Write true (T) or false (F): (i) Every rational number has a reciprocal. (ii) Subtraction of rational numbers is commutative. (iii) The additive inverse of is -4. (iv) is a rational number. (v) 0 is the smallest rational number. (vi) The multiplicative inverse of is . (vii) There are countless rational numbers between .

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KnowledgeBoat · Accepted Answer

(i) There is no rational number which when multiplied by 0 gives 1. So the rational number 0 has no reciprocal. Hence, the given statement is false. (ii) For any two rational numbers , we have . For example, , but . Hence, the given statement is false. (iii) The additive inverse is obtained by changing the sign of the rational number. So the additive inverse of and not -4. Hence, the given statement is false. (iv) A rational number is of the form where . Here the denominator is 0 and division by 0 is not defined. Hence, the given statement is false. (v) Every negative rational number is smaller than 0, and there is no smallest rational number as we can always find a rational number smaller than any given one. For example, -1 0 and -2 -1 and so on. Hence, the given statement is false. (vi) First express the rational number with a positive denominator. To find the multiplicative inverse of a non-zero rational number, interchange its numerator and denominator. Interchanging the numerator and the denominator of , we get , which is the same as the multiplicative inverse given in the statement. Hence, the given statement is true. (vii) Between any two different rational numbers we can always insert their mean, and this process can be repeated as many times as we want. Hence, the given statement is true.

Mathematics

Rational Numbers

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