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I am looking for the easiest (elementary) proof that $\mathbb R$ is infinite dimensional as a $\mathbb Q$-vector space, without using cardinality. It should be understandable at highschool level.

So I guess the question could be reformulated as: what is the easiest infinite family of reals that can be showed to be independent ? So far square roots of primes seems a good candidate, but the proof is still a little intricate, is it the easiest possible ?

The goal of this is to show students that we can prove the result by different ways, and see that understanding cardinals is useful. But I still want the students to be able to understand the other proof.

QuantumSuperfield
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Denis
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2 Answers2

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Consider the set:

$$\{\log p\}_p$$

of logs of prime numbers. They are linearly independent over $\Bbb Q$ by the Fundamental Theorem of Arithmetic--i.e. unique factorization of integers greater than $1$ into primes. As there are infinitely many primes, the set $\Bbb R$ contains an infinite dimensional subspace, and is so itself infinite dimensional. (This is the standard proof number theorists love).

Adam Hughes
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  • Awesome, exactly what I was looking for, thanks ! It is almost better than the cardinality proof... – Denis Jul 16 '14 at 11:40
  • Glad to help! As a number theorist I love this kind of thing. :-) – Adam Hughes Jul 16 '14 at 11:41
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    in a strict sense you need FTA, because some of the $a_p$ can be negative, and then you have to say that the decomposition is unique. – Denis Jul 16 '14 at 11:44
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    I like recycling this fact a lot in algebra it takes the form of presenting $\Bbb Q$ as a $\Bbb Z$-module of the form:

    $$\Bbb Q\cong \bigoplus_p \Bbb Z$$

    where the index represents the exponent of the prime.

     – Adam Hughes Jul 16 '14 at 11:47
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Try $\{1, \pi, \pi^2, \pi^3,\ldots\}$ (or use any transcendental number instead of $\pi$). Since if $$\sum_{i=0}^na_i\pi^i=0, \text{ }a_i\in\mathbb Q$$then $\pi$ must be a root of $$a_0 + a_1x + \ldots + a_nx^n$$which would contradict the transcendence of $\pi$.

Mathmo123
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    Thanks it is nice, but transcendance of $\pi$ is a hard result which must be admitted at this level... It is harder than the proof with square root of primes. – Denis Jul 16 '14 at 11:34
  • It would work with any transcendental number. Are there any that you could prove a result for? – Mathmo123 Jul 16 '14 at 11:35
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    The only argument I know for existence of transcendental numbers is cardinality ;) – Denis Jul 16 '14 at 11:35
  • I see the issue... there is Liouville's theorem (http://en.wikipedia.org/wiki/Liouville_number#Liouville_numbers_and_transcendence) but that is hardly high school level. I'll have a think – Mathmo123 Jul 16 '14 at 11:37
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    Transcendence of $e$ is more easily proven than for $\pi$. – hardmath Jul 16 '14 at 14:12