Square And Square Roots, Cube and Cube Roots

Square And Square Roots, Cube and Cube Roots

Types of numbers :
i)  Natural numbers :  Conting numbers 1,2,3,4... are called natural numbers. Natural numbers are denoted by the letter N.
    N = {1,2,3....}
ii) Whole numbers : Natural numbers along with zero are called whole numbers. Whole numbers are denoted by the letter W.
    W = {0,1,2,3,4,....}
iii) Integers : Set of whole numbers along with the negative natural nubers are called integers. Integers are denoted by the letter I or Z.
    I (or) Z = {...-3, -2, -1, 0,1,2,3...}
iv) Rational numbers : All numbers of the form \( \frac{p} {q}\left( {p,q \in z,\,\,q \ne 0} \right) \)  p and q are coprimes are called rational numbers
    Ex :\( Q = \left\{ {\frac{p} {q}/q \ne 0,\,\,p,q \in z,\,\,p\,\,and\,\,\,q\,\,\,are\,\,co - primes\,} \right\} \)
    Note : a) \( \frac{5} {0},\frac{7} {0} \) are not rational numbers
    b) Zero is a rational numbers \( 0 = \frac{0} {1} = \frac{0} {2} = \frac{0} {{\sqrt 7 }} \) etc.
    c) Every integer is a rational number but every rational number need not be an Integer
    \( 2 = \frac{2} {1} \in Q,\,\,\,but\,\,\frac{3} {7} \ne \,\,\operatorname{int} eger \)
    d) Every rational number can be  expressed as a decimal :  The decimal         representation of a rational number is either teminating (or) non-terminating and repeating.
    Terninating decimal has factors of its denominator as 2 and 5 only 
    In terminating decimal the “DR” should be of the form 2^m or 5ⁿ or both 2^m x 5ⁿ
    Ex : \( \frac{3} {{100}} = \frac{3} {{2^2 \times 5^2 }} = 0.03 \)
    e) A rational number lying between two given rational numbers \( \frac{p} {q}\,\,and\,\,\frac{r} {s}\,\,is\,\,\frac{1} {2}\left( {\frac{p} {q} + \frac{r} {s}} \right) \). There exist infinite rational numbers between any two given rational numbers.
    f) The operatins of addition, subtraction multiplication are formed on rational numbers as follows
    Ex :  a)\( - \frac{4} {9} + \frac{{15}} {{12}} + \frac{{ - 7}} {{18}} = \frac{{ - 16 + 45 - 14}} {{36}} = \frac{{15}} {{36}} = \frac{5} {{12}} \)   b)\( - \frac{3} {4} - \left( { - \frac{2} {7}} \right) = \frac{3} {4} + \frac{2} {7} = \frac{{29}} {{28}} \)
            c)\( \frac{7} {9},\frac{{11}} {2} = \frac{{77}} {{18}} \)   
Irrational number - A number which cannot be expressed as a terminating decimal decimal (or) repeating decimal is called an irrational number it is represented by  Q* (or) Q¹
Examples for types of irrational numbers
Type I :  a) Clearly 0.010010001....... which non terminating are non repeating
Type II :  If ‘m’ is apositive integer, which is not a perfect square then \( \sqrt m \) is an irrational number
    Ex : \( \sqrt {2,} \,\,\sqrt {3,} \,\,\sqrt 5 ,\sqrt 7 ,\sqrt 8 ,.... \) if m is a positive integer which is not perfect cube. then \( \sqrt[3]{m} \) is irrational
    \( \sqrt[3]{2},\sqrt[3]{4},\sqrt[3]{{10}} \).....
Type III: \( \pi \) is an irrational number
     \( \pi = 3.\overline {142857} \)        
    ‘e’ is also an irrational number e = 2.718....
Note :  1), e are called transcendental numbers
    2) If a and b are any two irrational numbers
    3) If a and b are any two positive real numbers such that a x b is not a perfect square of rational number then \( \sqrt {ab} \) is an irrational number lying between a and b \( \sqrt {ab} = \sqrt {12 \times 3} = \sqrt {36} = 6 \)  \( \therefore \sqrt {ab} \)is not an irrational number
     \( \sqrt {ab} = \sqrt {7 \times 11} = \sqrt {77} \) is not lperfect square   \( \therefore \sqrt {77} \) is an irrational number
Even and odd numbers  The numbers which are exactly divisible by 2 or the last digit  i.e units place should have 0, 2,4,6,8,.... etc are called even numbers. 
    Those natural numbers which are not divisible by 2 are called odd numbers
    Ex : 1,3,5,7.........
Prime numbers and composite numbers : A natural number which has only two factors i.e 1 and itself is called a prime number
    Ex : 2(even prime), 3,5,7,11,13,17,19,......
    The numbers which have more than two factors are called composite number 
    Ex : 4,6,8, 9, 10, 12,.....
Twin Primes :  Prime numbers differ by ‘2’ are called twin primes.
    Ex : (5,7), (3,5), (17,19) (29,31), etc..
CO - Primes :  Every pair of two natural numbers having no. common factor other than ‘1’ is called a pair of co-primes 
    Ex :  set of factors of 16 =- {1,2,4,8,16}
             set of  factors of 15 = {1,3,5,15}
     Clearly ‘1’ is common factors. 
      are co-primes
    we write it as (15,16)= 1
Perfect number : A number in which sum of all factors is equal to twice the number is called perfect number
Ex : 1) factors of 6 are 1,2,3,6
          1+2+3+6+=12 = 2 
      2) factors of 28 = 1,2,4,7,14,28
          1+2+4+7+714+28 = 56  = 2 x 28
Note :  Facotrs of 15 are 1,3,5,15
    1+3+5+15+= 24  2 x 15       \( \therefore 15 \) is not perfect number.