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Note on Rational and Irrational Number

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• Things to remember
• Exercise
• Quiz

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.

• 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.
.

Very Short Questions

Solution:

1. 0.5
2. 0
3. -100
4. $$\frac{3}{5}$$

Solution:

1. √5 and 5-√5
2. √3+2 and 3-√3

Solution:

1. √3 and -√3
2. √5 and -√5

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:

-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:

(-25/9) × (-18/15)

= (-25) × (-18)/9 × 15

= 450/135

= 10/3

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.

0%
• Change the following decimal number into fraction:0.(overline{5})

(frac{9}{4})
(frac{5}{9})
(frac{6}{29})
(frac{3}{9})

;i:3;s:15:
;i:2;s:15:
;i:4;s:15:
;i:1;s:15:
• Change the following decimal number into fraction:0.(overline{24})

(frac{7}{16})
(frac{9}{41})
(frac{6}{10})
(frac{8}{33})
• Change the following decimal number into fraction:0.(overline{132})

(frac{44}{333})
(frac{55}{444})
(frac{14}{153})
(frac{36}{863})
• Change the following decimal number into fraction:0.(overline{27})

(frac{3}{11})
(frac{6}{18})
(frac{4}{20})
(frac{1}{10})
• Change the following decimal number into fraction:1.(overline{57})

(frac{43}{12})
(frac{52}{33})
(frac{12}{32})
(frac{2}{19})
• Change the following decimal number into fraction:0.(overline{365})

(frac{162}{222})
(frac{365}{999})
(frac{305}{125})
(frac{265}{888})
• Change the following decimal number into fraction:4.(overline{78})

(frac{189}{41})
(frac{111}{91})
(frac{158}{33})
(frac{135}{12})
• Change the following decimal number into fraction:0.(overline{445})

(frac{142}{669})
(frac{565}{555})
(frac{325}{189})
(frac{445}{999})
• Change the following decimal number into fraction:1.(overline{525})

(frac{500}{554})
(frac{508}{333})
(frac{458}{444})
(frac{226}{289})
• The sum of the rational numbers (frac{– 8}{19}) and (frac{-4}{57}) is?

(frac{7}{22})
(frac{-5}{57})
(frac{4}{27})
(frac{-28}{57})

-2
10
8
-7

17
-21
-17
20

5
10
2
9

3
2
4
1