Chapter: Algebraic and Transcendent Numbers

The b-adic fractions used for the representation of real numbers are dependent on the basis $b.$ For instance, a real number like for instance $\pi$ can be written in the decimal system $3.14159 26536\ldots$, the binary system $11.001001000011\ldots,$ the hexadecimal system $3.243F6A8886\ldots$, etc.

Therefore, a real number $x\in\mathbb R$ can have different b-adic representations, and each b-adic representation provides only a little (if any) information about the nature of $x.$ Also, the distinction between rational numbers $\mathbb Q$ and irrational numbers $\mathbb R\setminus\mathbb Q$ does not reveal all aspects of the nature of real numbers, except the fact that an irrational number cannot be written as a fraction of two integers.

Motivations: 1

  1. Definition: Continued Fractions
  2. Lemma: Continuants and Convergents

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  1. Kraetzel, E.: "Studienb├╝cherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981