Real numbers | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-05

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Real_number


00:02:48 1 History
00:06:45 2 Definition
00:07:35 2.1 Axiomatic approach
00:09:31 2.2 Construction from the rational numbers
00:10:08 3 Properties
00:10:17 3.1 Basic properties
00:12:47 3.2 Completeness
00:16:13 3.3 "The complete ordered field"
00:19:16 3.4 Advanced properties
00:24:20 4 Applications and connections to other areas
00:24:31 4.1 Real numbers and logic
00:26:14 4.2 In physics
00:27:10 4.3 In computation
00:29:15 4.4 "Reals" in set theory
00:29:44 5 Vocabulary and notation
00:31:43 6 Generalizations and extensions
00:33:58 7 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.984682016936729
Voice name: en-GB-Wavenet-D


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number).  Included within the irrationals are the transcendental numbers, such as π (3.14159265...).   In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; ), up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.
The set of all real numbers is uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted 
  
    
      
        
          
            c
          
        
      
    
    {\displaystyle {\mathfrak {c}}}
   and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted 
  
    
      
        
          ℵ
          
            0
          
        
      
    
    {\displaystyle \aleph _{0}}
   'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than 
  
    
      
        
          ℵ
          
            0
          
        
      
    
    {\displaystyle \aleph _{0}}
   and strictly smaller than 
  
    
      
        
          
            c
          
        
      
    
    {\displaystyle {\mathfrak {c}}}
   is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel ...