Uniqueness of the Binary Operation on Rational Numbers (Q)

When discussing the binary operation defined on the rational numbers (Q), we often deal with well-defined functions. Specifically, the question posed is whether the operation ( frac{a}{b} circ frac{c}{d} frac{ac}{bd} ) is unique. This article will explore the concept, the implications of using a non-uniform representation, and the broader mathematical significance.

Interpreting the Binary Operation

The binary operation in question is ( frac{a}{b} circ frac{c}{d} frac{ac}{bd} ). However, the input forms (frac{a}{b} frac{c}{d}) and (frac{ac}{bd}) imply a misunderstanding or misinterpretation of the operation. The correct notation for the operation should be shown as (frac{a}{b} circ frac{c}{d} frac{a}{b} cdot frac{c}{d} frac{ac}{bd}). The symbolism ( frac{a}{b} frac{c}{d} ) without clear delimiters can lead to confusion. A clearer notation would be to use parentheses to group the fractions, for example, (left(frac{a}{b}right) circ left(frac{c}{d}right) frac{a}{b} cdot frac{c}{d} frac{ac}{bd}).

The Nature of Rational Numbers

Rational numbers are typically defined as equivalence classes of ordered pairs of integers, where ( frac{a}{b} ) is equivalent to ( frac{c}{d} ) if and only if ( ad bc ). This implies that the same rational number can be represented in multiple forms, such as ( frac{1}{2} ) being equivalent to ( frac{1000}{2000} ).

The Well-Defined Nature of the Function

When considering the function ( F left( frac{a}{b} circ frac{c}{d} right) frac{ac}{bd} ), this function is well-defined if it consistently maps equivalent fractions to the same result. However, the given binary operation ( frac{a}{b} circ frac{c}{d} frac{ac}{bd} ) fails the well-defined test. Let’s take the example of ( frac{1}{2} circ frac{3}{2} ):

Using ( frac{1}{2} ) and ( frac{3}{2} ), we get ( frac{1 cdot 3}{2 cdot 2} frac{3}{4} ). Using the equivalent representations ( frac{1000}{2000} ) and ( frac{3000}{2000} ), we get ( frac{1000 cdot 3000}{2000 cdot 2000} frac{3000000}{4000000} ), which simplifies to ( frac{3}{4} ).

This example does not illustrate the issue clearly because the simplified forms ultimately yield the same result. However, consider the representation ( frac{1}{2} circ frac{3000}{2000} ) and ( frac{1000}{2000} circ frac{3}{2} ). Here, we get:

Using ( frac{1}{2} ) and ( frac{3000}{2000} ), we get ( frac{1 cdot 3000}{2 cdot 2000} frac{3000}{4000} ), which simplifies to ( frac{3}{4} ). Using ( frac{1000}{2000} ) and ( frac{3}{2} ), we get ( frac{1000 cdot 3}{2000 cdot 2} frac{3000}{4000} ), which simplifies to ( frac{3}{4} ).

The discrepancy arises when considering non-simplified representations, where the intermediate steps yield different forms before simplification. For instance, using the non-simplified forms directly:

Using ( frac{1}{2} ) and ( frac{3000}{2000} ), we get ( frac{1 cdot 3000}{2 cdot 2000} frac{3000}{4000} ). Using ( frac{1000}{2000} ) and ( frac{3}{2} ), we get ( frac{1000 cdot 3}{2000 cdot 2} frac{3000}{4000} ).

Both operations yield ( frac{3000}{4000} ), but the forms themselves are different. This indicates that the operation is not well-defined with the given representation. To ensure the operation is well-defined, the fractions must be reduced to their simplest form before performing the multiplication.

Implications and Uniqueness

The fact that the binary operation requires the fractions to be reduced to their simplest form before mapping to the operation suggests that the definition is not elegant or unique. This interpretation implies that the operation ( frac{a}{b} circ frac{c}{d} frac{ac}{bd} ) is not unique in the broad sense. For example:

Let ( F left( frac{1}{2} circ frac{3}{2} right) 1 ) as simplified fractions result in 1. Let ( F left( frac{1000}{2000} circ frac{3000}{2000} right) ) produces a value very close to but not exactly 1 due to the intermediate steps.

This demonstrates that the function ( F ) can map inputs to different outputs, depending on the chosen representation before simplification. Additionally, the operation is not injective, meaning there can be multiple pairs of fractions that yield the same result under the given operation.

Conclusion

Thus, the binary operation ( frac{a}{b} circ frac{c}{d} frac{ac}{bd} ) is not unique when considering the non-uniform representation of rational numbers. It fails the well-defined test and is not injective, indicating that the operation is not robust or unique in the mathematical sense. Ensuring the operation is well-defined requires reducing fractions to their simplest form, which adds complexity and reduces the elegance of the operation.