Sunday, 22 November 2015

Bank Jobs Quantitative Aptitude - Study Notes : NUMBER SYSTEM




A.    TYPES OF NUMBERS

1.     Natural Numbers :Counting numbers 1, 2, 3, 4, 5,..... are called natural  numbers.

2.     Whole Numbers :All counting numbers together with zero form the set of whole numbers. Thus,
(i)    0 is the only whole number which is not a natural number.
(ii)   Every natural number is a whole number.

3.     Integers : All natural numbers, 0 and negatives of counting numbers i.e.,
{…, - 3 , - 2 , - 1 ,  0, 1, 2, 3,…..} together form the set of integers.

(i)    Positive Integers : {1, 2, 3, 4, …..} is the set of all positive integers.
(ii)   Negative Integers : {- 1, - 2, - 3,…..} is the set of all negative integers. (iii)  Non-Positive  and Non-Negative Integers : 0 is neither positive nor negative. So, {0, 1, 2, 3,….} represents the set of non-negative integers, while {0, - 1 , - 2 , - 3 , …..} represents the set of non-positive integers.

4.     Even Numbers :A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, etc.

5.     Odd Numbers :A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

6.     Prime  Numbers :A number  greater than 1 is called a  prime number, if it has exactly two  factors, namely 1 and the number itself. Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,  53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Prime numbers Greater than 100 :Let be a given number greater than 100. To find out whether it is prime or not, we use the following method :
Find a whole number nearly greater than the square root of p. Let k >square root of p. Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.

e.g,,We have to find whether 191 is a prime number or not. Now, 14 >square root of 191.Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.
191 is not divisible by any of them. So, 191 is a prime number.

7.     Composite Numbers :Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12.
Note :
(i)    1 is neither prime nor composite.
(ii)   2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.

Co-primes :Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,

B.    MULTIPLICATION BY SHORT CUT METHODS

1.    Multiplication By Distributive Law :
(i) a* (b + c) = a * b + a * c                         (ii) a * (b-c) = a * b-a * c.

Ex.(i) 567958 x 99999 = 567958 x (100000  - 1)      = 567958 x 100000 - 567958 x 1
= (56795800000 - 567958) = 56795232042.

(ii) 978 x 184 + 978 x 816 = 978 x (184 + 816) = 978 x 1000 = 978000.

2.    Multiplication  of  a  Number  By 5n  :   Put  n zeros to the right  of the multiplicand  and  divide the number so formed by 2n
Ex. 975436 x 625 = 975436 x 54= 9754360000 =609647600

A.    BASIC FORMULAE

(i)   (a + b)2  = a2 + b2 + 2ab
(ii)   (a - b)2 = a2 + b2 - 2ab
(iii) (a + b)2 - (a - b)2 = 4ab
(iv) (a + b)2 + (a - b)2 = 2 (a2 + b2)
(v) (a2 - b2) = (a + b) (a - b)
(vi) (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
(vii)      (a3 + b3) = (a +b) (a2 - ab + b2)
(viii)     (a3 - b3) = (a - b) (a2 + ab + b2)
(ix) (a3 + b3 + c3 -3abc) = (a + b + c) (a2 + b2 + c2 - ab - bc - ca)
(x)  If a + b + c = 0, then a3 + b3 + c3 = 3abc.

B.    DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM
If we divide a given number by another number, then :
Dividend = (Divisor x Quotient) + Remainder

(i)       (xn- an ) is divisible by (x - a) for all values of n.
(ii)     (xn- an) is divisible by (x + a) for all even values of n.
(iii)   (xn + an) is divisible by (x + a) for all odd values of n.

C.    PROGRESSION - A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.

1.     Arithmetic Progression (A.P.) :If each term of a progression differs from its preceding term by a constant, then such a  progression  is  called  an  arithmetical  progression.  This  constant difference is called the common difference of the A.P.

An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),... The nth term of this A.P. is given by Tn =a (n - 1) d.

The sum of n terms of this A.P.

Sn  = n/2 [2a + (n - 1) d] = n/2          (first term + last term).

SOME IMPORTANT RESULTS :

(i)   (1 + 2 + 3 +…. + n) =n(n+1)/2
(ii) (l2 + 22 + 32 + ... + n2) = n (n+1)(2n+1)/6 (iii) (13 + 23 + 33 + ... + n3) =n2(n+1)2

2.     Geometrical Progression (G.P.) :A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression.

The constant ratio is called the common ratio of the G.P. A G.P. with first term a and common ratio r is : a, ar, ar2,
In this G.P. Tn = arn-1
Sum of the n terms, Sn= a(1-rn) / (1-r)

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Quantitative Aptitude - Notes for Latest Bank Jobs


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