In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

An algebraic number of degree 1 is a rational number.

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.

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.

For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.

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.

* The set of rational numbers q such that q < Ω is computably enumerable ; a real number with such a property is called a left-c. e.

* There exists a computable function which, given any positive rational error bound, produces a rational number r such that

Every real number, whether integer, rational, or irrational, has a unique location on the line.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that " ought " to have a given real limit is identified with that real number.

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.

The converse to this observation is that every recurring decimal represents a rational number p / q.

This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number.

The field of definable numbers is not complete ; there exist convergent sequences of definable numbers whose limit is not definable ( since every real number is the limit of a sequence of rational numbers ).

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.

Because of this necessity to have relations with other rational beings in order to achieve consciousness, Fichte writes that there must be a ' relation of right ,' in which there is a mutual recognition of rationality by both parties.

These factors limit the extent to which agents can make a fully rational decision, thus they possess only “ bounded rationality ” and must make decisions by “ satisficing ,” or choosing that which might not be optimal but which will make them happy enough.

Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

So we can conclude that the original proposition, ¬ p, must be true — " there is no smallest rational number greater than 0 ".

Nozick says no, then asks whether we have reasons not to plug into the machine and concludes that since it does not seem to be rational to plug in, ethical hedonism must be false.

To conduct a frisk, officers must be able to point to specific and articulatory facts which, taken together with rational inferences from those facts, reasonably warrant their actions.

Second, only with this postulate is a rational interpretation of history possible, and we are justified in seeking — as scientists we must seek — such a rational interpretation.

* June 2 – Pope Paul III publishes the encyclical Sublimis Deus, which declares the natives of the New World to be rational beings with souls who must not be enslaved or robbed.

Since they are produced automatically without any rational analysis and verification ( see the modern idea of the subconscious ) of whether they are correct or not, they need to be confirmed ( epimarteresis: confirmation ), a process which must follow each assumption.

* the target of the strategy must behave based on rational self-interest to the extent that the threat will be effective in preventing the behavior.

In the years since, however, scholarly consensus has shifted to consider that ethnic groups may in fact be counted as rational actors, and the puzzle of their apparently irrational actions ( for example, fighting over territory of little or no intrinsic worth ) must therefore be explained some other way.

Realism, believing as it does in the objectivity of the laws of politics, must also believe in the possibility of developing a rational theory that reflects, however imperfectly and one-sidedly, these objective laws.

By the rational root theorem, this root must be 1 or − 1, but both are clearly not roots.

In contrast to many other buildings built in the modernisme style, however, it must also be said that the design of the Palau is eminently rational.

Hence the following dilemma: Either must be held to be an impossible expression in general, or else the meaning of the fundamental symbol of algebra must be extended so as to include rational fractions.

* A central argument of Teleology says that the world has clearly been constructed in a purposeful telic rather than a chaotic manner, and must therefore have been made by a rational being, i. e. God

A moral maxim must have universality, which is to say that it must be disconnected from the particular physical details surrounding the proposition, and could be applied to any rational being.

Every rational action must set before itself not only a principle, but also an end.

Simultaneous treatment of multiple disorders requires strict consideration of compatibility of drugs and detailed adherence of rules of rational drug therapy, based on E. M. Tareev ’ s principles, which state: “ Each non-indicated drug is contraindicated ” and B. E. Votchal said: “ If the drug does not have any side-effects, one must think if there is any effect at all ”.

There must instead be given a rational number r such that 0 < r < | x |.

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.

Similarly, the set of rational numbers in the closed interval is not compact: the sets of rational numbers in the intervals and cover all the rationals in for but this cover does not have a finite subcover.

As a result, most real numbers have no description ( in the same sense of " most " as ' most real numbers are not rational ').

More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets ( sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers ).

Most of Hume's followers have disagreed with his conclusion that belief in an external world is rationally unjustifiable, contending that Hume's own principles implicitly contained the rational justification for such a belief, that is, beyond being content to let the issue rest on human instinct, custom and habit.

Prohairesis allows us to act, and gives us the kind of freedom that only rational animals have.

In modern times, namely, on account of the reconciliation of the worldly principle with itself, the external world is at rest, is brought into order — worldly relationships, conditions, modes of life, have become constituted and organized in a manner which is conformable to nature and rational.

Many people will subsequently activate an idempotent button, even if consciously aware of the rational information that it will have no effect.

In probability, a subjectivism stands for the belief that probabilities are simply degrees-of-belief by rational agents in a certain proposition, and which have no objective reality in and of themselves.

We would dismiss it with some portentous words of Sir Kenelm Digby, in his observations on Browne's religio Medici: ' I have much ado to believe what he speaketh confidently ; that he is more beholding to Morpheus for learned and rational as well as pleasing dreams, than to Mercury for smart and facetious conceptions '.

A value pluralist might, for example, contend that both a life as a nun and a life as a mother realize genuine values ( in a universalist sense ), yet they are incompatible ( nuns may not have children ), and there is no purely rational way to measure which is preferable.

Many scholars have described rationalisation and the question of individual freedom in an increasingly rational society, as the main theme of Weber's work.

Tax reductions likewise, as rational consumers would predict that taxes would have to increase later to balance public finances.

However, when decimal representation is used for the rational or real numbers, such numbers in general have an infinite number of representations, for example 2. 31 can also be written as 2. 310, 2. 3100000, 2. 309999999 …, etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown.

# People have rational preferences among outcomes that can be identified and associated with a value.

Objectivism's central tenets are that reality exists independent of consciousness, that human beings have direct contact with reality through sense perception, that one can attain objective knowledge from perception through the process of concept formation and inductive logic, that the proper moral purpose of one's life is the pursuit of one's own happiness ( or rational self-interest ), that the only social system consistent with this morality is full respect for individual rights embodied in laissez-faire capitalism, and that the role of art in human life is to transform humans ' metaphysical ideas by selective reproduction of reality into a physical form — a work of art — that one can comprehend and to which one can respond emotionally.