 ## Rational and Irrational Number

Subject: Compulsory Maths

#### Overview

The union of the set of rational numbers and the set of irrational numbers is called the real numbers.The Number in the form $\frac{p}{q}$, where p and q are integers and q≠0 are called rational numbers.Numbers which can be expressed in decimal form are expressible neither in terminating nor in repeating decimals, are known as irrational numbers.

#### Rational Numbers

The number can be in different form. Some numbers can be in a form of fraction, ratio, root and with the decimal. If the number is in the form of $\frac{p}{q}$ (fraction) ,of two integer p and q where numerator p and q≠0 are called rational numbers.

5 $\frac{2}{3}$, $\frac{7}{4}$, $\frac{3}{4}$, $\frac{3}{5}$ etc are the examples of rational numbers.

Rational number can be:

• All natural number
• All whole number
• All integer
• All fraction

#### Irrational Numbers

Numbers which cannot be expressed in a ratio (as a fraction of integer) or it can be expressed in decimal form is known as irrational numbers. It can neither be terminated nor repeated. For example,

√7 = 2.64575131.............

√5 = 2.23620679....... etc are irrational numbers.

√2,√3,√5,√6,√7, etc. are the examples of irrational number where the numbers are a non-terminating and a non-repeating number.

### Some Results on Irrational Numbers

1. If we made an irrational number negative then it is always an irrational number.
For example, -√5

2. If we add a rational number and an irrational number then a result is always an irrational number.
For example, 2 +√3 is irrational.

3. If we multiply a non-zero rational number with an irrational number then it is always an irrational number.
For example, 5√3 is an irrational number.

4. The sum of two irrational number is not always an irrational number.
For example, (2 +√3) + (2 -√3) = 4, which is irrational.

5. The product of two irrational number is not always an irrational number.
For example, ( 2 +√3) x (2 -√3) = 4 -3 =1, which is rational.
##### Things to remember
• The number in the form $\frac{p}{q}$, where p and q are integers and q≠0 are called rational number.
• A rational number is a number that can be written as a ratio.
• An irrational number is a real number that cannot be expressed as a ratio of integers.
• Irrational numbers cannot be represented as terminating or repeating decimals.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

$\frac{1}{2}$ and $\frac{3}{5}$

0.50 0.60

0.51 0.61

0.52 0.62

0.54 0.64

0.56 0.66

0.58 0.68

a) π is an irrational number. ( True)

b) -√3 is an irrational number. (True)

c) Irrational numbers cannot be represented by points on the number line. (False)

d) All real number are rational ( False)

e) Every real number is not a rational number. (True)

Solution:

1) $\sqrt{3 }$ and 3 -$\sqrt{3}$

2) $\sqrt{7 }$+ 2 and 7 -$\sqrt{7}$

Solution:

1) $\sqrt{5}$ and -$\sqrt{5}$

2) $\sqrt{7 }$ and -$\sqrt{7}$

Solution:

( 2+$\sqrt{3}$) + ( 2 - $\sqrt{3}$) = 4, which is rational.

The sum of two irrational numbers is not always an irrational number.

Solution:

( 2 + $\sqrt{3}$) $\times$ ( 2 - $\sqrt{3}$)

= 22 + ($\sqrt{3}$)2

= 4 - 3

= 1, which is rational.

So, the product of two irrational numbers is not always irrational numbers.

Solution:

a) $\frac{4}{7}$ (Rational)

b) -$\frac{2}{5}$ (Rational)

c) $\sqrt{4}$ (Rational)

d) $\sqrt{8}$ (Rational)

e) $\sqrt{3}$ (Irrational)

f) 2 - $\sqrt{3}$ (Irrational)

g) - $\sqrt{7}$ (Irrational)

h) $\sqrt{25}$ (Rational)

i) 0.75 (Rational)

Solution:

9/16 ÷ 5/8

= 9/16 × 8/5

= (9 × 8)/(16 × 5)

= 72/80

= 9/10

Solution:

-6/25 ÷ 3/5

= -6/25 × 5/3

= {(-6) × 5}/(25 × 3)

= -30/75

= -2/5

Solution:

11/24 ÷ (-5)/8

= 11/24 × 8/(-5)

= (11 × 8)/{24 × (-5)}

= 88/-120

= -11/15

Solution:

Let the other number be x.

Then, x × (-4)/9 = -28/27

or, x = (-28)/27 ÷ (-4)/9

or, x = (-28)/27 × 9/-4

or, x = {(-28) × 9}/{27 × (-4)}

or, x = -(28 × 9)/-(27 × 4)

or,x =(287×91)/(273×41)

$\therefore$x = 7/3

Hence, the other number is 7/3.

Solution:

(-25/9) × (-18/15)

= (-25) × (-18)/9 × 15

= 450/135

= 10/3

Solution:

6/11 × (-55)/36

= 6 × -55/11 × 36

= -330/396

= -5/6

Solution:

The additive inverse of 2/5 is -2/5

4/7 - 2/5

= 4/7 + (-2/5)

= 4 × 5/7 × 5 + (-2) × 7/5 × 7

= 20/35 + -14/35

= 20 + (-14)/35

= 6/35

Solution:

(-11)/3 is not a positive rational. Since both the numerator and denominator are of the opposite sign.

Solution:

25/(-27) is not a positive rational. Since both the numerator and denominator are of the opposite sign.