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what are the differences between rational and irrational numbers Informational
- What are the sets of rational and irrational numbers denoted by? What is the union of the set of rational numbers and the set of irrational numbers?
- Do the continued fraction representations of rational and irrational numbers differ? Are the continued fraction expansions of rational numbers finite or infinite?
- How are irrational numbers used in the construction of specific fractal sets (e.g., Julia sets)? How do irrational numbers appear in the study of wavelets and their applications?
- Does the existence of irrational numbers have implications for the completeness or consistency of formal systems?
- Explain the method of Cauchy sequences for defining irrational numbers.
- Why do some musical intervals involve irrational ratios in certain tuning systems? How are irrational numbers used in computer graphics and image processing?
- How can we help students understand the difference between rational and irrational numbers?
- Are logarithms of rational numbers to rational bases sometimes irrational?
- How are irrational numbers used in lattice-based cryptography? What is the connection between the security of lattice-based cryptosystems and the properties of irrational numbers in high dimensions?
- Are trigonometric functions of rational multiples of π sometimes irrational?
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