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[6]: 51–5, 698 His prime example of value-rational action was institutionalised rituals found in all societies: culturally prescribed but eternally legitimate ends. [6]: 467, 675–9, 717 [7] Rational humans pursue socially legitimate value-rational ends by using operationally efficient instrumental means.
In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...
Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π. Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number.
(1) instrumentally rational (zweckrational), that is, determined by expectations as to the behavior of objects in the environment and of other human beings; these expectations are used as "conditions" or "means" for the attainment of the actor's own rationally pursued and calculated ends; (2) value-rational (wertrational), that is, determined ...
Values of trigonometric functions of rational multiples of π (except when undefined): for example, cos π / 7 , cos 3 π / 7 , and cos 5 π / 7 satisfy 8x 3 − 4x 2 − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers.
Things are deemed to have instrumental value (or extrinsic value [2]) if they help one achieve a particular end; intrinsic values, by contrast, are understood to be desirable in and of themselves. A tool or appliance, such as a hammer or washing machine, has instrumental value because it helps one pound in a nail or clean clothes, respectively.
A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
Rationalization (or rationalisation) is the replacement of traditions, values, and emotions as motivators for behavior in society with concepts based on rationality and reason. [2] The term rational is seen in the context of people, their expressions, and or their actions.