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A rational number is one that can be expressed as a fraction, with integer numerator and denominator. This includes all "normal" fraction, as well as integers.
An irrational number is one that can not be expressed in such a way. This includes, for example, square roots of any natural number (except the square root of a perfect square) - like square root of 2, of 3, of 5, of 6, etc.; as well as two numbers that are of extreme importance in math: pi, and e.
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Is there a function that is continuous everywhere differentiable at rationals but not differentiable at irrationals?
No.
Is a real number either a rational or an irrational number?
Yes, reals are rationals and irrationals.
Diff kind of real numbers?
Natural (or counting) numbers Integers Rationals Irrationals Transcendentals
Are There are fewer rational numbers then irrational numbers?
Yes, there are countably infinite rationals but uncountably infinite irrationals.
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.
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.
Does the set of rational number overlap the set of irrational numbers why?
By definition, the two sets do not overlap. This is because the irrationals are defined as the set of real numbers that are not members of the rationals.
What is the intersection of the rational numbers and the irrational numbers?
Some would say that there is no intersection. However, if the set of irrational numbers is considered as a group then closure requires rationals to be a proper subset of the irrationals.
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.
What product is true about the irrational and rational numbers?
The product of 2 rationals must be rational. The product of a rational and an irrational is irrational (unless the rational is 0) The product of two irrationals can be either rational or irrational.
0 belongs to what family of real numbers?
0 belongs to the reals. It is a member of the irrationals, the rationals. It is also a member of the integers; It is a member (the identity) of the group of even integers, 3*integers, 4*integers etc with respect to addition.
Is 67 divided by a rational or irrational?
You can divide 65 by rationals and irrationals: Divided by a rational: 65 ÷ 13/2 = 10 Divided by an irrational: 65 ÷ √13 = 5√13
