Building Q from Z

Abstract Algebra · Axiom Academy

Constructing the rational numbers as equivalence classes of integer pairs Excellent work! You've constructed the rational numbers from the integers. Here's what we established: Foundation: We built from ( \ 0\ ) using equivalence classes, mirroring how we think of fractions intuitively Equivalence Relation: The relation captures the idea that a/b = c/d , and we proved it satisfies reflexivity, symmetry, and transitivity Well-Defined Operations: Addition and multiplication on equivalence classes are well-defined and follow familiar fraction arithmetic rules Field Structure: With these operations, forms a field satisfying all field axioms (associativity, commutativity, identity elements, inverses, distributivity) Embedding : The map embeds into , preserving addition and multiplication, showing integers are a special case of rationals This construction is fundamental in abstract algebra. It demonstrates how to rigorously build new algebraic structures from existing ones using equivalence relations—a technique that extends to constructing real numbers from rationals, quotient groups, and many other structures!

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