The same argument proves that no subfield of the real field is algebraically closed ; in particular, the field of rational numbers is not algebraically closed.

* The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because is the root of.

These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots.

* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers.

The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.

In the cases of the rational numbers ( Q ) and the real numbers ( R ) there are no nontrivial field automorphisms.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

* The algebraic closure of the field of rational numbers is the field of algebraic numbers.

* There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers ; these are the algebraic closures of transcendental extensions of the rational numbers, e. g. the algebraic closure of Q ( π ).

Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.

Later abstractive and rational processes may indicate errors of judgment in these apprehensions of value, but the apprehensions themselves are the primary stuff of experience.

I think it must be said that, contrary to metaphysical insistence, these are questions so framed as to defy either empirical exploration or rational solutions.

However, one can argue that no such control is necessary as long as one pretends that the anti-trust laws are effective and rational.

Quite clearly the anti-trust laws are neither effective nor rational -- and yet the argument goes that they should be extended to the labor union.

Namely, these are rotations by angles which are rational multiples of π.

Note that it is often rational to improve a system in an order that is " non-optimal " in this sense, given that some improvements are more difficult or consuming of development time than others.

Among the five-sensed beings, the rational ones ( humans ) are most strongly protected by Jain ahimsa.

The term acquired a special significance in the philosophy of Immanuel Kant ( 1724 – 1804 ), who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of reason that are proper to the universe of sensible perception or experience ( phenomena ).

The backward nature of expectation formulation and the resultant systematic errors made by agents ( see Cobweb model ) was unsatisfactory to economists such as John Muth, who was pivotal in the development of an alternative model of how expectations are formed, called rational expectations.

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers ( whose terms are the successive truncations of the decimal expansion of x ) has the real limit x.

" It is a rational calculation where states fight for their interests ( whether they are economic, security related, or otherwise ) once normal discourse has broken down.

Cantor's construction of the real numbers is similar to the above construction ; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances.

Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield.

Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q.

Albert Ellis ' system, originated in the early 1950s, was first called rational therapy, and can ( arguably ) be called one of the first forms of cognitive behavioral therapy.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Diophantus was the first Greek mathematician who recognized fractions as numbers ; thus he allowed positive rational numbers for the coefficients and solutions.

However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms ( i. e. the elliptic integrals of the first, second and third kind ).

One of the more popular arguments for internalism begins with the observation, perhaps first due to Stewart Cohen, that when we imagine subjects completely cut off from their surroundings ( thanks to a malicious Cartesian demon, perhaps ) we do not think that in cutting these individuals off from their surroundings, these subjects cease to be rational in taking things to be as they appear.

1308 ), a Friar Minor Conventual like Saint Bonaventure, argued, on the contrary, that from a rational point of view it was certainly as little derogatory to the merits of Christ to assert that Mary was by him preserved from all taint of sin, as to say that she first contracted it and then was delivered.

He manages to find some rational points on these curves – elliptic curves, as it happens, in what seems to be their first known occurrence — by means of what amounts to a tangent construction: translated into coordinate geometry

From the 8th century, the Mutazilite school of Islam, compelled to defend their principles against the orthodox Islam of their day, looked for support in philosophy, and are among the first to pursue a rational Islamic theology, called Ilm-al-Kalam ( scholastic theology ).

* Baruch Spinoza: Set forth the first analysis of " rational egoism ", in which the rational interest of self is conformance with pure reason.

For a simple example of the first kind, consider the proposition, ¬ p: " there is no smallest rational number greater than 0 ".

Economists of the Austrian school argue that socialist systems based on economic planning are unfeasible because they lack the information to perform economic calculation in the first place, due to a lack of price signals and a free price system, which they argue are required for rational economic calculation.

As the first philosophers, they emphasized the rational unity of things, and rejected mythological explanations of the world.

Also in 1979, in the first revised edition of " Real Magic ", Bonewits defined " thealogy " in his Glossary as " Intellectual speculations concerning the nature of the Goddess and Her relations to the world in general and humans in particular ; rational explanations of religious doctrines, practices and beliefs, which may or may not bear any connection to any religion as actually conceived and practiced by the majority of its members.

Preference utilitarianism was first put forward in 1977 by John Harsanyi in Morality and the theory of rational behaviour but it is more commonly associated with R. M. Hare, Peter Singer and Richard Brandt.

The crucial first point is that the choice is hers, its quirkiness another sign of her much-prized individuality .” “ Bhaer has all the qualities Bronson Alcott lacked: warmth, intimacy, and a tender capacity for expressing his affection — the feminine attributes Alcott admired and hoped men could acquire in a rational, feminist world .”

It implies in the first place a broad, generous sympathy with every form of honest, rational and disinterested study or research.

The experiences with the quality movement made it clear that information flows are fundamentally different from the mass and energy flows ; this inspired the first forms of rational workflows.

* Rabbinic scholar Maimonides, suggested that " prophecy is, in truth and reality, an emanation sent forth by Divine Being through the medium of the Active Intellect, in the first instance to man's rational faculty, and then to his imaginative faculty.

Beyond their efforts to write better Latin, to copy and preserve patristic and classical texts and to develop a more legible, classicizing script, the Carolingian minuscule that Renaissance humanists took to be Roman and employed as humanist minuscule, from which has developed early modern Italic script, the secular and ecclesiastical leaders of the Carolingian Renaissance for the first time in centuries applied rational ideas to social issues, providing a common language and writing style that allowed for communication across most of Europe.

Despite Louie sometimes fails to understand people's aims, he has proven to be a rational thinker ; when the trio come up with a plan for a particular situation Louie is usually the first of the boys to have second thoughts and rethink the plan and compare it to the situation.

A first order rational difference equation has the form.

At present it is maintained by a considerable number-of distinguished naturalists, such as Blumenbach, Cuvier, Bory de St. Vincent, R. Brown, & c. " The notion or spontaneous generation ," says Bory, " is at first revolting to a rational mind, but it is, notwithstanding, demonstrable by the microscope.