| 词汇 | example_english_irrational-number |
| 释义 | Examples of irrational 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. Every infinite continued fraction converges to an irrationalnumber. Conversely, every irrationalnumber is decomposable, in one and only one way, into a continued fraction which is necessarily infinite. Recall that each number has at most two binary expansions and any irrationalnumber has exactly one. Thus, for example, an irrationalnumber is the limit of different fractions which have values approaching it more and more. Incidentally, this "irrational flow on a torus," where 1 /2 is an irrationalnumber, determines one of the simpler examples of chaotic behavior. Let x be an irrationalnumber. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrationalnumber as a number. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrationalnumber has an infinite non-repeating representation in all integer bases. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. However, considered as a sequence of real numbers, it does converge to the irrationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The former is a rational number; the latter is an irrationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Thus, the notion of an irrationalnumber is meaningless to even the most powerful floating-point computer. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. One way to consider this is that the real value often has the characteristics of an irrationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The golden ratio has the slowest convergence of any irrationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Every location on the number line continuum contains either a rational or an irrationalnumber. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrational cut is equated to an irrationalnumber which is in neither set. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrationalnumber stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrationalnumber is any number that can not be expressed as a ratio of two integers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The decimal expansion of an irrationalnumber continues without repeating. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. It can't be an algebraic irrationalnumber like 2. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Informally, this means that an irrationalnumber can not be represented as a simple fraction. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrationalnumber is badly approximable if and only if the partial quotients of its continued fraction are bounded. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An infinite continued fraction representation for an irrationalnumber is useful because its initial segments provide rational approximations to the number. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrationalnumber being approximated. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Since the numerical value of an irrationalnumber can not be stored exactly in a computer, an approximation of the incommensurate frequencies by all rational numbers is required in implementation. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Since is an irrationalnumber (see proof that e is irrational), it can not be represented as a fraction, but it can be represented as a continued fraction. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. An irrationalnumber is any real number that can not be expressed as a fraction "a" / "b", where "a" is an integer and "b" is a non-zero integer. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. This article uses "irrational" in the music theory sense, not the mathematical sense, where an irrationalnumber is one that "can not" be written as a ratio of whole numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Now we are in a position to formulate a basic statement on the action of the modular group on the irrational numbers. Moreover, the infinite series of digits of an irrationalnumber does not exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In this manner, an irrationalnumber can give an infinite sequence of notes where each note is a digit in the decimal expression of that number. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. However, if an assertion were to be made about every irrationalnumber, there would be no way to enumerate all the conjuncts, since irrationals can not be enumerated. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Under these conditions negative numbers, irrational numbers, imaginary numbers, algebraic numbers, etc., became progressively more accepted and broke up the canonical meaning of the number concept. Notions such as prime numbers and rational and irrational numbers are introduced. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. While the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Moreover, the repeated addition model must be substantially modified when irrational numbers are brought into play. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Quadratic irrational numbers are the only numbers that have these. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Meanwhile, there is an "uncountably" infinite set of strings which do not end in such repetition; these correspond to the irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. For instance, exponentiation takes the term "hopping", and the fictional term "unreasonable numbers" was coined for irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. He used this method to provide a proof of the existence of irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. What can be said about the irrational numbers? From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. They are credited with numerous mathematical advances, such as the discovery of irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The real numbers consist of irrational numbers and rational numbers, as well as the integers, whole numbers, and the natural numbers (the counting numbers). From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedrons. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. It has been conjectured that all algebraic irrational numbers are normal. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Besides counting fruits, subtraction can also represent combining other physical and abstract quantities using different kinds of objects: negative numbers, fractions, irrational numbers, vectors, decimals, functions, matrices and more. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. However, some irrational numbers are not transcendental. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. In mathematics, the definition of number has been extended over the years to include such numbers as, negative numbers, rational numbers, irrational numbers, real numbers, and complex numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. These irrational numbers are called normal. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Here variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Besides counting fruits, addition can also represent combining other physical and abstract quantities using different kinds of objects: negative numbers, fractions, irrational numbers, vectors, decimals, functions, matrices and more. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. It means that the conformal weights of its primary fields and the central charge are irrational numbers which makes the theory harder to solve and understand. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Let 0 < < 1 be an irrationalnumber. Being an irrationalnumber, can not be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. He is noted as the first to publish an arithmetical theory of irrational numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. First, even though rational numbers all have a finite or ever-repeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers. 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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