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The real numbers contain the integers and the rational numbers as well as irrational numbers such as the square root of 2 and transcendental numbers such as pi. |
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The real numbers contain the integers and the rational numbers as well as irrational numbers such as the square root of 2 and transcendental numbers such as pi. |
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computers can only approximate most real numbers; these approximations are known as floating point numbers. [Computer algebra]? systems are able to treat some real numbers exactly by storing their algebraic description (such as sqrt(2)) rather than their decimal approximation. |
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Computers can only approximate most real numbers; these approximations are known as floating point numbers. [Computer algebra]? systems are able to treat some real numbers exactly by storing their algebraic description (such as sqrt(2)) rather than their decimal approximation. Mathematicians use the symbol R (or more properly, a double-barred R represented by the unicode character ℝ if your browser supports unicode display) to represent the set of all real numbers. |
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Let ℝ denote the set of all real numbers. Then: * The set ℝ is a field (i.e., addition, multiplication? and division? are defined and have the usual properties) * The field ℝ is ordered, i.e. for all real numbers x, y and z: |
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Let R denote the set of all real numbers. Then: * The set R is a field (i.e., addition, multiplication? and division? are defined and have the usual properties) * The field R is ordered, i.e. for all real numbers x, y and z: |
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* Every non-empty subset S of ℝ with an [upper bound]? in ℝ has a [least upper bound]? (also called supremum) in ℝ. |
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* Every non-empty subset S of R with an [upper bound]? in R has a [least upper bound]? (also called supremum) in R. |
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Let ℝ be the set of Cauchy sequences of rational numbers. Cauchy sequences (xn) and (yn) can be added, multiplied and compared as follows: |
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Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences (xn) and (yn) can be added, multiplied and compared as follows: |
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This does indeed define an equivalence relation, it is compatible with the operations defined above, and the equivalence classes can be shown to satisfy all the axioms of the real numbers given above. We can embed the rational numbers into the reals by identifying the rational number r with the sequence (r, r, r, ...). |
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This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the axioms of the real numbers given above. We can embed the rational numbers into the reals by identifying the rational number r with the sequence (r, r, r, ...). |
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Every non negative real number has a square root. This shows that the order on ℝ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make ℝ the premier example of a [real closed field]?. |
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Every non negative real number has a square root. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a [real closed field]?. |
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The reals are one of the two [local fields]? of characteristic 0 (the other one being the complex numbers). |
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The reals are one of the two [local fields]? of characteristic 0 (the other one being the complex numbers). |
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The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the [Lowenheim-Skolem theorem]? implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of [hyperreal number]?s is much bigger than ℝ but also satisfies the same first order sentences as ℝ. |
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The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the [Lowenheim-Skolem theorem]? implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of [hyperreal number]?s is much bigger than R but also satisfies the same first order sentences as R. |
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Fractions had been used by the Egyptians a thousand years BC; the Greek mathematicians around [500 BC]? realized the need for irrational numbers. Negative numbers begun to be generally accepted in the 1600s. The development of the calculus in the 1700s used the real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. |
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Fractions had been used by the Egyptians a thousand years BC; the Greek mathematicians around [500 BC]? realized the need for irrational numbers. Negative numbers begun to be generally accepted in the 1600s. The development of the calculus in the 1700s used the real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. |
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