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Rational Numbers

State True or...

Question

State True or False.
A rational number can always be written in a fraction $\frac{a}{b}$ , where a and b are not integers $(b \neq 0)$ .

A

True

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B

False

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Solution

The correct option is B False
A number that can always be written in the form of $p / q$ , where p is any integer and q is a non-zero integer, is a rational number.
The given statement is false.

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Q. In each of the following state if the statement is true (T) or false (F):
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every fraction is a rational number.
(v) Every rational number is a fraction
(vi) If

\frac{a}{b}

is a rational number and m any integer, then

\frac{a}{b} = \frac{a \times m}{b \times m}

(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x)

\frac{2}{3} is equal to \frac{4}{6} .

Integers are closed under division, i.e. for any two integers, a and b, a $\div$ b will be an integer. State true or false.

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.

Question 01

A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$ , where p and q are integers and

(a) q = 0

(b) q = 1

(c) q $\neq$ 1

(d) q $\neq$ 0

State true or false.

a^{n} - b^{n}

is always divisible by a - b where n is a positive integer

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