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Introduction to Rational Numbers

State whether...

Question

State whether the following statements are true or false? Justify your answer.
(i) There are infinitely many integers between any two integers.
(ii) Number of rational numbers between 15 and 18 is finite.

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Solution

(i) False. Because between two consecutive integers ( like 1 and 2 ) there does not exist any other integer.

(ii) False. Because between any two rational numbers there exist infinitely many rational numbers.

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Q. Question 3
State whether the following statements are true or false? Justify your answer.
(i)

\frac{\sqrt{2}}{3}

is a rational number.
(ii) There are infinitely many integers between any two integers.
(iii) Number of rational numbers between 15 and 18 is finite.
(iv) There are numbers which cannot be written in the form

\frac{p}{q}, q \neq 0

p and q both are integers.
(v) The square of an irrational number is always rational.
(vi)

\frac{\sqrt{12}}{\sqrt{3}}

is not a rational number as

\sqrt{1} 2

\sqrt{3}

are not integers.
(vii)

\frac{\sqrt{15}}{\sqrt{3}}

is written in the form

\frac{p}{q}

, so it is a rational number.

Q. State whether the following statement is True of False ?Justify your answer.

\frac{\sqrt{72}}{\sqrt{2}}

is not a rational number as

\sqrt{72} a n d \sqrt{2}

are not integers.

Q. State true or false:
(i) Between any two distinect integers there is always an linteger.
(ii) Between any two distinct rational numbers there is always a rational number.
(iii) Between any two distinct rational numbers there is always a rational number.

Q. State whether the following statements are true or false. Justify your answers.
Every rational number is a real number

State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

(ii) Every integer is a whole number.

(iii) Every rational number is a whole number.

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