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text: 'A **real number** is any number that can be used to represent a physical quantity.\n\nIntuitively, real numbers are any number that can be found between two [48l integers], such as $0,$ $1,$ $-1,$ $\\frac{3}{2},$ $\\frac{-7}{2},$ [49r $\\pi,$] [e $e$], $100 \\cdot \\sqrt{2},$ and so on. The set of real numbers is written $\\mathbb R.$ You can think of $\\mathbb R$ as [4zq $\\mathbb Q$] extended to include the [54z irrational numbers] like $\\pi$ and $e$ which can be found between rational numbers but which cannot be completely written out in [-4sl].\n\n## Definitions of the real numbers\n\nThe most commonly used definitions of the real numbers are constructions as extensions of the [4zq rational numbers], which involve either [53b Cauchy sequences] or [dedekind_cut Dedekind cuts].\n\n### Cauchy sequences\n\nBroadly speaking, a Cauchy sequence is a sequence where as the sequence goes on, all the elements past that point get closer and closer together. In the real numbers, every Cauchy sequence [convergence_analysis converges] to a real number. However, in the set of rational numbers, not all Cauchy sequences converge to a rational number.\nIn the set of rationals, a Cauchy sequence which does not converge to a rational number cannot really be said to "converge" at all: the set of rationals is "missing some of the points" that would be required to make every Cauchy sequence converge.\n\nFor example, the sequence of fractions of consecutive Fibonacci numbers $1/1, 2/1, 3/2, 5/3, 8/5, \\ldots$ gets closer and closer to $\\frac{1 + \\sqrt{5}}{2}$, but cannot be said to converge to that number because it is not in the set of rational numbers.\n\nFor each of these non-convergent Cauchy sequences, we define a new irrational number to "fill in the gap", and for the Cauchy sequences that do converge, we define a real number equal to that rational number.\n\n### Dedekind cuts\n\nA Dedekind cut of a [-540] is a [-partition] of that set into two sets so that every element in the first set is [ less than] every element in the second set, and the second set has no smallest element. The latter restriction requires that the set also be a [ perfect set] (have no [isolated_point isolated points]), in the sense used in topology.\n\nIn the real numbers, such a partition will always have the first set having a greatest element, which is known as the least-upper-bound property. However, in the rational numbers, we might come across a partition where the first set does not have such an element.\n\nFor example, define a Dedekind cut $(A, B)$ of the rational numbers such that $B = \\{x \\in \\mathbb{Q} \\ | \\ x > 0 \\wedge x^2 > 2\\}$ and $A$ is the complement of $B$. In plainer language, $B$ consists of all the numbers greater than $\\sqrt{2}$, but because $\\sqrt{2}$ doesn't exist in the space of rational numbers, we can't use that to formulate our definition. Obviously every element of $A$ is less than every element of $B$, but $A$ has no greatest element either, because we can create a sequence of numbers in $A$ that gets bigger and bigger (as it approaches $\\sqrt{2}$) but never stops at a maximum value.\n\nFor each of these "strict cuts" where neither set has a "boundary element", we define a new irrational number to "fill in the gap", just like with the Cauchy sequences. For the Dedekind cuts where one of the sets does have a least or greatest element, we define a real number equal to that rational number.\n\nThis definition has the advantage that each real number is represented by a unique Dedekind cut, unlike the Cauchy sequences where multiple sequences can converge to the same number.',
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