Does ChatGPT know modern math concepts?
Another prompt from a friend, Matt Mohler.
There are infinitely many rational numbers. There are infinitely many irrational numbers. Explain why there are more irrational numbers than rational numbers.
To understand why there are more irrational numbers than rational numbers, it’s essential to understand what the terms “rational” and “irrational” mean.
A rational number is any number that can be expressed as the ratio of two integers, that is, a fraction of the form p/q, where p and q are integers and q is not equal to zero.
On the other hand, an irrational number is a number that cannot be expressed as a ratio of two integers. In other words, its decimal representation goes on forever without repeating.
Now, let’s consider the set of rational numbers. While there are infinitely many rational numbers, they can be arranged in a countable set, which means that there is a one-to-one correspondence between the set of rational numbers and the set of positive integers. This is because every rational number can be expressed as a fraction of two integers, and we can create a bijection between the set of positive integers and the set of all pairs of integers.
On the other hand, the set of irrational numbers is uncountable, which means that there is no one-to-one correspondence between the set of irrational numbers and the set of positive integers. This was first proven by Georg Cantor in the late 19th century, using his diagonal argument. Essentially, Cantor showed that no matter how you try to list all the possible decimal expansions of irrational numbers, there will always be an infinite number of expansions that are not on the list.
Therefore, since there are more irrational numbers than rational numbers, it can be said that there are “more” real numbers that are irrational than those that are rational.
Georg Cantor is sometimes called the Father of Infinity and Modern Mathematics, and ChatGPT knows who he is because humans have written about him for some time and there’s a lot of information available about his life and work. Another example of a question that has a finite response based on well-documented knowledge, ChatGPT is able to coalesce vast amounts of information into a concise summary.