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irrational numbers

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Indicate which of the following statements about sets under the specified operations are true. For the ones that are false, provide one counter-example.

(a) The set of irrational numbers under addition forms an abelian group.
(b) The set of complex numbers under multiplication forms a group.
(c) The set of rational numbers under multiplication and addition forms a ring.
(d) The set of rational numbers under multiplication and addition does not form a field.

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(a) This statement is false, as the addition of two irrational numbers does not guarantee another irrational number. Ex: (1 + √2) + (1 - √2) = 2, which is not an irrational number.

(b) The ...

Solution Summary

This post examines irrational numbers and other concepts.

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