# Number System

## 01 Number System

#### CLASSIFICATION OF NUMBERS

class 9 math

Natural numbers:

Whole numbers:

Integers

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

Irrational numbers: Irrational numbers can not be expressed as $\frac{\text{p}}{\text{q}}$ form. Numbers of the type  are called irrational numbers.

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

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

2) $0.010010001...$

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

Equivalent fractions:

*Number  =….  are called equivalent fractions.

* There are infinitely many rational numbers between two integers.

e.g.   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

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

e.gBetween 2and 3 lie the irrational numbers $\sqrt{7},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[3]{17},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[4]{54},\text{\hspace{0.17em}}...$

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  are is irrational number.

e.g.  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.

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

e.g.

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

If Q and R are two is irrational numbers, then  are irrational numbers.

e.g.

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

$\text{R}×\text{Q}$ 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.

is rational number,it is division of $\text{2}$ irrational numbers.

Rationalise:

Rationalising factor of  is .

So

=  (rationalising)

=

Laws of exponents:

i)

ii)

iii)

iv)

v)