Rationalization

Resources

Rationalization is a process of converting an irrational number into a rational number, which is one which can be expressed as the ratio of two integers. The numbers 1.003, -11/3, and 22/7 are all rational numbers. Irrational numbers are those that cannot be so expressed. The ratio pi, the square root of 5, and the cube root of 4 are all irrational numbers.

Rationalization is a process applied most often to the denominators of fractions, such as 5/(1 +ǀ ). There are two reasons for this. If someone wanted to compute a rational approximation for such an expression, doing so would entail dividing by a many-place decimal, in this case 2.41421. . . With a calculator it would be easy to do, but if it must be done without a calculator, the process is long, tedious, and subject to errors. If the denominator were rationalized, however, the calculations would be far shorter.

The second and mathematically more important reason for rationalizing a denominator has to do with “fields,” which are sets of numbers which are closed with respect to addition, subtraction, multiplication, and division. If one is working with the field of rational numbers and if one introduces a single irrational square root into the field, forming all possible sums, differences, products, and quotients, what happens? Are the resulting numbers made more complex in an unlimited sort of way, or does the complexity reach a particular level and stop?

The answer with respect to sums, differences, and products is simple. If the irrational square root which is introduced happens to be , then any possible sum, difference, or product can be put into the form p + q , where p and q are rational. The cube of 1 + , for example, can be reduced to 7 + 5 .

To check quotients, one can first put the numerator and denominator in the form p + q (thinking of a quotient as a fraction). Then one rationalizes the denominator. This will result in a fraction whose numerator is in the form p + q , and whose denominator is a simple rational number. This can in turn be used with the distributive law to put the entire quotient into the form p + q .

How does one rationalize a denominator? The procedure relies on the algebraic identity (x + y) (x - y) = x² -y², which converts two linear expressions into an expression having no linear terms. If x or y happens to be a square root, the radical will disappear.

Using this identity can be illustrated with the example given earlier:

The procedure is not limited to expressions involving .

If any irrational square root, is introduced into the field of rational numbers, expressions involving it can be put into the form . Then quotients involving such a form as a divisor can be computed by multiplying numerator and denominator by , which will turn the denominator into p² -nq², a rational number. From there, ordinary arithmetic will finish the job.

Fields can be extended by introducing more than one irrational square root, or by introducing roots other than square roots, but everything becomes more complicated.

One analogous extension that is of great mathematical and practical importance is the extension of the field of real numbers to include or i. A process similar to the one used to rationalize denominators is used to convert a denominator from a complex number involving i into a real number.

Resources

BOOKS

Bittinger, Marvin L., and David Ellenbogen. Intermediate Algebra: Concepts and Applications. 7th ed. Reading, MA: Addison-Wesley Publishing, 2005.

Niven, Ivan. Numbers: Rational and Irrational. Washington, DC: The Mathematical Association of America, 1961.

J. Paul Moulton

Rationalization

Rationalization is a process of converting an irrational number into a rational number , which is one which can be expressed as the ratio of two integers . The numbers 1.003, -11⁄3, and 22/7 are all rational numbers. Irrational numbers are those that cannot be so expressed. The ratio pi , the square root of 5, and the cube root of 4 are all irrational numbers.

Rationalization is a process applied most often to the denominators of fractions, such as 5/(1 + √+2). There are two reasons for this. If someone wanted to compute a rational approximation for such an expression, doing so would entail dividing by a many-place decimal, in this case 2.41421... With a calculator it would be easy to do, but if it must be done without a calculator, the process is long, tedious, and subject to errors. If the denominator were rationalized, however, the calculations would be far shorter.

The second and mathematically more important reason for rationalizing a denominator has to do with "fields," which are sets of numbers which are closed with respect to addition, subtraction , multiplication , and division . If one is working with the field of rational numbers and if one introduces a single irrational square root into the field, forming all possible sums, differences, products, and quotients, what happens? Are the resulting numbers made more complex in an unlimited sort of way, or does the complexity reach a particular level and stop?

The answer with respect to sums, differences, and products is simple. If the irrational square root which is introduced happens to be √2, then any possible sum, difference, or product can be put into the form p + q √2, where p and q are rational. The cube of 1 + √2, for example, can be reduced to 7 + 5 √2.

To check quotients, one can first put the numerator and denominator in the form p + q √2 (thinking of a quotient as a fraction). Then one rationalizes the denominator. This will result in a fraction whose numerator is in the form p + q √2, and whose denominator is a simple rational number. This can in turn be used with the distributive law to put the entire quotient into the form p + q √2 .

How does one rationalize a denominator? The procedure relies on the algebraic identity (x + y)(x - y) = x2 - y2, which converts two linear expressions into an expression having no linear terms. If x or y happens to be a square root, the radical will disappear.

Using this identity can be illustrated with the example given earlier:

The procedure is not limited to expressions involving √2.

If any irrational square root, √7, √80, or √n is introduced into the field of rational numbers, expressions involving it can be put into the form p + q √n . Then quotients involving such a form as a divisor can be computed by multiplying numerator and denominator by p - q √n, which will turn the denominator into p2 - nq2, a rational number. From there, ordinary arithmetic will finish the job.

Fields can be extended by introducing more than one irrational square root, or by introducing roots other than square roots, but everything becomes more complicated.

One analogous extension that is of great mathematical and practical importance is the extension of the field of real numbers to include √−1 or i. A process similar to the one used to rationalize denominators is used to convert a denominator from a complex number involving i into a real number.

Resources

books

Bittinger, Marvin L., and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Niven, Ivan. Numbers: Rational and Irrational. Washington, DC: The Mathematical Association of America, 1961.

J. Paul Moulton

rationalization Just as it is impossible to understand Karl Marx's concerns without seeing the centrality of labour power and its alienation into capital, so also it would be equally difficult to grasp the intellectual coherence of Max Weber's writings without understanding what Alvin Gouldner has termed the ‘metaphysical pathos’ associated with his vision of the rationalization of everyday life. This progressive disenchantment of the world, the eradication of mystery, emotion, tradition, and affectivity, and its replacement by rational calculation, informs much of his research and writing. It has created a whole industry among students of his work, who continue to debate the issue of whether or not Weber offers a fully developed theory of rationalization, and (if so) where precisely in his writings it is to be found (see, for example, S. Lash and and S. Whimster ( eds.) , Max Weber, Rationality and Modernity, 1987
).

The process of rationalization is said by Weber to affect economic life, law, administration, and religion. It underpins the emergence of capitalism, bureaucracy, and the legal state. The essence of the rationalization process is the increasing tendency by social actors to the use of knowledge, in the context of impersonal relationships, with the aim of achieving greater control over the world around them. However, rather than increasing freedom and autonomy, rationalization makes ends of means (slavish adherence to the rules within modern bureaucracies are an obvious example), and imprisons the individual within the ‘iron cage’ of rationalized institutions, organizations, and activities.

Commentators have remarked that, in his concern over the rationalizing tendencies of modern societies, Weber was more pessimistic in the prognosis for human freedom than any of his contemporaries. Marx at least foresaw an emancipating revolution, whereas for Weber the only antidote to rationalization was the emergence of the charismatic figure. Socialism, Weber claimed, would create an even more confining cage, since it would combine formal rationality with substantive rationality. Whereas the market had counteracted bureaucratic state power under capitalism, socialism would see the two combined. That it was toppled in its Soviet variant by the charismatic individuals and movements of Wałęsa, Solidarity, Havel, and Civic Forum would seem to provide some optimistic respite, in the face of all-encompassing disenchantment. See also REFLEXIVE MODERNIZATION.

RATIONALIZATION

A rationalization is a logical or moral justification for an action or attitude that is provided by a subject whose (unconscious) motives are inaccessible. Two examples are justifying a fear of cancer by referring to other family members who suffered from the disease and justifying one's compulsive washing by offering sanitary concerns. The term was introduced in psychoanalysis by Ernest Jones (1908).

Rationalization is not really a symptom. It is more a way of masking and denying the symptom. Nor is it a compromise formation, since within certain limits it satisfies the drive. Nor is it a defense mechanism, since it is not directed toward any libidinal satisfaction. It is more of a way to keep from recognizing neurotic conflicts. It is the conscious secondary thought process of covering the symptom with a screen.

Rationalization is primarily found in cases of neurosis: "Compulsive acts [that occur] in two successive stages, of which the second neutralizes the first, are a typical occurrence in obsessional neuroses. The patient's consciousness naturally misunderstand them and puts forward a set of secondary motives to account for them—rationalizes them, in short" (Freud 1909d, p. 192).

Can the notion of rationalization be applied to delusion in particular, the logical delusion of paranoiacs? Some psychiatric studies have made use of an analogous concept to show how megalomania is caused by a need to explain and justify the feeling of persecution. In his essay on Judge Schreber, Freud rejects this formulation: "to ascribe such important affective consequences to a rationalization is, as it seems to us, an entirely unpsychological proceeding; and we would consequently draw a sharp distinction between our opinion and the one which we have quoted from the textbooks. We are making no claim, for the moment, to knowing the origin of the megalomania" (1911c [1910], p. 49).

Rationalization is sometimes compared to intellectualization, but the two concepts must be distinguished. In intellectualization, one distances oneself from psychic processes by cathecting one's own intellectual processes and thought. Rationalization, in contrast, primarily finds support in systems of thought, representations, and beliefs that are socially constituted and accepted.

MichÈle Bertrand

See also: Intellectualization; Jones, Ernest; Negative, work of the; Paranoia; Secondary revision; Thought.

Bibliography

Freud, Sigmund. (1909d). Notes upon a case of obsessional neurosis. SE, 10: 151-318.

——. (1911c [1910]). Psycho-analytic notes on an autobiographical account of a case of paranoia (Dementia Paranoides ). SE, 12: 1-82.

Jones, Ernest. (1908). Rationalization in everyday life. Journal of Abnormal Psychology.

rationalization (rash-ŏn-ă-ly-zay-shŏn) n. (in psychology) the explanation of events or behaviour in terms that avoid giving the true reasons.