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How can I prove that the basis of the vector space $\mathbb{R}$ over $\mathbb{Q}$ is uncountable.

PLz help !!

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If the basis were counatable, so would be the space. We yould have $\mathbb R\cong \mathbb Q[X]$. –  Hagen von Eitzen Oct 27 '12 at 16:30
    
What would be that basis, I can't seem to find one? –  Manjil P. Saikia May 9 '13 at 15:05
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1 Answer

up vote 2 down vote accepted

HINT: Suppose that $B$ is a countable base for $\Bbb R$ over $\Bbb Q$. Then every real number can be written in the form

$$q_1b_1+q_2b_2+\ldots+q_nb_n$$

for some positive integer $n$ and $n$-tuples $\{b_1,b_2,\dots,b_n\}\in B^n$ and $\langle q_1,q_2,\dots,q_n\rangle\in\Bbb Q^n$.

  1. How many positive integers are there?
  2. How many elements does $\Bbb Q^n$ have?
  3. How many elements does $B^n$ have?
  4. How many combinations of $n\in\Bbb Z^+$, $\{b_1,b_2,\dots,b_n\}\in B^n$ and $\langle q_1,q_2,\dots,q_n\rangle\in\Bbb Q^n$ are there altogether?
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