Rational and Irrational Number

Subject: Compulsory Maths

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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 and Irrational Number

Rational Numbers

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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

example for Irrational Numbers
Example for 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.

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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.
Questions and Answers

Solution:

  1. √5 and 5-√5
  2. √3+2 and 3-√3

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:

√2 = 1.41421356.......

Solution:

1) 0.75

2) -100

3) \(\frac{7}{20}\)

4) 0

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) \(\sqrt[3]{8}\) (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.

Quiz

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