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 (108) | Crores (107) | Ten Lacs (Milions) (106) | Lacs (105) | Ten Thousands (104) | Thousands (103) | Hundreds (102) | Tens (101) | Units (100) |
| 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 108 = 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. 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 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.g. , 2, 4, 6, 8, 10, etc.
5. Odd Numbers : A number not divisible by 2 is called on 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 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 5n : Put n zeros to the right of the multiplicand and divide the number so formed by 2n
- Ex. 975436 × 625 = 975436 × 54 = 9754360000/16 = 609647500
BASIC FORMULAE
1 (a + b)2 = a2 + b2 + 2ab
2 (a - b)2 = a2 + b2 – 2ab
3 (a + b)2 – (a – b)2 = 4ab
4 (a + b)2 + (a – b)2 = 2 (a2 + b2)
5 (a2 – b2) = (a + b) (a – b)
6 (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
7 (a3 + b3) = (a + b) (a2 – ab + b2)
8 (a3 – b3) = (a – b) (a2 + ab + b2)
9 (a3 + b3 + c3 – 3abc) = (a + b + c) (a2 + b2 + c2 – ab – bc – ca )
10 If a + b + c = 0, then a3 + b3 + c3 = 3abc.
DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM
If we divide a given number by another number, then :
Dividend = (Divisor × Quotient) + Remainder
(i) (xn – an) is divisible by (x – a) for all values of n
(ii) (x2 – a2) 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.
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 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) (12 + 22 + 32 +… + n2 = n(n+1)(2n+1)/6
(iii) (13 + 23 + 33 +… + n3) = n2(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, ar2, ar3,……
In this G.P. Tn = arn = 1.
Sum of the n terms, Sn = a(1-rn) / (1-r)
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