Are there more rational or irrational numbers?
I am thinking irrational, because the product of a rational and irrational number is irrational. Is this enough of a reason to prove it?
There are many many more irrationals than rationals. In fact, there are so many more that if you threw a dart at the number line, the chance of hitting a rational is zero.
The proof is relatively simple, and involves the idea of countability. You can count the rationals, but not the irrationals.
Your reasoning is a good start, but it is not enough to prove that there are more irrational numbers than rational numbers. To determine which set is larger, we need to use a different approach.
To compare the cardinality of two sets, we can use a concept from set theory called "cardinality." The cardinality of a set represents the size or number of elements in that set.
The set of rational numbers is countable, which means its cardinality is the same as that of the set of natural numbers (positive whole numbers). This is because we can list all the rational numbers in a systematic way, such as using fractions in lowest terms.
On the other hand, the set of irrational numbers is uncountable, meaning its cardinality is greater than that of any countable set. This has been proven mathematically using Cantor's diagonal argument.
Based on this, we can conclude that there are more irrational numbers than rational numbers, even though the product of a rational and an irrational number is indeed irrational.
So, in summary, the fact that the product of a rational and an irrational number is irrational may support the idea that there are more irrational numbers, but the actual proof lies in the concept of cardinality and Cantor's diagonal argument.