# Explaining there's an infinite amount of irrationals to differential calculus freshmen

I'm currently working as a teacher assistant. My job now is to orientate differential calculus' students and try to solve any questions that may rise in the development of the course and exercises. They're instructed to solve a particular set of problems; in that set you can find the following question:

Are there infinitely many irrational numbers?

They're asked to answer either yes or no and to justify their answer. I can't seem to find an easy way of explaining that there is indeed an infinite amount of irrational numbers. To do so, I only imagine talking about Cantor's diagonal argument.

Resuming, my question's this: is there an easy, naive way of explaining that there's an infinite amount of irrationals?

• I think the question may be poorly worded and miscue the students. "Infinite irrational numbers" does not mean the same thing as "Infinitely many irrational numbers". Students may think the question is asking them whether there are irrational numbers that are infinite (to which the correct answer is "No"). – mweiss Aug 20 '14 at 3:28
• @mweiss I'm deeply sorry: my english is not the best and I'm translating it from spanish directly. You're right, that's the right question to ask. – Miguelgondu Aug 20 '14 at 3:30
• Is it possible that in Spanish a similar confusion could exist? – mweiss Aug 20 '14 at 3:33
• The short answer is a quick no. The question states Hay infinitos números irracionales, which is different from Hay números irracionales infinitos. The placement of the adjetive is everything. Thanks for the remark. – Miguelgondu Aug 20 '14 at 3:35
• How are you defining irrational numbers? Do you mean uncountably many irrational numbers? If you just want to show there are infinitely many, then you could show irrational + rational is irrational; take the classic fact (or proof...) that $\sqrt{2}$ is irrational, and generate infinitely many irrational numbers as: $\sqrt{2} + n$ for all $n \in \mathbb{Z}$. – Benjamin Dickman Aug 20 '14 at 4:10

## 1 Answer

(Migrated from comments at request of OP.) The question is a bit difficult to answer without knowing how the topics (e.g., infinitude, irrational number) have been defined. Nevertheless, since the question does not broach countability and is strictly about explaining the infinitude of irrational numbers, one might proceed as follows:

1. Show (or recall) that the sum and difference of any two rationals is rational.

2. Observe that the sum of any irrational and any rational is irrational. To prove this, suppose $a$ is irrational and $b$ is rational. Now: If $a+b$ were rational, then we could write $a = (a+b) - b$, i.e., we could write $a$ as the difference of rational numbers, hence $a$ would need to be rational; a contradiction. Thus, it must be the case that $a+b$ is irrational.

3. Show (or recall) that $\sqrt{2}$ is irrational.

4. Using #2, observe that $\sqrt{2} + q$ is irrational for all $q \in \mathbb{Q}$. This gives "infinitely many irrational numbers" as desired. If preferable, you could even modify and use $\sqrt{2} + n$ for all $n \in \mathbb{N}$.

But since this is MESE (not MSE): Ask students what they have already tried. Have they seen the proof that $\sqrt{2}$ is irrational? If so, has anyone suggested $\sqrt{p}$ is irrational for all primes $p$? Maybe someone has even gone and proved such a thing. Has someone suggested $\sqrt{n}$ is irrational for all natural numbers $n$? If so, where does such an argument break down? (And why can it be rescued by looking at, e.g., the square roots of primes?) If someone has tried the trick suggested above: Can they prove the different building blocks? Can they state anything meaningful about the sum of two irrationals? Can they solve related problems (e.g., given an irrational number and a rational number, prove there is always an irrational number strictly between them)?

Of course, if the irrationals have been defined in some other way, then your explanation may differ. For example, if one defines them as the non-repeating decimals, then can they prove $0.1234567891011\ldots$ is irrational? If they can prove this does not repeat, then can the proof be adapted to show the same about $1.1234567891011\ldots$? What about $2.1234567891011\ldots$? And so forth.

You write that your job is to help the students in this Calculus course to solve the problems, and that this particular one showed up. I am not convinced that showing them "an easy way" to answer this problem is the best approach with regard to helping them further down the road. Let them struggle with the problem; let them spend some time with the problem; demand to see some evidence that they have given the problem thought before you start to scaffold and assist them in figuring out a solution!