Rational number | Wikipedia audio article
Channel: wikipedia tts
Date: 26/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Rational_number


00:03:24 1 Terminology
00:04:55 2 Arithmetic
00:05:05 2.1 Irreducible fraction
00:05:37 2.2 Embedding of integers
00:05:58 2.3 Equality
00:07:25 2.4 Ordering
00:08:01 2.5 Addition
00:09:02 2.6 Subtraction
00:09:59 2.7 Multiplication
00:10:50 2.8 Inverse
00:12:41 2.9 Division
00:14:05 2.10 Exponentiation to integer power
00:16:33 3 Continued fraction representation
00:17:59 4 Other representations
00:20:28 5 Formal construction
00:29:05 6 Properties
00:30:48 7 Real numbers and topological properties
00:31:43 8 ip/i-adic numbers
00:32:44 9 See also
00:33:22 10 References
00:34:17 11 External links



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SUMMARY
=======
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  , Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian 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 any other integer base (e.g. binary, hexadecimal).
A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) such that q ≠ 0, for the equivalence relation defined by (p1, q1) ~ (p2, q2) if, and only if p1q2 = p2q1. With this formal definition, the fraction p/q becomes the standard notation for the equivalence class of (p, q).
Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. 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.