| 词汇 | example_english_rational-number |
| 释义 | Examples of rational numberThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. We then completely describe the transformations having a given rationalnumber explicitly and count the number of periodic attractors. More precisely, suppose that p/q is a rationalnumber, and we search for a continued fraction representation for it. We now show that this affine function can be replaced by a single rationalnumber using convex minimisation. Such precise computation over the rationals is unnecessary because the resulting rationalnumber will just be fed into compress, and a simpler rationalnumber found. In the following code, rat to stream builds the infinite sequence of digits for a rationalnumber given by its numerator and denominator. Now we distinguish two cases: (1) /2 is a rationalnumber. They would have to be eliminated in order to have a rationalnumber system. Each incommensurable magnitude of a geometrical figure is necessarily reduced to a convergent rationalnumber according to some set 'degree of precision' - digitization. Evidently, every such fraction represents a rationalnumber. In fact, each time we evaluate polynomials at some rationalnumber, we are never interested in the exact value of the result, but only in its sign. We are thinking, for instance, of the sequences of rational numbers converging towards an irrational, for example, 2: this theoretical limit produces 2, which is not a rationalnumber. This is because every rationalnumber has a recurring decimal expansion. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In fact, they are only defined when the exponent is a rationalnumber with the denominator being an odd integer. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. All the rational numbers are equivalent, because each rationalnumber is equivalent to zero. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In both cases, for input systems with rationalnumber coefficients, routines for isolating the real solutions are available. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. More generally, the sequence of digits in the decimal expansion of any rationalnumber is eventually periodic (see below). From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. For the orbits to be closed, must be a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The topological index is by definition a rationalnumber, but it is usually not at all obvious from the definition that it is also integral. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. It can be shown that any algebraic integer that is also a rationalnumber must actually be an integer, whence the name algebraic integer. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The strategy is to construct the decimal representation of a number that differs from the decimal representation of every rationalnumber in our sequence. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Any positive rationalnumber can be written as a sum of unit fractions in infinitely many ways. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. No rationalnumber is transcendental and all real transcendental numbers are irrational. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Software packages that perform rational arithmetic represent numbers as fractions with integral numerator and denominator, and can therefore represent any rationalnumber exactly. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An algebraic number of degree 1 is a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Every rationalnumber is also a real number. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Though it may not be possible to assign a rational value to a ratio, it is possible to compare a ratio with a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. But if the helix's coiling and internal repeating are incommensurable (ratio not a rationalnumber), then "n" is effectively. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The rationalnumber "a" / "b" is not an algebraic integer unless "b" divides "a". From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. For any zero-dimensional input system with rationalnumber coefficients it returns those solutions whose coordinates are real algebraic numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Every positive rationalnumber appears exactly once in the tree. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The former is a rationalnumber; the latter is an irrational number. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. On the other hand, this set of irrationals is not closed because every rationalnumber lies in its closure. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A rationalnumber may be expressed as the quotient of two integers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Each rationalnumber is equivalent to 0; thus the rational numbers are an equivalence class for this relation. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Since "q" may be equal to 1, every integer is a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Whether a rationalnumber has a finite representation or requires an infinite repeating representation depends on the base. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In fact, any rationalnumber can be written uniquely as the product of primes if negative exponents are allowed. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Let bcode(x) be the rationalnumber formed by adding a decimal point before a binary code. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The rationalnumber line does not satisfy the nested interval theorem. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The simplest example is the pentagram, which corresponds to the rationalnumber 5/2. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The strategy is to construct a sequence of nested intervals such that every rationalnumber is excluded from the interior of some interval. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Now "r" /2 is a rationalnumber greater than 0 and smaller than "r". From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. As for the polynomials with integer coefficients, this defines a factorization into a rational number and a primitive polynomial with integer coefficients. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. If a rationalnumber is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In general, however, will not be such a rationalnumber, and thus the orbit will not be closed. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The abundancy of "n" is the rationalnumber ("n") / "n", in which denotes the sum of divisors function. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In some case, every rationalnumber may be replaced by all rational numbers except a finite number of them, which amounts to multiplying by some constant depending on. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In 1960, he proved that every normal number in base "r" is normal in base "s" if and only if log "r" / log "s" is a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The more terms and the larger the rational numbers, the longer the computation takes. Their arguments form periodic orbit(s) of the map (so, they are rational numbers). For example, a rationalnumber is a square of a rationalnumber if it is a square of a "p" -adic number for almost all primes "p". From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Another implementation that could be efficient would be rational numbers with lazy computations. The next rationalnumber (ordered by size of denominator) that is a better rational approximation of is, still only correct to 6 decimal places and hardly closer to than. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. With the exception of zero, reciprocals of every real number are real, reciprocals of every rationalnumber are rational, and reciprocals of every complex number are complex. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The 9 product of rational numbers is computed simply by putting them in the same solution. For this purpose, it is reasonable to assume that we have the ability to generate rational numbers of user-specified arbitrary precision. The prototypical apartness relation is that of the real numbers: two real numbers are said to be apart if there exists (one can construct) a rationalnumber between them. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Our implementation is modular, parameterised by the implementation of rational numbers. Quotient and remainder answers were multiplied by divisor inverses, 1/3, 1/7, 1/10, 1/11 and 1/13, exactly returning the beginning rationalnumber (64/64). From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A rational interval is one with both its endpoints rational numbers. Similarly, rational numbers are traditionally formalised as pairs consisting of an integer, and a positive natural number. Odds can also be expressed as a number, by dividing the terms in the ratio in this case it is unique (different fractions can represent the same rationalnumber). From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Nevertheless, the rotation number is obtained by going to the limit over rational numbers and the result is still a result of existence. These powers of eigenvalues all are in the intersection field, the rational numbers, and their product is 1 by the determinant. A more efficient solution is to represent a sequence of these three symbols as a rationalnumber in base 3 where each digit represents a symbol. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The situation is similar when we get a real number by completion from rational numbers. We describe the extended signature needed for such an implementation of rational numbers, and let the users choose which one they want to compute with. Therefore, with appropriate representations, rational numbers can, in fact, be a faithful copy of integers. What's more, it must be the "same" rationalnumber for all radii, since can not change continuously; the rational numbers are totally disconnected from one another. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Suppose that "e" is a rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A similar phenomenon occurs when rational numbers are defined as quotients of integers: 1/2 and 2/4 are different representations of the same rationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. For ease of argument, it actually made sense to allow negative or rational numbers of pearls at any place. Chapter 10 discusses exact arithmetic over natural, integer and rational numbers. It only takes a polynomial of a few degrees to cause the exact evaluation on large (in terms of simplicity) rational numbers to become very slow. This algorithm is interesting because it provides us with ways to add two real numbers, to multiply them by rational numbers, and to encode rational numbers as real numbers. When implementing this recursive process as a co-inductive function, we propose to represent rational numbers with pairs of integers and to replace comparisons of rational numbers with comparisons of integers. In this approach real numbers are represented by functions on some fixed countable set, where the codomain of the function (the elements of the sequence) need not be rational numbers. About the combination of trees and rational numbers in a complete firstorder theory. This fact can be used to prove that there is a 1-1 correspondence between the vertices of tree and the positive rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. It maps quadratic irrationals to non-dyadic rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The rational numbers are therefore the prime field for characteristic zero. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In mathematical analysis, the rational numbers form a dense subset of the real numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. For example, rational numbers also form a field. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Similarly, if the sum diverges, then for almost all real numbers, there are infinitely many such rational numbers "p" / "q". From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Another approach is the metric completion of the rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. If the coefficients are integers or rational numbers exactly represented, the intermediate entries can grow exponentially large, so the bit complexity is exponential. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The question mark function has the interesting property of having vanishing derivatives at all rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In general, this is an operation on fractions rather than on rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. If one possessed an algorithm to determine solvability in non-negative rational numbers, it could easily be used to determine solvability in the rationals. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The rational numbers, while dense in the real numbers, are meagre as a subset of the reals. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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