Number System

Natural numbers: $\text{1, 2, 3, 4…}$

Whole numbers: $\text{0, 1, 2, 3, 4,…}$

Integers $\text{…-4, -3, -2, -1, 0 , 1, 2, 3, 4…}$

Rational numbers: A number   “ $\text{a}$ ” is called rational number if “ $\text{a}$ ” can be written as $\frac{p}{q}$,

Where $\text{q}\ne \text{0}$.

*Every rational number has terminating  or recurring decimalexpansion.

e.g $\frac{\text{3}}{\text{4}}\text{ = 0}\text{.75, }\frac{\text{1}}{\text{3}}\text{ = 0}\text{.33333 = 0}\text{.}\overline{\text{3}}$

Irrational numbers: Irrational numbers can not be expressed as $\frac{\text{p}}{\text{q}}$ form. Numbers of the type $\sqrt{\text{2}}\text{, }\sqrt[\text{3}]{\text{5}}\text{, π}$ are called irrational numbers.

*Every irrational number has non-terminating,non recurringdecimal expansion.

e.g 1) $\text{0}\text{.052631578947368421052631578947368421…,}$

       2) $\mathrm{0.010010001...}$

Real numbers:Real numbers are rational and irrational numbers together.

Equivalent fractions:

*Number $\frac{\text{3}}{\text{4}}\text{ = }\frac{\text{12}}{\text{16}}\text{ = }\frac{\text{15}}{\text{20}}$ =….  are called equivalent fractions.

More about real numbers

* There are infinitely many rational numbers between two integers.

e.g.  $\text{2}\text{.1, 2}\text{.2, 2}\text{.3, 2}\text{.4,…}$ lie between the integers $\text{2}$ and $\text{3}$.

*There are infinitely many rational numbers between two irrational numbers.

e.g.between $\sqrt{\text{2}}\approx \text{1}\text{.41}$ and $\sqrt{3}\approx \text{1}\text{.71}$, lie the numbers  $\text{1}\text{.42, 1}\text{.521, 1}\text{.522, 1}\text{.523,…}$

*There are infinitely many irrational numbers between two rational numbers.

e.gBetween 2and 3 lie the irrational numbers $\sqrt{7},\sqrt[3]{17},\sqrt[4]{54},\mathrm{...}$

Number line:

Every point on line, there can be assigned a real number.

Hence the line is called number line.

(fig of no. line)

Operations on Real numbers:

 1) Addition and subtraction of rational numbers with irrational numbers is irrational number.

      If $\text{P}$ is rational number, $\text{Q}$ is irrational number, then $\text{P + Q, P - Q}$ are is irrational number.

      e.g. $\text{2 + }\sqrt{\text{3}}$ is irrational number. It is addition of $\text{2}$ (rational) and $\sqrt{\text{3}}$ (irrational number).

 2) Multiplication and division of a rational and an irrational numbers mustbeanirrational number.

      $\text{P}\times \text{Q}$ isirrational number.

      e.g. $\text{2}\times \sqrt{\text{3}}\text{ = 2}\sqrt{\text{3}}$

3)   Addition or subtraction of  two irrational number is irrational number.

      If Q and R are two is irrational numbers, then $\text{Q + R}\text{.Q - R}$ are irrational numbers.

      e.g. $\sqrt{\text{5}}\text{ + 2}\sqrt{\text{5}}\text{ = 3}\sqrt{\text{5}}\text{, }\sqrt{\text{2}}\text{ + }\sqrt{\text{7}}$

4)  Multiplication or division of two irrational numbers will be either rational or irrational.

      $\text{R}\times \text{Q}$ may result in rational or irrational numbers.

      e.g $\frac{\sqrt{\text{2}}}{\sqrt{\text{3}}}$ is irrational,  it is division of $\text{2}$ irrational numbers.

      $\frac{\sqrt{\text{3}}}{\sqrt{\text{3}}}\text{ = 1}$ is rational number,it is division of $\text{2}$ irrational numbers.

Rationalise:

Rationalising factor of $\text{a + }\sqrt{\text{b}}$ is $\text{a - }\sqrt{\text{b}}$.

So $\frac{\text{1}}{\text{a + }\sqrt{\text{b}}}$

       = $\frac{\text{a - }\sqrt{\text{b}}}{\left(\text{a + }\sqrt{\text{b}}\right)\left(\text{a - }\sqrt{\text{b}}\right)}$ (rationalising)

      =  $\frac{\text{a - }\sqrt{\text{b}}.}{{\text{a}}^{\text{2}}\text{ - b}}$

Laws of exponents:

      i) ${\text{a}}^{\text{p}}{\text{ × a}}^{\text{q}}{\text{ = a}}^{\text{p + q}}$

      ii) ${\left({\text{a}}^{\text{p}}\right)}^{\text{q}}{\text{ = a}}^{\text{pq}}$

iii) $\frac{{\text{a}}^{\text{p}}}{{\text{a}}^{\text{q}}}{\text{ = a}}^{\text{p - q}}$

iv) ${\text{a}}^{\text{p}}·{\text{b}}^{\text{p}}\text{ = }{\left(\text{ab}\right)}^{\text{p}}$

      v) $\sqrt[\text{n}]{\text{a}}\text{ = }{\left(\text{a}\right)}^{\frac{\text{1}}{\text{n}}}$