Types Of Rational Numbers Types Of Rational Numbers
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode ℚ), which stands for quotient.+ The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base. A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Know More About Use The Product Rule To Simplify The Expression
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The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z ∖ {0})) / ~, where the cartesian product Z × (Z ∖ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not zero (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0. In abstract algebra, the rational numbers together with certain operations of addition and multiplication form a field. This is the archetypical field of characteristic zero, and is the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals. The equivalence relation (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set (Z × (Z ∖ {0})) / ∼, i.e. we identify two pairs (m1,n1) and (m2,n2) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by [(m1,n1)] the equivalence class containing (m1,n1). If (m1,n1) ~ (m2,n2) then, by definition, (m1,n1) belongs to [(m2,n2)] and (m2,n2) belongs to [(m1,n1)]; in this case we can write [(m1,n1)] = [(m2,n2)]. Given any equivalence class [(m,n)] there are a countably infinite number of representation. Read More About Number Sets Algebra 1 Worksheets
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The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x,y) = |x − y|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x,y) = |x − y|, above.
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