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Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.
Some Related Sentences
Thus and real
Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.
Thus, many a creativity-oriented aspirant for a career in architecture, drama, or journalism, resigns himself to a real estate business ; ;
Thus, he believed real charity supplied the means for those who wish to help themselves, achieve their goals.
Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.
Thus criminal law grew out what 21st-century lawyers would call torts ; and, in real terms, many acts and omissions classified as crimes actually overlap with civil-law concepts.
Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.
Thus, on a real computer display, each pixel is actually composed of separate red, green, and blue subpixels.
Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning.
Thus, any real object cannot remain the same, also because of movement of its physical parts ( even much bigger than molecules ).
Thus, it has generally been used to describe something which, while unreal, is so in a very specific or unusual fashion, usually one emphasizing not just the " not real ," but some form of estrangement from our generally accepted sense of reality.
Thus, Sterne lost his chances for clerical advancement but discovered his real talents ; until the completion of this first work, " he hardly knew that he could write at all, much less with humour so as to make his reader laugh ".
Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement " for all ... there exists ..."
Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:
Thus the ( real < ref > The complex spinors are obtained as the representations of the tensor product H ⊗< sub > R </ sub > C = Mat < sub > 2 </ sub >( C ).
" Thus without holding a proposition up against the real world, we cannot tell whether the proposition is true or false.
Thus rill, reel and real fall together in Cockney as ; while full and fool are and may rhyme with cruel.
Thus, a circuit with a low power factor will use higher currents to transfer a given quantity of real power than a circuit with a high power factor.
Thus a " Conversation for Action " can be readily tracked and facilitated by a device with little or no ability to model circumstances in the real world other than the ability to register claims by specific agents about a domain.
Thus, opportunity costs are not restricted to monetary or financial costs: the real cost of output forgone, lost time, pleasure or any other benefit that provides utility should also be considered opportunity costs.
Thus, some objects will end in a real reference, and some with the referenced object, and the machine must be able to tell by reading the final cell which one it is.
Thus and number
Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.
Thus, in the last few years, a number of programs which looked very promising at the time their development was commenced have since been completely eliminated.
Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.
Thus, casework involving a limited number of interviews is still to be regarded in terms of the quality of service rendered rather than of the quantity of time expended.
Thus a simple count of the number of distinct symbols is an important clue to the nature of an unknown script.
Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.
Thus, almost all positive integers are composite ( not prime ), however there are still an infinite number of primes.
The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:
If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.
Thus, when the fluid finally does strike the object, it is forced to change its properties -- temperature, density, pressure, and Mach number -- in an extremely violent and irreversible fashion called a shock wave.
Thus the discovery report by the Berkeley group reads: " It is suggested that element 97 be given the name berkelium ( symbol Bk ) after the city of Berkeley in a manner similar to that used in naming its chemical homologue terbium ( atomic number 65 ) whose name was derived from the town of Ytterby, Sweden, where the rare earth minerals were first found.
Thus, a PICMG backplane can provide any number and any mix of ISA, PCI, PCI-X, and PCI-e slots, limited only by the ability of the SBC to interface to and drive those slots.
Thus, a number of Allied soldiers received the Bronze Star Medal in World War II, as well as U. N. soldiers in the Korean War, Vietnamese and allied forces in the Vietnam War, and coalition forces in recent military operations such as the Gulf War, Operation Enduring Freedom and the Iraq War.
Thus, all carbon isotopes have nearly identical chemical properties because they all have six protons and six electrons, even though carbon atoms may differ in number of neutrons.
Thus, a poison that covers surface randomly will tend to reduce the number of uncontaminated large planes but leave proportionally more smaller sites free, thus changing the hydrogenation vs. hydrogenolysis selectivity.