Reals in a sentence

For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions.

Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers.

A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.

Explicitly, the Brauer group of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another.

The empty set and the set of all reals are open intervals, while the set of non-negative reals, for example, is a right-open but not left-open interval.

There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow ; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow.

They are also known as the recursive numbers or the computable reals or recursive reals.

As a result, over the next five years, SUS saved nearly $500 million reals in expenses by buying generic versions of efavirenz for use in the country.

CBS News, in an article by someone named ‘Tucker Reals’, is equally as dishonest, claiming Li “was threatened by his government” when he “tried to raise the alarm about the new coronavirus”, again totally misrepresenting the facts.

Judge Renato Borelli ordered Bolsonaro, who has famously compared the new coronavirus to a "little flu," to stop ignoring the capital city's mask decree, or face a fine of 2,000 reals (US$390).

The National Climate Change Fund, for example, did not spend any of the 8 million reals (US$ 2 million) authorized by the federal government, according to a report in Estado de Sao Paulo newspaper.

As a result, it did not have to close any shops and will continue to open 25 to 30 doors a year — investing 200 million reals annually in the process, according to Picinini.

California became a sanctuary state yesterday, for reals.

A finer equivalence relation, Solovay equivalence, can be used to characterize the halting probabilities among the left-c.e. reals.

All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals.

Along with the above remarks about measure, it shows that the set of Liouville numbers and its complement decompose the reals into two sets, one of which is meagre, and the other of Lebesgue measure zero.

Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis ) imply the axiom of choice and hence a well order of the reals.

Ball, p. 31 In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were.

By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum.

By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals.