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No. The set of Irrational Numbers has the same cardinality as the set of real numbers, and so is uncountable.
The set of rational numbers is countably infinite.
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What do you mean by countably infinite and infinite?
Countably infinite means you can set up a one-to-one correspondence between the set in question and the set of natural numbers. It can be shown that no such relationship can be established between the set of real numbers and the natural numbers, thus the set of real numbers is not "countable", but it is infinite.
What do rational and irrational numbers have in common?
Rational and irrational numbers are part of the set of real numbers. There are an infinite number of rational numbers and an infinite number of irrational numbers. But rational numbers are countable infinite, while irrational are uncountable. You can search for these terms for more information. Basically, countable means that you could arrange them in such a way as to count each and every one (though you'd never count them all since there is an infinite number of them). I guess another similarity is: the set of rational numbers is closed for addition and subtraction; the set of irrational numbers is closed for addition and subtraction.
There are fewer rational numbers than irrational numbers?
We can categorize all infinite sets into two categories. The first set is made up of a an infinite number of countable elements and the second consists of a infinite number of elements that cannot be counted. The set of rational numbers is countably infinite. This comes from that fact that we can easily count integers and natural numbers. Just remember to think of rationals as a ratio of two integers, a/b. The set of irrational numbers is uncountably infinite. There is no way to find a correspondence between irrational numbers and integers. By the definition of irrational, they can't be written as fractions. The fact that the natural numbers and rational numbers can be counted and the irrationals cannot be gives some intuitive understanding of why the latter is bigger. In general, it turns out that an countably infinite set is smaller. The German mathematician Georg Cantor did extensive work on the size ( cardinality) of sets including the these two. He provided some amazing proofs in two papers 1895 and 1897 and titled 'Beiträge zur Begründung der transfiniten Mengenlehre' . This is an interesting topic and a link is attached to help the reader understand it better. The proofs require a good deal of set theory background So I have not presented them here, but will give link to those too.
What is infinite and finite set in math?
A set is finite if there exists some integer k such that the number of elements in k is less than k. A set is infinite if there is no such integer: that is, given any integer k, the number of elements in the set exceed k.Infinite sets can be divided into countably infinite and uncountably infinite. A countably infinite set is one whose elements can be mapped, one-to-one, to the set of integers whereas an uncountably infinite set is one in which you cannot.
What is a set of rational and irrational numbers?
It is the set of Real numbers.
