# Number System Class 9 Notes - Chapter 1

## Introduction to Number Systems

### Numbers

Number: Arithmetical value representing a particular quantity. The various types of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers etc.

### Natural Numbers

Natural numbers(N) are positive numbers i.e. 1, 2, 3 ..and so on.

### Whole Numbers

Whole numbers (W) are 0, 1, 2,..and so on. Whole numbers are all Natural Numbers including ‘0’. Whole numbers do not include any fractions, negative numbers or decimals.

### Integers

Integers are just like whole numbers, but they also include negative numbers. They are denoted by Z. Examples: -3, -2, -1, 0, 1, 2

### Rational Numbers

A number ‘r’ is called a rational number if it can be written in the form pq, where p and q are integers and q ≠ 0.

### Irrational Numbers

Any number that cannot be expressed in the form of pq, where p and q are integers and q≠0, is an irrational number. Examples: √2, 1.010024563…, e, π

### Real Numbers

Any number which can be represented on the number line is a Real Number(R). It includes both rational and irrational numbers. Every point on the number line represents a unique real number.

## Irrational Numbers

### Representation of Irrational numbers on the Number line

Let √x be an irrational number. To represent it on the number line we will follow the following steps:

• Take any point A. Draw a line AB = x units.
• Extend AB to point C such that BC = 1 unit.
• Find out the mid-point of AC and name it ‘O’. With ‘O’ as the centre draw a semi-circle with radius OC.
• Draw a straight line from B which is perpendicular to AC, such that it intersects the semi-circle at point D.

Length of BD=√x.

• With BD as the radius and origin as the centre, cut the positive side of the number line to get √x.

## Identities for Irrational Numbers

### Operations on Rational and Irrational numbers

Arithmetic operations between:

• rational and irrational will give an irrational number.
• irrational and irrational will give a rational or irrational number.

Example : 2 × √3 = 2√3 i.e. irrational. √3 × √3 = 3 which is rational.

### Identities for irrational numbers

If a and b are real numbers then:

• √ab = √a√b
• √ab = √a√b
• (√a+√b) (√a-√b) = a – b
• (a+√b)(a−√b) = a²−b
• (√a+√b)2 = a+2√ab+b

### Rationalisation

Rationalisation is converting an irrational number into a rational number. Suppose if we have to rationalise 1/√a.
1/√a × 1/√a = 1/a

Rationalisation of 1/√a+b:

(1/√a+b) × (1/√a−b) = (1/a−b²)

### Laws of Exponents for Real Numbers

If a, b, m and n are real numbers then:

• a× an= am+n
• (am) = amn
• am/a= am−n
• ambm=(ab)m

Here, a and b are the bases and m and n are exponents.

### Exponential representation for irrational numbers

If a > 0 and n is a positive integer, then: n√a=a1n Let a > 0 be a real number and p and q be rational numbers, then:

• ap × aq = ap + q
• (ap)= apq
• ap/ aq= ap−q
• apb= (ab)p

## Decimal Representation of Rational Numbers

### Decimal expansion of Rational and Irrational Numbers

The decimal expansion of a rational number is either terminating or non- terminating and recurring.

Example: 1/2 = 0.5 , 1/3 = 3.33…….
The decimal expansion of an irrational number is non terminating and non-recurring.
Examples: √2 = 1.41421356..

### Expressing Decimals as rational numbers

Case 1 – Terminating Decimals

Example – 0.625

Let x=0.625

If the number of digits after the decimal point is y, then multiply and divide the number by 10y.

So, x = 0.625 × 1000/1000 = 625/1000 Then, reduce the obtained fraction to its simplest form.

Hence, x = 5/8

Case 2: Recurring Decimals

If the number is non-terminating and recurring, then we will follow the following steps to convert it into a rational number:

Example –

Step 1. Let x =   (1)

Step 2. Multiply the first equation with 10y, where y is the number of digits that are recurring.

• Thus, 100x = (2)Steps 3. Subtract equation 1 from equation 2.On subtracting equation 1 from 2, we get99x = 103.2

x=103.2/99 = 1032/990

Which is the required rational number.

Reduce the obtained rational number to its simplest form Thus,

x=172/165