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Earn +20 ptsQ: Is the difference of root2 and root3 irrational?
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Is the set of irrational numbers closed under division?
No, it is not. Root2 and root 8 are each irrational. Root8 / root2 =2. 2 is not a member of the set.
How do you show that 2 plus root2 is an irrational number?
Assume it's rational. Then 2 + root2 = some rational number q. Then root2 = q - 2. However, the rational numbers are well-defined under addition by (a,b) + (c,d) = (ad + bc, bd) (in other words, you can add two fractions a/b and c/d and always get another fraction of the form (ad + bc)/bd.) Therefore, q - 2 = q + (-2) is rational, since both q and -2 are rational. This implies root2 must be rational, which is a contradiction. Therefore the assumption that 2 + root2 is rational must be false.
Is the sum or difference of a rational number and an irrational number is irrational?
Yes.
Is the difference of any two irrational numbers rational or irrational?
It could be either.
Is the difference of a rational number and an irrational number always ratioal?
No, it is always irrational.
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