Published: 05 Aug 2019

**1. Numeral :** In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called digits to represent any numbers.

A group of digits, denoting a number is called a numeral.

*We represent a number, say 689745132 as shown below :*

We read it as : ‘Sixty- eight crores, ninety-seven lacs, forty-five thousand, one hundred and thirty-two’.

Ten Crores (10^{8}) |
Crores (10^{7}) |
Ten Lacs (Milions) (10^{6}) |
Lacs (10^{5}) |
Ten Thousands (10^{4}) |
Thousands (10^{3}) |
Hundreds (10^{2}) |
Tens (10^{1}) |
Units (10^{0}) |

6 | 8 | 9 | 7 | 4 | 5 | 1 | 3 | 2 |

**2. ** **Place Value or Local Value of a Digit in a Numeral :**

In the above numeral :

Place value of 2 is (2 x 1) = 2;

Place value of 3 is (3 x 10) = 30;

Place value of 1 is (1 x 100) = 100 and so on.

Place value of 6 is 6 x 10^{8} = 600000000.

**3.** **Face VALUE** : The ** face value **of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above numeral is the face value of 2 is 2; the face value of 3 is and so on.

**4. ** **TYPES OF NUMBERS**

**Natural numbers**: Counting numbers 1, 2, 3, 4, 5, ………… are called.*natural numbers***Whole Numbers**: All counting numbers together with zero form the set of.. Thus,*whole numbers*

(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 from 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, 3,……} represents the set of non-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.*

**5. Odd Numbers : **A number not divisible by 2 is called on odd number*. e.*

**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 number 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 number Greater than 100 :** Let p 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 > √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 > √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 only even number which is prime.

(iii) There are 25 prime numbers between 1 and 100

**8. 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.

**TESTS OF DIVISIBILITY**

**1. Divisibility By 2 :** A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8

**Ex**. 84932 is divisible by 2, while 65935 is not.

**2. Divisibility By 3 :** A number is divisible by 3, if the sum of its digits =is divisible by 3.

**Ex**. 592428 is divisible by 3 since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.

But, 864329 is not divisible by 3, since sum of its digits = (8 + 6 + 4 + 3 + 2 + 9) = 32, which is divisible by 3.

**MULTIPLICATION BY SHORT CUT METHODS**

**Multiplication By Distributive Law :**

- a × (b + c) = a × b + a × c (ii) a × (b – c) = a × c.

**Ex**. (i) 567958 × 99999 = 567958 × (100000 – 1)

= 567958 × 100000 – 567958 × 1

= (56795800000 – 567958) = 56795232042.

- 978 × 184 + 978 × 816 = 978 × 1000 = 978000.
**Multiplication of a Number By 5**Put n zeros to the right of the multiplicand and divide the number so formed by 2^{n}:^{n}- Ex. 975436 × 625 = 975436 × 5
^{4}= 9754360000/16 = 609647500

*BASIC FORMULAE *

*1 (a + b) ^{2} = a^{2} + b^{2} + 2ab*

*2 (a - b) ^{2} = a^{2 }+ b^{2} – 2ab*

*3 (a + b) ^{2} – (a – b)^{2 } = 4ab*

*4 (a + b) ^{2} + (a – b)^{2} = 2 (a^{2} + b^{2})*

*5 (a ^{2} – b^{2}) = (a + b) (a – b)*

*6 (a + b + c) ^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + bc + ca)*

*7 (a ^{3} + b^{3}) = (a + b) (a^{2} – ab + b^{2}) *

*8 (a ^{3 }– b^{3}) = (a – b) (a^{2} + ab + b^{2})*

*9 (a ^{3} + b^{3} + c^{3} – 3abc) = (a + b + c) (a^{2} + b^{2} + c^{2} – ab – bc – ca )*

*10 If a + b + c = 0, then a ^{3} + b^{3} + c^{3} = 3abc.*

**DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM**

If we divide a given number by another number, then :

**Dividend = (Divisor × Quotient) + Remainder**

** (i) **(x

** (ii)** (x

* (iii)* (x

**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 + b), (a + 2d),(a + 3d),……

**The** **nth term of this A.P. is given by T _{n} = a (n – 1) d.**

**The sum of n terms of this A.P.**

**S _{n} = n/2**

**SOME IMPORTANT RESULTS :**

(i) (1 + 2 + 3 + … + n) = n(n+1)/2

(ii) (1^{2} + 2^{2} + 3^{2} +… + n^{2} = n(n+1)(2n+1)/6

(iii) (1^{3} + 2^{3} + 3^{3} +… + n^{3}) = n^{2}(n+1)^{2}/4

**Geometrical progression (G.P.)** : A **geometric progression**, also known as a **geometric** sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. 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 a geometrical progression.

The constant ratio is called the common ratio of the G.P.

G .P with first term a and common ratio r is :

*a, a, ar ^{2}, ar^{3},……*

In this G.P*. T_{n} = ar^{n} = 1.*

Sum of the n terms, **S _{n} = a(1-r^{n}) / (1-r)**

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