0
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.
Wiki User
∙ 13y ago
Add your answer:
Earn +20 pts
There are fewer rational numbers than irrational numbers.?
Yes, there are.
Which numbers is not rational?
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
Numbers existing between two rational numbers?
There are more irrational numbers between any two rational numbers than there are rational numbers in total.
How many rational numbers can be inserted between two irrational numbers?
Infinitely many! There are an infinite number of rational numbers between any two irrational numbers (they will more than likely have very large numerators and denominators), and there are an infinite number of irrational numbers between any two rational numbers.
Which is a real number that is not a rational number?
Actually there are more irrational numbers than rational numbers. Most square roots, cubic roots, etc. are irrational (not rational). For example, the square of any positive integer is either an integer or an irrational number. The numbers e and pi are both irrational. Most expressions that involve those numbers are also irrational.
There are fewer rational numbers than irrational numbers.?
Yes, there are.
Is There are fewer rational than irrational numbers?
Yes, fewer by an order of infinity.
Do irrational numbers contain fewer numbers?
No, the set of irrational numbers has a cardinality that is greater than that for rational numbers. In other words, the number of irrational numbers is of a greater order of infinity than rational numbers.
Are there fewer rational numbers than irrational numbers?
For any given subset, yes, because there are an infinite number of irrational numbers for each rational number. But for the set of ALL real numbers, both are infinite in number, even though the vast majority of real numbers would be irrational.
List of rational and irrational numbers?
-- There's an infinite number of rational numbers. -- There's an infinite number of irrational numbers. -- There are more irrational numbers than rational numbers. -- The difference between the number of irrational numbers and the number of rational numbers is infinite.
What is the number of rational numbers between square root 3 and square root 5?
Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.
Are there more rational numbers than irrational numbers true or false?
In between any two rational numbers there is an irrational number. In between any two irrational numbers there is a rational number.
Are more rational numbers than irrational numbers true or false?
In between any two rational numbers there is an irrational number. In between any two Irrational Numbers there is a rational number.
Is There fewer rational numbers than irrational numbers.?
Yes. The infinity of rational numbers has the same size as the natural numbers, said to be "countable". The infinity of real numbers (and therefore, also of irrational numbers) is a larger infinity, said to be "uncountable".
How are rational and irrational numbers similar?
Both are part of the real numbers; both are infinite sets. (However, there are more irrational than rational numbers.)Both are part of the real numbers; both are infinite sets. (However, there are more irrational than rational numbers.)Both are part of the real numbers; both are infinite sets. (However, there are more irrational than rational numbers.)Both are part of the real numbers; both are infinite sets. (However, there are more irrational than rational numbers.)
Is there more rational numbers then irrational?
No. There are infinitely many of both but the number of irrational numbers is an order of infinity greater than that for rational numbers.
Which numbers is not rational?
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
