section_path: Arithmetic/Integers and fractions as real numbers
section_name: Integers and fractions as real numbers
paragraph_text: Integers and fractions as real numbers
  
  The real numbers 0 and 1 are commonly identified with the whole numbers 0 and 1. This allows identifying any natural number n with the sum of n real numbers equal to 1.
  
  This identification can be pursued by identifying a negative integer − n {\displaystyle -n} (where n {\displaystyle n} is a natural number) with the additive inverse − n {\displaystyle -n} of the real number identified with n. {\displaystyle n.} Similarly a rational number p / q {\displaystyle p/q} (where p and q are integers and q ≠ 0 {\displaystyle q\neq 0}) is identified with the division of the real numbers identified with p and q.
  
  These identifications make the set Q {\displaystyle \mathbb {Q} } of the rational numbers an ordered subfield of the real numbers R. {\displaystyle \mathbb {R}.} The Dedekind completeness described below implies that some real numbers, such as 2, {\displaystyle {\sqrt {2}},} are not rational numbers; they are called irrational numbers.
  
  The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties (axioms). So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function.
  
  Formally, one has an injective homomorphism of ordered monoids from the natural numbers N {\displaystyle \mathbb {N} } to the integers Z, {\displaystyle \mathbb {Z},} an injective homomorphism of ordered rings from Z {\displaystyle \mathbb {Z} } to the rational numbers Q, {\displaystyle \mathbb {Q},} and an injective homomorphism of ordered fields from Q {\displaystyle \mathbb {Q} } to the real numbers R. {\displaystyle \mathbb {R}.} The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write
  
  These identifications are formally abuses of notation (since, formally, a rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming. In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by the compiler.