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If a and b are rational, then a + p(b-a) where p is any number between 0 and 1, is rational and lies between a and b. So pick any 3 values for p.
If a and b are not rational, then find rationals u and v such that
a < u < v < b and repeat as above.
Cantor proved that between any two real numbers there are an infinite number of rationals which ensures the existence of u and v.
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Are all number between 1 and 2 rational numbers?
No, there are more irrational numbers between 1 and 2 than there are rational numbers.
Are all numbers between 1 and 2 rational numbers?
No, not at all. There are more irrational numbers between 1 and 2 than there are rational numbers in total!
How many rational numbers less than 0.001?
Infinitely many.
When you multiply two rational numbers will you always get a number that is smaller than the two factors?
No. A rational number is ANY number that can be represented as one integer over a second integer (which cannot be zero). There is no requirement that the top integer is less than the bottom integer (an improper fraction is still a rational number - all integers are rational numbers as they can be represented as an improper fraction with a 1 as the denominator). Only if both rational numbers are less than 1 will the result of multiplying them together be less than both of them. If one rational number is greater than 1 and the other less than 1, then the result of multiplying them together is greater than the number less than 1 and less than the number greater than 1. If both rational numbers are greater than 1, then the result of multiplying them together is greater than both of them.
What is the smallest digit that's common in all subsets of the rational numbers?
There is no such number. The empty set is a subset of rational numbers and, by definition, it contains no numbers so nothing that can be common to any other subset.Alternatively, all rational numbers less than -1 and all rational numbers greater than 1 are subsets of rational numbers. There is no number common to them.
