Note on Rational and Irrational Number

  • Note
  • Things to remember
  • Exercise
  • Quiz

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.

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.
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Very Short Questions

Solution:

  1. √5 and 5-√5
  2. √3+2 and 3-√3

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{3}{9})
    (frac{6}{29})
    (frac{5}{9})
    (frac{9}{4})
  • Change the following decimal number into fraction:

    0.(overline{7})

    ;i:1;s:15:
    ;i:2;s:15:
    ;i:3;s:15:
    ;i:4;s:15:
  • Change the following decimal number into fraction:

    0.(overline{24})

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

    0.(overline{132})

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

    0.(overline{27})

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

    1.(overline{57})

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

    0.(overline{365})

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

    4.(overline{78})

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

    0.(overline{445})

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

    1.(overline{525})

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

    (frac{4}{27})
    (frac{-28}{57})
    (frac{7}{22})
    (frac{-5}{57})
  • Write down the numerator of  rational numbers:

    (frac{(-7)}{5})      

    10
    8
    -7
    -2
  • Write down the numerator of  rational numbers:

     (frac{(-17)}{(-21)})   

    20
    -17
    17
    -21
  • Write down the denominator of  rational numbers:

    (frac{(-4)}{5})    

    2
    9
    5
    10
  • Write down the denominator of  rational numbers:

     15

    2
    4
    1
    3
  • You scored /15


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