system
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human
Do irrational numbers "exist"?

We were studying radicals today in Maths session, and this whole unit in maths is about arithmetic and numbers and sequences. The teacher, as a preamble to the contents of the unit, gave us a brief revision over numbers, and laid the foundation for number theory. Previously, I had been told that irrational numbers were numbers that cannot be put into ratios between whole numbers.

As the teacher was explaining irrational numbers, he said that there were only three cases of them. The first is the square root of a number which does not have a square root. The second is the cubic root of a number that does not have a cubic root. The third is when this number is pi, or is some multiple of pi, or is added to pi. Of course, I knew that there were higher roots, and that there existed countless other irrational numbers, such as e (Euler's number). When he talked about the first and second cases, I assumed he meant that it doesn't have a whole, or rational, root, but when I asked, he said it simply doesn't exist. I showed him a value on the calculator, and he said that it was just an approximation to a never-ending number, which  I understand. But, I argued, this never-ending number does exist, we simply cannot write it down or give an exact value for it. His counterargument was that, since we can never place for it an exact number, it doesn't exist. The whole class had sided with him, and so I relented, but is this really true? Do irrational numbers not exist, or do the majority of mathematicians agree that it doesn't?
gpt
They exist. Your teacher is wrong. There is no restriction for the digits of a number to be finite.

There is a definition of real numbers using cuts (dedekind cut). And you can see how with this definition we can should that the square root of 2 is a number.