**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 5****n****:**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.

**S**

**n**

**= 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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