Operations on Real Numbers

1. The sum, difference, product and quotient of two rational numbers will be rational.

Example:

 \Rightarrow \dfrac{3}{4}+\dfrac{7}{4}=\dfrac{10}{4}=\dfrac{5}{2}

\Rightarrow \dfrac{7}{4}-\dfrac{3}{4}=\dfrac{4}{4}=1

\Rightarrow \dfrac{7}{4}\times\dfrac{3}{4}=\dfrac{21}{16}

=\dfrac{7}{4} \div\dfrac{3}{4}=\dfrac{7}{3}

2. If we add or subtract a rational number with an irrational number then the result will be irrational.

Example:

If  5 is a rational number and \sqrt7 is an irrational number then  5+\sqrt7 and 5-\sqrt7 are irrational numbers.

3. If we multiply or divide a non-zero rational number with an irrational number then the result will be irrational.

Example:

If 7 is a rational number and \sqrt5 is an irrational number then 7\sqrt7 and \dfrac{7}{\sqrt5} are irrational numbers.

4. The sum, difference, product and quotient of two irrational numbers could be rational or irrational.

Example:

\sqrt3+\sqrt3=2\sqrt3\qquad\mathrm{(irrational + irrational =irrational)}

\sqrt2-\sqrt2=0\qquad\quad\mathrm{(irrational-irrational=rational)}

(\sqrt6).(\sqrt6)=6\qquad\mathrm{(irrational\times irrational=rational)}

\dfrac{\sqrt{13}}{\sqrt{13}}=1\qquad\qquad\mathrm{(irrational\div irrational=rational)}

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