Question

I once met a respectable professor who believed that there is no such thing as an irrational number. Unfortunately, he never stated his reasons behind said belief, but why do you think anyone would ever want to think that? He taught topology, so perhaps that had to do something with it.

Answer 1

\[\text{coffee cup}=\text{donut}\]

Answer 2

lol

Answer 3

The closest thing I can think of is the representation of an irrational number as a fraction with infinite, yet different, numerators and denominators:\[\frac{31415...}{10000...},\]but that still makes no sense.

Answer 4

Sounds too strange to me, I can't really think of a reason.

Answer 5

i am sure he/she was making some sort of philosophical point, and knew full well that almost all numbers are irrational.

Answer 6

He sounded very factual about it, almost empirical. But according to density in \(\mathbb{R}\), there are as many rational numbers as there are irrational numbers. :P\[\]

Answer 7

I guess he just wasn't very fond of real analysis.

Answer 8

I know there are more irrational numbers than natural numbers!

Answer 9

Or had an ongoing debate with a faculty member thereof. lol

Answer 10

rational*

Answer 11

Yeah, the cardinality of \(\mathbb{N}\) is \(\aleph_0\) whereas the cardinality of \(\mathbb{Q}\) is \(c=\aleph_1\), if I recall correctly. xd\[\]

Answer 12

^that's what I was typing.

Answer 13

Yeah, and the cardinality of rational numbers is the same as the cardinality of natural numbers.

Answer 14

You should have asked him ... what is \( π \) then?

Answer 15

Yeah, but I guess I didn't want to begin an argument with a faculty member over a (pretty much) personal belief of theirs. xd I believe certain aspects of math become ambiguous as you grow in it.

Answer 16

Didn't the Greeks have a great proof for the sqrt(2) not being the ratio of two natural numbers? I don't think your professor has a choice of what facts to believe.

Answer 17

phi is right. Exoduxus' theory of incommensurables, presented in geometrical form in Euclid's elements, is a masterpiece of Greek mathematics, though it is usually omitted in the diluted high-school syllabus.