1.4. Natural Numbers, Integers, and Rational Numbers

Examples 1.4.3(c):

Let A be the set N x N and define an equivalence relation r on N x N and addition of the equivalence classes as follows:
  1. (a,b) is related to (a,b) if a + b = a + b
  2. [(a,b)] + [(a',b')] = [(a + a', b + b')]
  3. [(a,b)] * [(a, b)] = [(a * b + b * a, a * a + b * b)]
What is the best symbol to use for the resulting equivalence classes ?
Since two pairs (a,b) and (a', b') are related if we might as well choose the symbol b - a to denote their equivalence classes. Hence: By the above rules, if the symbols 2 and -3 are added together we get the class and if the symbols 2 and -3 are multiplied together we get the class Hence, these equivalence classes, together with the definition of addition and multiplication, give a mathematically precise meaning to the symbol -2, and explains in fact the meaning, the addition, and the multiplication of the integers Z
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