Revisiting Rational and Irrational numbers

1. History of Irrational Numbers : Hippassus (around 500BC), a Greak philosopher, is widely regarded as the first person to recognise the existence of irrational numbers. See, how he got the idea of irrational numbers.
        
    Let us consider, a right angled trianlge with AB = 1unit and AC = 1unit. Definitely, the length of BC can not be a positive integer i.e., it should be some number that lie between 1 and 2.
2. The meaning of “Irrational” is not having a ratio or no ratio can be written for that numbers.

3. Irrational Number (Definition):    A number which cannot be expressed in the form of  \( \frac{p} {q} \), where p and q are integers and \( q \ne 0 \) is called an irrational number.
4. Examples of irrational number: 
(i) All non-terminating and non-repeating decimal expansions are irrational number.
    i.e.,  2.314231..........
           1.4142318.........
            6.1101011.......
(ii)  \( \sqrt 2 ,\sqrt 3 ,\sqrt 8 ,..... \)  
(iii) \(\ \sqrt {prime\text{ }number} \)is always an irrational number.

(iv)\( \pi \) is an irrational number where as \( \frac{{22}} {7} \) is a rational number.
(v) 
 

5. How to find irrational numbers?
    Let us find irrational numbers between 2 and 3.
    We know that \( \sqrt 4 = 2 \) and \( \sqrt 9 = 3 \). Therefore, the irrational numbers between 2 and 3 are \( \sqrt 5 ,\sqrt 6 ,\sqrt 7 and\sqrt 8 \), as these are not perfect squares and cannot be         simplified further.

6. Dense property : Between two irrational numbers, there exists infinite number of irrational numbers.

7. The letters ‘P’ or ‘Q’ is used to denote the set of irrational numbers.

8. Logarithms of primes with prime base are irrationals i.e.,\( \log _3^2 ,\log _2^5 ,\log _3^7 ....... \) are irrational numbers.

9. Infinite continued fractions: This is one of the ways to represent irrational numbers 
       \( a_0 + \frac{1} {{a_1 + \frac{{b_0 }} {{a_2 + \frac{{b_1 }} {{a_3 + .....}}}}}} \) eg., \( 2 + \frac{1} {{2 + \frac{1} {{2 + 3.....}}}} \)
    This is an irrational number.

10.  Properties of Irrational Number : Since irrational numbers are the subsets of real numbers, Irrational numbers will obey all the properties of the real numbers.
(i)     The addition of an irrational number and a rational number gives an irrational number i.e., \( \sqrt 3 + 2,\sqrt 3 - 2 \)

(ii)    The multiplication of an irrational number with non-zero rational number results an irrational number i.e., \( 2\sqrt 3 ,\frac{1} {2}\sqrt 5 ,6\sqrt 7 ,..... \)

(iii)    The L.C.M of two irrational numbers may or may not exists.

(iv)    The addition or the multiplication of two irrational numbers may or may not irrational number.
For example \( \left( {2 + \sqrt 3 } \right) + \left( {4 + \sqrt 3 } \right) = 6 + 2\sqrt 3 \) is an irrational number.
But , \( \left( {2 + \sqrt 3 } \right) + \left( {2 - \sqrt 3 } \right) = 4 \) is not an irrational number
Also, \( 2\sqrt 2 \times 4\sqrt 2 = 16 \)  is not an irrational number.

11. Representing Irrational Numbers on number line :
            
        
    Let OA = OB = 1 unit
    By pythogorous theorem
        AB² = OA²+OB²
            = 1²+1²
            = 2
        \( \therefore AB = \sqrt 2 \)
Now, take a point on C such that AB = OC = \( \sqrt 2 \)
    Hence, the point C represents \( \sqrt 2 \) on the number line.
    Similarly by joining B and C and continuing the same process, we can indicate \( \sqrt 3 ,\sqrt 5 ,\sqrt 6 ....etc \) on the number line.

12.    Rational numbers

12.1    History of rational numbers : Rational numbers were invented in the sixth century BCE. Pythagoras is indeed an early Greek mathematician who has best known for inventing rational numbers.
12.2    Rational number (Definition) : The numbers of the form \( \frac{p} {q} \), where both p and q are integers and \( q \ne 0 \) are called rational numbers.

12.3.   Classification of Rational Numbers: Rational numbers are classified as positive, zero or negative rational numbers.
    When the numerator and the denominator both are either positive or negative integers, then it is called positive rational numbers, otherwise they are called negative rational numbers.

NOTE : 0 is a rational number.

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12.4 Types of rational numbers : There are four types of rational numbers
* Integers
Ex : -3, 2, 4, 0, -5, .....etc
* Fractions made up of integers.
Ex : \( \frac{2} {3},\frac{{ - 1}} {2},\frac{4} {3},\frac{{ - 7}} {8},....etc \)
* Terminating decimal fractions
Ex : 6.12, 6.3748, 1.234893
* Non-terminating decimal numbers with infinitely repeating patterns.
    Ex : 2.143214321432.…
1.0111011101110111....…

13. Dense Property: There exists an infinite number of ratonal numbers between any two rational numbers.

14.  Representing Rational Numbers on Number Line :
                           
    Let us consider a rational number \( \frac{2} {7} \). Divide unit length between 0 and 1 into 7 equal parts, call them subdivisions. The point at the line indicating the second-sub division from 0 which represents \( \frac{2} {7} \). In this way any rational number can represented on the number line.

15. Laws of Algebra for Rational Numbers :

(i) Closure property : If a, b \( \in \)Q, where Q is the set of rational numbers then 
    a+b\( \in\) Q and a.b\( \in \) Q

(ii) Commutative Property :     a+b = b+a and a.b = b.a

(iii) Associative property :     a+(b+c) = (a+b)+c    
                                                 a.(bc) = (ab)c

(iv) Additive Identity : There exists a rational number 0 such that a+0 = 0+a = a where a is any rational number.

    Hence, zero(0) is considered as the additive identity in the set of rational numbers.

(v) Additive Inverse : If a and b are two rational numbers such that a+b = b+a = 0, then each one is called additive inverse of other.

Ex: \( \left( {\frac{{ - 1}} {2}} \right) + \left( {\frac{1} {2}} \right) = 0 \) hence  are additive inverse of each other.

(vi)  Distributive Law : a x (b+c) = a x b + a x c

(vi)  Multiplicative Identity : We know for any rational number a x 1 = 1 x a = a.

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16. Componendo and Dividendo: Consider two rational numbers \( \frac{a} {b} \) and \( \frac{c} {d} \) such that \( \frac{a} {b} = \frac{c} {d} \)
Then, 
(i) \( \frac{b} {a} = \frac{d} {c}\left( {invertendo} \right) \)
(ii) \( \frac{{a + b}} {b} = \frac{{c + d}} {d}\left( {componendo} \right) \)
(iii) \( \frac{{a - b}} {b} = \frac{{c - d}} {d}\left( {dividendo} \right) \)
(iv) \( \frac{{a + b}} {{a - b}} = \frac{{c + d}} {{c - d}}\left( {componendo\text{ }and\text{ }dividendo} \right) \)

17. SURDS : If a is a positive rational number, which is not the nth power (here n is a natural number) of any rational number, then the irrational number \( \pm \sqrt[n]{a} \) are called simple surds or monomial surds.

Ex : \( \sqrt 3 ,\sqrt[3]{5},\sqrt[7]{8} \)
NOTE: Every surd is an irrational number ( but every irrational number is not a surd)

Ex: (i) \( \sqrt 3 \) is a surd and irrational number
     (ii) \( \sqrt[3]{5}\) is a surd and irrational numeber.
      (iii) \( \pi \) is an irrational number but it is not a surd.
      (iv)\( \sqrt[3]{{3 + \sqrt 2 }} \) is an irrational number but it is not a surd since \( 3 + \sqrt 2 \) is not a rational number.

18. Laws of radicals : If a>0, b>0 and n is a positive rational number, then
(i) \( \left( {\sqrt[n]{a}} \right)\left( {\sqrt[n]{b}} \right) = \sqrt[n]{{ab}} \)
(ii) \( \frac{{\left( {\sqrt[n]{a}} \right)}} {{\left( {\sqrt[n]{b}} \right)}} = \sqrt[n]{{\frac{a} {b}}} \)
(iii) \( \sqrt[m]{{\sqrt[n]{a}}} = \sqrt[n]{{\sqrt[m]{a}}} = \sqrt[{mn}]{a} \)
(iv) \( \sqrt[n]{{a^p }} = a^{\frac{p} {n}} \)
(v) \( \sqrt[n]{{a^p }} = \sqrt[n]{{\sqrt[m]{{\left( {a^p } \right)^m }}}} \)
(vi) \( \sqrt[n]{{a^n }} = a \)

19.  Rationalising Factor : If the product of two irrational numbers or surds is a rational number, then each surd is a rationalising factor (RF) for each other.

NOTE : 
1) RF is not unique 
2) If one RF of a surd is known then the product of this factor and any non-zero rational number is also the RF of the given surd.
3) It is convenient to use the simplest of all RF’s of the given surd to convert it to a rational number.

Ex: (1) \( \left( {3\sqrt 3 } \right)\left( {\sqrt 3 } \right) = \left( 3 \right)\left( 3 \right) = 9, \) a rational number.
        \( \therefore \sqrt 3 \) is a RF of \( 3\sqrt 3 \)
(2) \( \left( {\sqrt 3 + \sqrt 2 } \right)\left( {\sqrt 3 - \sqrt 2 } \right) = \left( {\sqrt 3 } \right)^2 - \left( {\sqrt 2 } \right)^2 = 3 - 2 = 1 \) a rational number.
     \( \therefore \sqrt 3 - \sqrt 2 \) is a RF of \( \sqrt 3 + \sqrt 2 \) and \( \sqrt 3 + \sqrt 2 \) is a RF of \( \sqrt 3 - \sqrt 2 \)
(3) \( \sqrt[n]{a} \) is a RF of \( \sqrt[n]{{a^{n - 1} }} \) and vice versa.
(4) \( \sqrt[n]{{a^m }} \) is a RF of \( \sqrt[n]{{a^{n - m} }} \) and vice versa.
(5) \( \sqrt a + \sqrt b \) is a RF of  \( \sqrt a + \sqrt b \) and vice versa.
(6) \( \sqrt[3]{a} + \sqrt[3]{b} \) is a RF of \( a^{\frac{2} {3}} - a^{\frac{1} {3}} .b^{\frac{1} {3}} + b^{\frac{2} {3}} \) and vice versa.
(7) \( \sqrt[3]{a} - \sqrt[3]{b} \) is a RF of \( a^{\frac{2} {3}} + a^{\frac{1} {3}} .b^{\frac{1} {3}} + b^{\frac{2} {3}} \) and vice versa.

20. Decimal representation of Rational Numbers : 
i)   The decimal representation of a rational number is converting a rational number into a decimal number that has the same mathematical value as the rational number.
ii)  A rational number can be represented as a decimal number with the help of the long division method.
iii) A rational number can have two types of decimal representations (expansion)
    * Terminating
    * Non-terminating but repeating
NOTE: Any decimal representation that is non-terminating and non-reccuring, will be an irrational number.
    Ex : 2.314231423142..... is a rational number.

21. Theorem :  Let ‘a’ be a terminating decimal then ‘a’ can be expressed as \( \frac{p} {q} \)(where \( q \ne 0 \) ) , where p and q are co-primes and the prime factorization of q is of the form 2^m.5ⁿ

Ex : (i) \( \frac{1} {4} = \frac{1} {{2^2 }} = \frac{{1 \times 5^2 }} {{2^2 \times 5^2 }} = \frac{{25}} {{100}} = 0.25 \)
(ii) \( \frac{7} {{25}} = \frac{7} {{5^2 }} = \frac{{7 \times 2^2 }} {{5^2 \times 2^2 }} = \frac{{28}} {{100}} = 0.28 \)
(iii) \( \frac{{23}} {{125}} = \frac{{23}} {{5^3 }} = \frac{{23 \times 2^3 }} {{5^3 \times 2^3 }} = \frac{{184}} {{1000}} = 0.184 \)
(iv) \( \frac{{147}} {{50}} = \frac{{147}} {{2 \times 5^2 }} = \frac{{147 \times 2}} {{2^2 \times 5^2 }} = \frac{{294}} {{100}} = 2.94 \)

22. Theorem : If \( \frac{p} {q} \) is a rational number where q is of the form \( 2^m .5^m \left( {m \in w} \right) \), then \( \frac{p} {q} \) has a terminating decimal expansion.
Examples :
i) \( 0.375 = \frac{{375}} {{1000}} = \frac{{375}} {{10^3 }} \)
ii) \( 0.104 = \frac{{104}} {{1000}} = \frac{{104}} {{10^3 }} \)

iii) \( 0.0875 = \frac{{875}} {{1000}} = \frac{{875}} {{10^4 }} \)

23. Theorem : If \( \frac{p} {q} \) is a rational number and q is not of the form \( 2^m .5^n \) (where \( m,n \in w \)) , then \( \frac{p} {q} \) has a non-terminating repeating decimal expansion.