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What are subsets of rational numbers?
There are an infinite number of subsets: All rationals other than 1 All rationals other than 2, etc All rationals other than 1.1 All rationals other than 2.1, etc, etc. All integers
Is fraction the densest subset of real numbers?
No. Fractions do not include irrational numbers. And although there are an infinite number of both rationals and irrationals, there are far more irrational numbers than rationals.
How many rational numbers are there between 1 and 6?
There are an infinite number of rationals between 1 and 6.
Between which two rational numbers does the square root of 85 lie?
It is not possible to answer this question sensibly, since rational numbers form a continuum. So for any pair of rational numbers surrounding sqrt(85), it is possible to give another pair of rationals that surrounds it but such that the rationals are closer together. And this sequence is infinite.
Can you multiply two rationals?
yes
Are There are fewer rational numbers then irrational numbers?
Yes, there are countably infinite rationals but uncountably infinite irrationals.
What are subsets of rational numbers?
There are an infinite number of subsets: All rationals other than 1 All rationals other than 2, etc All rationals other than 1.1 All rationals other than 2.1, etc, etc. All integers
Is fraction the densest subset of real numbers?
No. Fractions do not include irrational numbers. And although there are an infinite number of both rationals and irrationals, there are far more irrational numbers than rationals.
How many rational numbers are there between 1 and 6?
There are an infinite number of rationals between 1 and 6.
Are there more rational number than irrational numbers?
There are more irrational numbers than rational numbers. The rationals are countably infinite; the irrationals are uncountably infinite. Uncountably infinite means that the set of irrational numbers has a cardinality known as the "cardinality of the continuum," which is strictly greater than the cardinality of the set of natural numbers which is countably infinite. The set of rational numbers has the same cardinality as the set of natural numbers, so there are more irrationals than rationals.
Are half of irrational numbers rational?
No, nowhere near. Georg Cantor proved that the number of rational numbers is countably infinite whereas the irrationals are uncountably infinite. If you take the number of rationals to be N then the number of irrationals is of the order 2^N.
Between which two rational numbers does the square root of 85 lie?
It is not possible to answer this question sensibly, since rational numbers form a continuum. So for any pair of rational numbers surrounding sqrt(85), it is possible to give another pair of rationals that surrounds it but such that the rationals are closer together. And this sequence is infinite.
Is the cardinality of an infinitely countable set the same as the rational numbers?
Yes. There is an injective function from rational numbers to positive rational numbers*. Every positive rational number can be written in lowest terms as a/b, so there is an injective function from positive rationals to pairs of positive integers. The function f(a,b) = a^2 + 2ab + b^2 + a + 3b maps maps every pair of positive integers (a,b) to a unique integer. So there is an injective function from rationals to integers. Since every integer is rational, the identity function is an injective function from integers to rationals. Then By the Cantor-Schroder-Bernstein theorem, there is a bijective function from rationals to integers, so the rationals are countably infinite. *This is left as an exercise for the reader.
Can you multiply two rationals?
yes
Why do we use Z and Q to represent integers?
We use Q for Rationals... which is repreentative of Quocentia (Quotient), since rationals are RATIOs or fractions.
Is 83 a prime numbe?
Yep.
What is a set of rational numbers that begin at 0?
It could be the set denoted by Q- (the non-positive rationals) or Q+ (the non-negative rationals).
