numbers, quaternions and octonions, if a blind eye is turned on the nonassociativity of the latter.

The seven-dimensional cross product has the same relationship to octonions as the three-dimensional product does to quaternions.

Only recently new `` holes '' were discovered in our safety measures, and a search is now on for more.

When he discovered they had received from the Company's Court of Directors no permission to live in India, coupled with the fact that they were Americans who had been sent to Asia to convert `` the heathen '', he became more belligerent than ever.

From this point of view the `` militant mobs '' of the past, stirred into action by one ideology or another, were all composed of `` intellectuals '' -- and this is not the level on which the essence of mankind can be discovered.

It would be even more valuable because that same aircraft could immediately destroy any targets it discovered -- no need to wait for a missile to come all the way from the United States with the chance that the target, if it were mobile, would be gone.

I discovered that the girls had shrewdly vacated the kitchen, and were playing quietly in the living room.

One morning, we discovered not only that the pennies were missing from the idol but that a cigarette had been stubbed out in its lap.

Final ratings were made on the basis of a point system which was developed after studying the distributions of actual behaviors recorded and assigning weight values to each type of behavior that was deviant from the discovered norms.

In this manner, the factors measured by the intelligence test were controlled, allowing discovered differences in achievement to be interpreted as resulting from other variables.

Early in her life she had discovered that where there were men, there was money, and with the two came luxury and liquor.

I discovered in the course of a visit there that almost all the pupils were Negroes.

After returning to the United States and raising the money, he discovered on getting back to Japan that the hours were no longer available.

They lay months away from the nearest Earth star by jump drive, and no one knew what they were good for, although it was felt that they would probably be good for something if it could only be discovered -- much like the continent of Antarctica in ancient history.

Anton van Leeuwenhoek, the famous 17th century microscopist, discovered living organisms ( living, because they were motile ) in deposits on the teeth ( what we now call dental plaque ).

These terms have historically been applied to any astronomical object orbiting the Sun that did not show the disk of a planet and was not observed to have the characteristics of an active comet, but as small objects in the outer Solar System were discovered, their volatile-based surfaces were found to more closely resemble comets, and so were often distinguished from traditional asteroids.

The physicist J. J. Thomson, through his work on cathode rays in 1897, discovered the electron, and concluded that they were a component of every atom.

In 1909, Hans Geiger and Ernest Marsden, under the direction of physicist Ernest Rutherford, bombarded a sheet of gold foil with alpha rays — by then known to be positively charged helium atoms — and discovered that a small percentage of these particles were deflected through much larger angles than was predicted using Thomson's proposal.

Archaeologist Timothy A. Kohler excavated large Pueblo I sites near Dolores, Colorado, and discovered that they were established during periods of above-average rainfall.

In 1906, precipitation hardening alloys were discovered by Alfred Wilm.

The first few amino acids were discovered in the early 19th century.

These were discovered by Carl D. Anderson in 1932 and named positrons ( a contraction of " positive electrons ").

James I's courtiers discovered in " James Stuart " " a just master ", and converted " Charles James Stuart " into " Claims Arthur's seat " ( even at that point in time, the letters I and J were more-or-less interchangeable ).

In Lebanon more than 450 outcrops of Lower Cretaceous amber were discovered between the 1960s and 1990s, among which about 20 outcrops have yielded biological inclusions comprising the oldest representatives of several recent families of terrestrial arthropods.

Many remarkable insects and spiders were recently discovered in the amber of Jordan including the oldest zorapterans, clerid beetles, umenocoleid roaches, and achiliid planthoppers.