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1. What can be said of the dimension of the basis of the Reals over the Irrationals

2. Homework Equations

3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number. ie. sqrt(2)*sqrt(2)=2 + sqrt(51)*sqrt(51)=53, etc. Could you please help me figure out if this is the correct solution or what a better way to phrase the proof would be? Thanks!

2. Homework Equations

3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number. ie. sqrt(2)*sqrt(2)=2 + sqrt(51)*sqrt(51)=53, etc. Could you please help me figure out if this is the correct solution or what a better way to phrase the proof would be? Thanks!

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