Some articles on real numbers, numbers, number, reals, real:
0.999... - Proofs From The Construction of The Real Numbers
... Further information Construction of the real numbers Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory ... The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor ... One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers ...
... Further information Construction of the real numbers Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory ... The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor ... One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers ...
Reverse Mathematics - The Big Five Subsystems of Second Order Arithmetic - The Base System RCA0
... In particular, any set of natural numbers that can be proven to exist in RCA0 is computable, and thus any theorem which implies that noncomputable sets exist is not provable in RCA0 ... any noncomputable sets exist), RCA0 is sufficient to prove a number of classical theorems which, therefore, require only minimal logical strength ... The classical theorems provable in RCA0 include Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form an ordered field) ...
... In particular, any set of natural numbers that can be proven to exist in RCA0 is computable, and thus any theorem which implies that noncomputable sets exist is not provable in RCA0 ... any noncomputable sets exist), RCA0 is sufficient to prove a number of classical theorems which, therefore, require only minimal logical strength ... The classical theorems provable in RCA0 include Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form an ordered field) ...
0.999... - In Alternative Number Systems - Infinitesimals
... = 1 rely on the Archimedean property of the real numbers that there are no nonzero infinitesimals ... must be smaller than any positive rational number, so it must be an infinitesimal but since the reals do not contain nonzero infinitesimals, the difference is ... are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean ...
... = 1 rely on the Archimedean property of the real numbers that there are no nonzero infinitesimals ... must be smaller than any positive rational number, so it must be an infinitesimal but since the reals do not contain nonzero infinitesimals, the difference is ... are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean ...
Real Computation
... In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers ... given this name because they operate on the set of real numbers ... computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers ...
... In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers ... given this name because they operate on the set of real numbers ... computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers ...
Algebra Over A Field - Definition and Motivation - A Motivating Example: Quaternions
... The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself ... We saw above that the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimension algebra over the ... can similarly define a multiplication on a three-dimensional real vector space such that every non-zero element has an inverse ...
... The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself ... We saw above that the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimension algebra over the ... can similarly define a multiplication on a three-dimensional real vector space such that every non-zero element has an inverse ...
Famous quotes containing the words numbers and/or real:
“The land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.”
—Bible: Hebrew Numbers 35:33.
“Perpetual modernness is the measure of merit, in every work of art; since the author of it was not misled by anything short- lived or local, but abode by real and abiding traits.”
—Ralph Waldo Emerson (1803–1882)
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