Locate $\sqrt{5}$ and $-\sqrt{5}$ on number line.

1. Rational Number:

A number $x$ is called a rational number, if it can be written in the form $\frac{m}{n}$, where $m$ and $n$ are integers and $n\ne 0.$

2. Irrational Number:

A number $x$ is called an irrational number, if it cannot be written in the form $\frac{m}{n}$, where $m$ and $n$ are integers and $n\ne 0.$

3. Decimal Expansion:

(i) The decimal expansion of a rational number is either terminating or non-terminating recurring. Thus, we say that a number whose decimal expansion is either terminating or non-terminating recurring is a rational number.

(ii) The decimal expansion of an irrational number is non-terminating non-recurring. Thus, we say that a number whose decimal expansion is non-terminating non-recurring is an irrational number.

4. Real Numbers:

(i) All the rational and irrational numbers taken together make up the collection of real numbers.

(ii) A real number is either rational or irrational.

(iii) There is a real number corresponding to every point on the number line. Also, corresponding to every real number there is a point on the number line.

5. Properties of Real Numbers:

(i) If $r$ is rational and $s$ is irrational, then $r+s, r-s, rs$ and $\frac{r}{s}$ are irrational numbers.

(ii) If $n$ is a natural number other than a perfect square, then $\sqrt{n}$ is an irrational number.

(iii) If $a$ is a positive real number and $n$ is a positive integer, then the principal $n$th root of $a$ is the unique positive real number $x$ such that $x^n=a.$ The principal $n$th root of $a$ is denoted by $a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$ or $\sqrt[n]{a}$.

6. Laws of Exponents:

If $a, b$ are positive real numbers and $m, n$ are rational numbers, then

(i) $a^{\mathit{m}}\times a^{\mathit{n}}=a^{m+n}$

(ii) $a^m÷a^n=a^{m-n}$

(iii) ${\left(a^m\right)}^n=a^{mn}$

(iv) $a^{-n}=\frac{1}{a^{\mathit{n}}}$

(v) ${\left(ab\right)}^n=a^n{b}^n$

(vi) ${\left(\frac{a}{b}\right)}^n=\frac{a^{\mathit{n}}}{b^{\mathit{n}}}$

(vii) $a^{\frac{m}{n}}={\left(a^m\right)}^{\frac{1}{n}}={\left(a^{\frac{1}{n}}\right)}^m$ i.e., $a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m, m, n\in N$

7. Rationalisation of Real Numbers:

If $a, b$ are positive real numbers, then

(i) Rationalisation factor of $\frac{1}{\sqrt{a}}$ is $\sqrt{a}$

(ii) Rationalisation factor of $\frac{1}{a\pm \sqrt{b}}$ is $a\mp \sqrt{b}$

(iii) Rationalisation factor of $\mathrm{ }\mathrm{ }\frac{1}{\sqrt{a}\pm \sqrt{b}}$ is $\sqrt{a}\mp \sqrt{b}$