An infinite set of fractions is a collection of fractions where the number of elements (fractions) is limitless; you can always find another fraction within the set, no matter how many you've already identified.
Understanding Infinite Sets
An infinite set, in general, is a set that is not finite. This means you can't count its elements to reach a final number. With fractions, this often arises when considering fractions within a particular range or with certain properties.
Examples of Infinite Sets of Fractions
Here are a few examples to illustrate the concept:
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Fractions between 0 and 1: As noted in the reference, the set of all fractions between 0 and 1 is infinite. For any two fractions in that range, say a/b and c/d (where a/b < c/d), you can always find another fraction between them, for instance, their average: (a/b + c/d)/2. This process can be repeated indefinitely, proving the set's infinite nature.
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Fractions with a denominator of 2: Consider the set {1/2, 2/2, 3/2, 4/2, 5/2, ... }. This set extends infinitely because we can keep incrementing the numerator.
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Fractions with a specific numerator: Similarly, {1/1, 1/2, 1/3, 1/4, 1/5, ... } is an infinite set as we can keep incrementing the denominator.
Why are These Sets Infinite?
The key reason these sets are infinite stems from the properties of rational numbers. Between any two distinct rational numbers (fractions), there exists infinitely many other rational numbers. This "density" of rational numbers is what allows us to continually find new fractions within a defined range or structure.
Key Takeaway
An infinite set of fractions demonstrates the limitless possibilities within the realm of rational numbers. Whether confined to a range or defined by a specific pattern, the ability to always generate another unique fraction is what defines this infinite nature.
