State whether each of the following statements is true or false: (i) The difference of two rational numbers is always a rational number. (ii) (iii) 1 is the additive identity for rational numbers. (iv) Subtraction is commutative on rational numbers. (v) There exists a rational number which is its own additive inverse. (vi)

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KnowledgeBoat · Accepted Answer

(i) Rational numbers are closed under subtraction, i.e. if are two rational numbers, then is also a rational number. Hence, the given statement is true. (ii) Solving L.H.S., Solving R.H.S., Clearly, L.H.S. ≠ R.H.S. Hence, the given statement is false. (iii) For every rational number , we have , so 0 is the additive identity and 1 is the multiplicative identity. Hence, the given statement is false. (iv) For any two rational numbers , we have . For example, but . Hence, the given statement is false. (v) The additive inverse of the rational number 0 is 0 itself, as 0 + 0 = 0. Hence, the given statement is true. (vi) Solving L.H.S., Solving R.H.S., Clearly, L.H.S. ≠ R.H.S. Hence, the given statement is false.

Mathematics

Rational Numbers

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