Number Systems Test - 20

Rational numbers are those numbers which can be expressed in the form $$ \dfrac {p}{q} $$, where p and q are integers and $$ q \neq 0 $$
Numbers which are not rational numbers are called irrational numbers.
Since, $$ \sqrt {7} $$ cannot be written in $$ \dfrac {p}{q} $$, where $$p$$ and $$q$$ are integers and $$ q \neq 0 $$; it is an irrational number.

$$\sqrt2 $$ is an irrational number.

$$(3+\sqrt{23})-\sqrt{23}$$
$$=3+\sqrt{23}-\sqrt{23}$$
$$=3$$.
Here, $$3$$ is a rational number.

$${\textbf{Step 1: Consider, option A.}}$$

                $${\text{Every fraction is a rational number}}.$$

                $${\text{As we know that,}}$$ $${\text{ A rational number is a type of real numbers, }}$$ 

                $${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$ 

                $${\text{Any fraction with non - zero denominators is rational number}}{\text{.}}$$

                $${\text{Thus, option A}}{\text{. is true}}{\text{.}}$$

$${\textbf{Step 2: Consider, option B}}{\textbf{.}}$$

                $${\text{Every rational number is a fraction}}.$$  

                $${\text{As we know that,}}$$ $${\text{ A rational number is a type of real numbers, }}$$ 

                $${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$

                $${\text{The given statement is not true as 10 is a rational number but it is not a fraction}}{\text{.}}$$

                $${\text{Thus, option B}}{\text{. is false}}{\text{.}}$$

$${\textbf{Step 3: Consider, option C}}{\textbf{.}}$$

                $${\text{Every integer is a rational number}}.$$

                $${\text{A rational number is a type of real numbers, }}$$

                $${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$

                $${\text{And we know that an integer can be represented as }}\dfrac{p}{q}$$ 

                $${\text{form where denominator will be always 1}}{\text{.}}$$

                $${\text{For example: }}10 = \dfrac{{10}}{1}$$

                $${\text{Thus, option C}}{\text{. is true}}{\text{.}}$$

$${\textbf{Hence, option B is correct answer.}}$$  

$$2^{0.64}*2^{0.36} =2^{0.64+0.36} = 2^{1.00} = 2^1=2\\ (\because a^m*a^n=a^{m+n})$$ 

If we divide a positive integer by another positive integer, the resulting number is always a rational number.

$$\displaystyle 5-2\sqrt{6}$$

The sum and difference of two simple quadratic surds are said to be conjugate surds to each other.
So, conjugate surd of $$a-\sqrt{b}$$  is  $$a+ \sqrt{b}$$.

$$\dfrac{1}{4} = \dfrac{6}{24} = \dfrac{12}{48} $$