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.

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.

The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If it is rational, the proof is complete, and

If the suspect was under the influence of alcohol or drugs, or suffered from an emotional or mental condition that substantially impaired their capacity to make rational decisions, the courts may well decide that the suspect's waiver was not knowing and intelligent.

If human beings are rational animals of such-and-such a sort, then the moral virtues are ...( filling in the blanks is the difficult part ).

Rand summarizes: If chooses to live, a rational ethics will tell him what principles of action are required to implement his choice.

If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean.

If it were rational, it could be expressed as a fraction a / b in lowest terms, where a and b are integers, at least one of which is odd.

If the equation lacks a constant term a < sub > 0 </ sub >, then 0 is one of the rational roots of the equation.

If one uses a composite representation of an " overall " reaction, some may be rational fractions.

If all herders make this individually rational economic decision, the common will be depleted or even destroyed, to the detriment of all.

If we fail to justify simplicity considerations on the basis of the context in which we make use of them, we may have no non-circular justification: " just as the question ' why be rational?

If B has a smallest element among the rationals, the cut corresponds to that rational.

If the starting number is rational then this process exactly parallels the Euclidean algorithm.

) If formulated in von Neumann – Bernays – Gödel set theory, the surreal numbers are the largest possible ordered field ; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals ; it has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

If a chattel slave refuses to work, a number of punishments are also available ; from beatings to food deprivation – although economically rational slave owners practiced positive reinforcement to achieve best results and before losing their investment ( or even friendship ) by killing an expensive slave.

* If X is the Euclidean space, then the interior of the set of rational numbers is empty.

If offered a choice between 100 today or 100 in one year and there is a positive real interest rate throughout the year ceteris paribus, a rational person will choose 100 today.

If there is more than one possible equilibrium at any time then the more interesting implications of the theory of rational expectations do not apply.

* If d is a square free integer then the extension K = Q (√) is a quadratic field of rational numbers.

If the rest mass is imaginary this implies that the denominator is imaginary since the total energy is an observable and thus must be real ; therefore the quantity under the square root must be negative, which can only happen if v is greater than c. As noted by Gregory Benford et al., among others, special relativity implies that tachyons, if they existed, could be used to communicate backwards in time ( see Tachyonic antitelephone article ).

( If the denominator has a zero at z and the numerator does not, then the value of the function will be infinite ; if both parts have a zero at z, then one must compare the multiplicities of these zeros.

) If the denominator is a 1st-degree polynomial or a power of such a polynomial, then the numerator is a constant.

If the denominator is a 2nd-degree polynomial or a power of such a polynomial, then the numerator is a 1st-degree polynomial.

If 0 < e < 1, then the denominator of the equation of free orbits varies with the true anomaly θ, but remains positive, never becoming zero.

If any fractional coefficient exist, multiply every coefficient with the smallest number required to make them whole, typically the denominator of the fractional coefficient for a reaction with a single fractional coefficient.

* If Q ( x ) contains factors which are irreducible over the given field, then the numerator N ( x ) of each partial fraction with such a factor F ( x ) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant.

If the summation in the denominator converges, the posterior probabilities will still sum ( or integrate ) to 1 even if the prior values do not, and so the priors may only need to be specified in the correct proportion.

If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction.

( If the denominator contains only factors 2 and 5, the result is a purely non-circulating decimal, else it is a mixture of the two: a short non-circulating sequence of digits, followed by an endless repetition.

If we now constrain the denominator to be 1, the problem of maximizing is reduced to maximizing the numerator.

" If Laffoley's work within the Boston Visionary Cell can be said to have one principal preoccupation-a common denominator of his eclectic scholarship and practice-then that preoccupation would be to understand the process by which one goes from becoming to being.

If the standards vary, the user may either need to choose a password that complies with all the rules ( the lowest common denominator ) or this may be achieved behind the scenes within the password synchronization software ( e. g. padding or truncating passwords to conform to size constraints on certain systems ).

If a real number x is approximated by any rational number a / b that is not in the sequence of mediants found by the algorithm above, then the sequence of mediants contains a closer approximation to x that has a denominator at most equal to b ; in that sense, these mediants form the best rational approximations to x.

`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If the case is thus determined by us to be domestic, the court has no jurisdiction.

If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If design head has a deep cavity, clay lid will be quite thick at this point ; ;

If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.

If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.

If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.

If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.

If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.

If a statement has been assigned an address in the index word area

If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.