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Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.

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## 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

**.**

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