Category: Science
Question: This term denotes any algebraic expression written as the quotient of two polynomials. If the discriminant of a quadratic equation is a perfect square, then both roots of that equation are of this type. All numbers denoted by this term are algebraic. For 10 points, give this term that denotes any number that can be written as a over b, where a and b are integers.
Answer: rational

Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either arational, if it is outside the domain of rational evaluation, or irrational, if it belongs to this domain but does not fulfill its standards.
There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accounts, which define rationality as internal coherence among the agent's mental states. Many rules of coherence have been suggested in this regard, for example, that one should not hold contradictory beliefs or that one should intend to do something if one believes that one should do it. Goal-based accounts characterize rationality in relation to goals, such as acquiring truth in the case of theoretical rationality. Internalists believe that rationality depends only on the person's mind. Externalists contend that external factors may also be relevant. Debates about the normativity of rationality concern the question of whether one should always be rational. A further discussion is whether rationality requires that all beliefs be reviewed from scratch rather than trusting pre-existing beliefs.
Various types of rationality are discussed in the academic literature. The most influential distinction is between theoretical and practical rationality. Theoretical rationality concerns the rationality of beliefs. Rational beliefs are based on evidence that supports them. Practical rationality pertains primarily to actions. This includes certain mental states and events preceding actions, like intentions and decisions. In some cases, the two can conflict, as when practical rationality requires that one adopts an irrational belief. Another distinction is between ideal rationality, which demands that rational agents obey all the laws and implications of logic, and bounded rationality, which takes into account that this is not always possible since the computational power of the human mind is too limited. Most academic discussions focus on the rationality of individuals. This contrasts with social or collective rationality, which pertains to collectives and their group beliefs and decisions.
Rationality is important for solving all kinds of problems in order to efficiently reach one's goal. It is relevant to and discussed in many disciplines. In ethics, one question is whether one can be rational without being moral at the same time. Psychology is interested in how psychological processes implement rationality. This also includes the study of failures to do so, as in the case of cognitive biases. Cognitive and behavioral sciences usually assume that people are rational enough to predict how they think and act. Logic studies the laws of correct arguments. These laws are highly relevant to the rationality of beliefs. A very influential conception of practical rationality is given in decision theory, which states that a decision is rational if the chosen option has the highest expected utility. Other relevant fields include game theory, Bayesianism, economics, and artificial intelligence.