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Mathematics

State whether the given statement is true or false :

(i) The quotient of two integers is always a rational number.

(ii) Every rational number is a fraction.

(iii) Zero is the smallest rational number.

(iv) Every fraction is a rational number.

Rational Numbers

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Answer

(i) False
Reason — While a rational number is defined as the quotient of two integers pq\dfrac{p}{q}, the denominator q cannot be zero. If the divisor (the second integer) is zero, the quotient is undefined and is not a rational number.

(ii) False
Reason — Fractions typically refer to a part of a whole and are usually represented as a ratio of two natural numbers. Rational numbers include negative values (like 23-\dfrac{2}{3}) and integers (like -5), which are not traditionally considered fractions in their standard form.

(iii) False
Reason — Rational numbers include negative values. Therefore, any negative rational number (such as -1 or 12-\dfrac{1}{2}) is smaller than zero. Because the set of rational numbers extends infinitely in the negative direction, there is no "smallest" rational number.

(iv) True
Reason — A fraction is represented as ab\dfrac{a}{b} where a and b are whole numbers and b ≠ 0. Since whole numbers are also integers, every fraction fits the definition of a rational number.

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