Number Systems Class 9 Notes: CBSE Maths Chapter 1

• Real numbers

• Real numbers and imaginary numbers together form number systems.

• We will discuss imaginary numbers in higher classes, let us restrict our discussion to real numbers

• Real numbers are a set of natural numbers, whole numbers, integers, rational and irrational numbers. Denoted by R

• Natural numbers:

• These are counting numbers starting from $1$.

•  The set $\left\{ 1,2,3,4,5,6,7.... \right\}$ is called natural numbers.

•  Denoted by N

• Whole numbers:

• These are the set of natural numbers including $0$. 

• The set $\left\{ 0,1,2,3,4,5,6.... \right\}$ is called whole numbers.

• Denoted by W

• Integers: 

• These are the set of negative numbers, positive numbers and $0$ excluding fractions. 

• The set $\left\{ ....-3,-2,-1,0,1,2,3.... \right\}$ is called integers.

• Denoted by Z

• Rational numbers:

• These are those numbers which can be expressed in the form of fractions i.e., $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$. 

• For example: $\dfrac{3}{5},\dfrac{-2}{9},\dfrac{-3}{4},$ etc. 

• Denoted by Q

• There are infinitely many rational numbers between any two rational numbers.

• Irrational numbers:

• These are those which are not rational i.e., which cannot be expressed in the form of  $\dfrac{p}{q}$ where $p$ and $q$ are integers and$q\ne 0$.

• For example: $\sqrt{2},\sqrt{3},\sqrt{5},$ etc.

Real Numbers and Their Decimal Expansions

There are two cases of decimal expansions

1. Remainder becomes zero

• Decimal expansion of numbers whose remainder becomes zero after some step is called terminating.

• For example: $\dfrac{7}{8}=0.875$ , the remainder becomes zero after some steps

2. Remainder never becomes zero

• Decimal expansion of numbers whose remainder never becomes zero after some step is called non-terminating.

• It is further divided into non-terminating recurring and non-terminating non-recurring.

• Non-terminating recurring means numbers which keep on repeating the same value after the decimal point.

• For example: $\dfrac{9}{11}=0.818181....$ 

• Non-terminating non-recurring means numbers which do not keep on repeating the same value after the decimal point but the remainder never becomes zero.

• For example: value of $\pi =3.141592653589793283....$ 

• Decimal expansion of rational numbers is either terminating or non-terminating.

• Decimal expansion of irrational numbers is non-terminating non-recurring.

Representing Real Number on Number Line

• Representation of real numbers on the number line can be done by the process of successive magnification.

• For example: If we want to locate $4.377$ on the number line we proceed by successive magnification i.e., $4.37$ lies between $4$ and $5$ then locate $4.37$ between $4.36$ and $4.38$further divide this portion into ten equal parts then $4.377$ will lie between $4.376$ and $4.378$. The number line is shown below

Operations on Real Numbers

• Real numbers can be added, subtracted, multiplied and divided.

• For example: 

Add $2+\sqrt{3}$ and $2-2\sqrt{3}$

$2+\sqrt{3}+2-2\sqrt{3}$ 

$=4-\sqrt{3}$ 

Subtract $2+\sqrt{3}$ and $2-2\sqrt{3}$ 

$\left( 2+\sqrt{3} \right)-\left( 2-2\sqrt{3} \right)$ 

$=2+\sqrt{3}-2+2\sqrt{3}$ 

$=3\sqrt{3}$ 

Multiply $2\sqrt{2}$ and $3\sqrt{3}$ 

$2\sqrt{2}\times 3\sqrt{3}$

$=2\times 3\times \sqrt{2}\times \sqrt{3}$ 

$=6\sqrt{6}$

Divide $10\sqrt{15}$ by $\sqrt{5}$ 

$\dfrac{10\sqrt{15}}{\sqrt{5}}=\dfrac{10\sqrt{3}\times \sqrt{5}}{\sqrt{5}}=10\sqrt{3}$

• Some common facts of operation on real numbers are

1. The sum or difference between a rational number and an irrational number is irrational.

2. The product or quotient of a non-zero rational number with an irrational number is irrational.

3. If we add, subtract, multiply or divide two irrationals, then the result may be rational or irrational.

Rationalizing Denominator

• When the denominator is irrational then the process of converting the denominator to rational is called rationalizing the denominator.

• It is obtained by multiplying the numerator and denominator by the irrational term present in the denominator but with the opposite sign.

• For example: Rationalizing $\dfrac{1}{\sqrt{2}+3}$ 

$\dfrac{1}{\sqrt{2}+3}\times \dfrac{\sqrt{2}-3}{\sqrt{2}-3}$

$=\dfrac{\sqrt{2}-3}{{{\left( \sqrt{2} \right)}^{2}}-{{3}^{2}}}$ 

$=\dfrac{\sqrt{2}-3}{2-9}$ 

$=\dfrac{\sqrt{2}-3}{-7}$ 

Laws of Exponents for Real Numbers

There are some laws of exponent for real numbers such as

1. ${{x}^{m}}.{{x}^{n}}={{x}^{m+n}}$ 

2. $\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}$

3. ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$ 

4. ${{x}^{m}}{{y}^{m}}={{\left( xy \right)}^{m}}$ 

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