I have a bunch of questions that I don't know how to approach about whether a number is rational or irrational
Could someone give me an idea of how to approach questions like the following:
1 + srt(2) + srt(3/2)
Is this rational or irrational (and why)?
Let's go back to your original question.
If is rational, surely r - 1 is rational.
So let's focus on s = r - 1 instead. Again, if s is rational, surely s2 is rational as well.
If s2 is rational, then t = s2 - 7/2 must also be rational.
Now , and if t is rational then t/2 = (1/2)t is also surely rational, which means that the square root of 3 is rational.
But this is absurd.
Working backwards, we see that t cannot be rational, which means that s2 cannot be rational, which means that s itself cannot be rational.
But this means (finally!) that r cannot be rational, for assuming r rational leads to .
The way I would do this would be to note that .
Now multiply that by to get
Finally multiply by 4 to get
a fourth degree polynomial with integer coefficients.
Since that was created by multiplying by other terms, is a factor so is a zero of the polynomial.
Now, the "rational root theorem" says that if a polynomial with integer has rational zeros then they must be of the form where the denominator, n, evenly divides the leading coefficient, here "4", and the numerator, m, evenly divides the constant term, here 13.
The only integers that evenly divide 4 are 1, -1, 2, -2, 4, and -4. The only integers that evenly divide 13 are 1, -1, 13, and -13. Thus, the only rational number that could possibly satisfy the equation are 1, -1, 13, -13, , , , , , , and . But putting those into the polynomial show that, in fact, none of them are actually zeros. Therefore no zeros of that polynomial are rational. Since is a root, it cannot be rational.