NATURAL NUMBERS :
The counting numbers 1, 2, 3, 4, 5, 6…………… are called the natural numbers and are denoted by N . As you can recall that a child at his very young stage try to count the things. Eg. Chocolates, toys, sweets etc. He just know the numbers viz. ,1, 2, 3, 4, 5………. And has no idea about zero. Here we are discussing the numbers which are represented in international symbols. So N= {1, 2, 3, 4, 5………}
PROPERTIES OF NATURAL NUMBERS
1. Successor : The next natural number just after any natural number n is called its successor ‘n+ ‘
Where n+ = n + 1 for example the successor of 2 is 3, successor of 6 is 7 etc.
2. Closure law : For any two natural numbers a and b
( a + b ) € N
And ( a x b ) € N
e.g. 3 + 4 = 7 € N and 3 x 4 = 12 € N
3. Commutative law : For any two natural numbers a and b
( a + b ) = ( b + a )
And (a x b ) = (b x a )
5 + 6 = 6 + 5 = 11
5 x 6 = 6 x 5 = 30
4.Associative law : For any three natural numbers
( a + b ) + c = a +( b + c )
( a x b ) x c = a x (b x c )
(7 + 8 ) + 9 = 7 + (8 + 9 ) = 24
(7 x 8 ) x 9 = 7 x (8 x 9 ) 504
5. Multiplicative Identity : 1 is the multiplicative identity of every natural number as
4 x1 =4
5 x 1 =5
10 x 1 =10
17 x 1 =17
6. Cancellation law : For any three natural numbers a,b,c
A + b = c + b ======= a = c
And a x b = c x b ======= a = c
7. Distributive Law : For any three natural numbers a, b, c
(b + c) x a = a x b + a x c
(b x c ) + a = ( a + b ) x (a + c )
e. g., 3 x ( 4 + 5 ) = 3 x 4 + 3 x 5 = 27
Even Numbers : All the natural numbers which are divisible by 2 are known as even numbers e.g., 2, 4, 6, 8, ………..
Odd Numbers : All the numbers which are not divisible by 2 are known as ‘ Odd numbers’ e.g. , 1, 3, 5, 7, 9, ……….
IMPORTANT NOTE
EVEN + EVEN = EVEN
EVEN - EVEN = EVEN
EVEN X EVEN= EVEN
EVEN ÷ EVEN = EVEN OR ODD
ODD + ODD = EVEN
ODD – ODD = EVEN
ODD X ODD = ODD
ODD ÷ ODD = ODD
EVEN + ODD = ODD
EVEN – ODD = ODD
EVEN X ODD =EVEN
EVEN ÷ ODD = EVEN
ODD + EVEN = ODD
ODD – EVEN = ODD
ODD X EVEN = EVEN
ODD ÷ EVEN = (NEVER DIVISIBLE )
(EVEN )EVEN/ODD =EVEN
(ODD ) ODD/EVEN =ODD
Prime Numbers : Except 1 each natural number which is divisible by only 1 and itself is called as prime number e. g. , 2, 3, 5, 7, 11, 13, 17, 19,23, 31………….etc.
1. There are total 25 prime numbers upto 100.
2. There are total 46 prime numbers upto 200.
3. 2 is the only even prime number and the least prime number.
4. 1 is neither prime nor composite number.
5. There are infinite prime numbers.
6. A list of all the prime numbers upto 100 is given below
Table of Prime Numbers ( 1 -100 ) :
2 11 23 31 41 53 61 71 83 97
3 13 29 37 43 59 67 73 89
5 17 47 79
7 19
How to test whether a number is prime or not : To test a number n take the square root of n and consider as it is , if it is a natural number otherwise just increase the square root of it to the next natural number. Then divide the given number by all the prime numbers below the square root obtained. If the number is divisible by any of these prime numbers then it is not a prime number else it is a prime number.
EXAMPLE : Check that whether 241 is prime number.
SOLUTION : When we take the square root of 241 it is approximate 15, so we consider it 16 . Now we divide 241 by all the prime numbers below 16 viz. , 2, 3, 5, 7, 11, 13 .
Since 241 in not divisible by any one of the prime numbers below 16 . So it is a prime number.
NOTE : ANY DIGIT IF IT IS WRITTEN CONTINUOUSLY 3 TIMES, 6 TIMES, 9TIMES…….ETC.THEN IT IS DIVISIBLE BY 3 E.G. , 555555, 777777, 222222222, 888, 222 ETC.
CO-PRIME NUMBER : Two natural numbers are called co-prime (or relatively prime ) numbers it they have no common factor other than 1 or in other words. The highest common factor I.e. H.C.F. between co-prime numbers is 1. e.g. , (15, 16), (14, 25), ( 8, 9 ), (13, 15) etc.
NOTE : IT IS NOT NECESSARY THAT THE NUMBERS INVOLVED IN THE PAIR OF CO-PRIMES WILL BE PRIME EVEN THEY CAN BE COMPOSITE NUMBERS AS SEEN IN THE ABOVE EXAMPLES.
COMPOSITE NUMBERS : A number other than one which is not a prime number is called a composite number. It means it is divisible by some other number(s) other than 1 and the number itself. E.g. 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26………..
Every natural number except 1 is either prime or composite.
TRICHOTOMY LAW : If there are any two natural numbers a and b then there exists one and only one relation necessarily is
(i) ( a >b )
(ii) ( a = b )
(iii) (a < b)
WHOLE NUMBERS : The extended set of natural numbers in which ‘0’is also included is called as the set of whole numbers and is denoted by W = { 0, 1, 2, 3, 4,………}
‘0’ (zero ) is an even number
CONSECUTIVE NUMBERS : A series of numbers in which the next number is 1 more than the previous number or the predecessor number is 1 less than the successor or just they can be differed by 1 e.g., 10, 11, 12, or 17, 18, 19 or 717, 718 , 719 etc.
PERFECT NUMBER : When the sum of all the factors (including 1 but excluding the number itself ) of the given number is the same number then this number is called as Perfect Number. For example 6, 28, 496, 8128…….etc.
So far only 27 perfect number are known.
As the factors of 28 are 1, 2, 4, 7, 14, 28
Now , 1 + 2 + 4 + 7 + 14 =28
Hence 28 is a perfect Number.
Triangular Number : A triangular number is obtained by adding the previous number to the nth position in the sequence of triangular number is 1. The sequence of triangular number is given as follows
1, 3, 6, 10, 15, 21,28, 36, 45, 55, 66, 78……..etc.
INTEGERS : The extended set of whole numbers in which negative integers are also included is known as the set of integers and is denoted by
Z or I = { ……. -4, -3, -2, -1, 0, 1,2, 3, 4, 5, ………}
(a ) POSITIVE INTEGERS : The set of integers { 1, 2, 3, ……….}is known as positive integers.
(b) NEGATIVE INTEGERS : The set of integers { -1, -2, -3, ……..} is known as negative integers.
( c ) NON-NEGATIVE INTEGERS : The set of integers { 0, 1, 2, 3, ………} is called as non-negative integers.
(d ) NON-POSITIVE INTEGERS : The set of integers { 0, -1, -2, -3, ……….} is called as non-positive integers.
‘0’ is neither positive nor negative integer.
REPRESENTATION OF THE INTEGERS ON A NUMBER LINE
All the integers ( whole number, natural number etc. ) can be shown on this number line, where every integer is represented by some point on the line.
It can be said that :
( a )There is no any largest or smallest integer.
( b) Every integer has a predecessor and a successor.
( c ) An integer is smaller than all those integers which are on the right side of it and is greater than all those integers which are on the left side of it on the number line.
e.g., -4 > -5 , 3> 1 , 10 > -4 etc. or -6 < 0 , -2 < 1 etc.
EXAMPLE : Arrange the following integers in ascending order -3, -7, 8, 5, 0, 3, 17, -23.
SOLUTION : -23, -7, -3, 0, 3, 5, 8, 17
Properties of integers
1. Closure law is followed by all the integers
2. Commutative law and Associative law is not followed by all the integers for the subtraction.
E.g. 4 - 6 ≠ 6 - 4
And (4 - 6 ) - 2 ≠ 4 – (6 - 2 )
But it is valid for the addition and multiplication as
4 + 6 = 6 + 4
Or ( -3) + ( - 2 ) = ( -2 ) + ( - 3 )
And - 3 x - 2 = -2 x -3
And ( -2 ) + [ (- 3 ) + ( - 7 ) ] = [ ( - 2 ) + ( - 3 ) ] + ( - 7 )
3. Additive identity of all the integers is zero ( 0 ) and Multiplicative identity of all the integers is 1. E. g. , - 3 + 0 = - 3 8 + 0 = 8
-5 x 1 = -5 7 x 1 = 7 etc.
4. Additive inverse fo and integer a is -a .
e.g. the additive inverse of 7, 8, 9, -3, -5, etc. are -7, -8, -9, 3, 5. Respectively.
5. Distributive law of multiplication over addition or subtraction
( b ±c ) x a = a x b ± a x c
For example : 3 x ( 4± 6 ) = 3 x 4 ± 3x 6
NOTE : 1. DIVISION BY ZERO IS NOT DEFINED IN MATHEMATICS.
2. DIVISION BY 1 IS ACTUALLY UNIFICATION ( NOT DIVISION ) SO IT IS AN IMPROPER DIVISOR.
SOME IMPORTANT RULES REGARDING THE SIGN CONVENTION IN MATHEMATICAL OPERATIONS :
(I ) ( a ) + ( b ) = + ( a + b )
(ii ) (a ) + ( b ) = b – a
( iii ) ( a) + ( - b ) = a – b
(iv ) ( -a ) + (- b ) = - ( a + b )
i.e. ( + ) + (+ ) = ( +) ( - ) + ( - ) = (- )
( - ) + (+ ) = ( - ) if the numerical value of + is greater
( - ) + ( +) = (+ ) if the numerical value of – is greater
For example ( 4 ) + ( 7 ) = 11,
(-3 ) + (8) = 5
(-5 ) + ( -3 ) = -8
( -8) + (3) = -5
SOME IMPORTANT RULES REGARDING THE SIGN CONVENTION IN MATHEMATICAL OPERATIONS :
(I ) ( a ) x ( b ) = + ( ab )
(ii ) (-a ) x ( b ) = -ab
( iii ) ( a) + ( - b ) = -ab
(iv ) ( -a ) + (- b ) = + ( ab )
i.e. ( + ) x (+ ) = ( +)
( - ) x (+ ) = ( - )
( - ) x ( +) = (- )
( - ) x ( - ) = (- )
Numerical Expression : Collection of numbers connected by one or more operations of addition, subtraction, multiplication and division is called a numerical expression. It can also involve some brackets.
e.g. , 7 + 18 ÷ 3 ÷ 1 x 6
84 – 6 ÷ 2 - 3 x ( - 7 )
RULE OF SIMPLIFICATION OR CALCULATION
The order in which various mathematical operations must be done can be remembered with the word
‘ BODMAS ‘
WHERE B BRACKETS
O OFF
D DIVISION
M MULTIPLICATION
A ADDITION
S SUBTRACTION
So first of all we solve the inner most brackets moving outwards. Then we perform ‘of ‘ which means multiplication then, Division, Addition and Subtraction.
Addition and Subtraction can be done together or separately as required.
Between any two brackets if there is no any sign of ‘+’ or ‘-‘ it means we have to do
multiplication eg.
( 7 ) x ( 2 ) = 7 x 2 = 14
[ 3 ( 5 ) + 7 ] = 15 + 7 = 22
BRACKETS : They are used for the grouping of things or entities. The various kind of brackets are :
( I ) ‘ – ‘ is known as line ( or bar ) bracket or vinculum
(ii) ( ) is known as parenthesis or common brackets
(iii ) { } is known as curly bracket or brace.
(iv ) [ ] is known as rectangular ( or big ) bracket.
The order of eliminating brackets is
(I ) line bracket
(ii ) common bracket
(iii ) curly bracket
(iv ) rectangular bracket
ABSOLUTE VALUE OF AN INTEGER OR MODULUS :
The absolute value of an integer is its numerical value irrespective of its sign ( or nature )
The absolute value of an integer x is written as | x | and is defined as
|x | = {x if x ≥ 0
= { -x if x < 0
|-7 | = 7 , |-4 |= 4 | 3 |= 3 etc.
It means any integer whether it is whether it is positive or negative if it is operated upon modulus, it always gives a positive integer.
Max. { x, -x } = |x |
Min. { x, -x } = |x |
Sqrt of x2= | x |
PROPERTIES OF A MODULUS OR MOD
1. |a |=| -a |
2. |ab | =|a | | b |
3. |a/b |= |a |/|b |
4. |a + b |≤ |a | + | b| ( The sigh of equality holds only when the sign of a and b are same.) \
5. If |a |≤ k ===== -k ≤ a ≤ k
6. If |a – b | ≤ k ==== -k ≤ a – b ≤ k ===== b- k ≤ a ≤ b + k
EXAMPLE Solution set of the equation |x -2 | = 5 is :
(a) {3, -7 } ( b ) {-3 , 7 } (c ) {3, 6 } (d ) none of these
SOLUTION : | x – 2 |= 5 ===== x -2 === 5 or x -2 ==== (-5)
X = 7 or X = -3
Hence x = {-3, 7} (b ) is correct option.
NOTE { } CONSISTS ONLY THE MENTIONED VALUE PARTICULARLY,
( ) DOES NOT INCLUDE THE EXTREME VALUES
[ ] INCLUDES THE VALUES OF EXTREMITY
( ] INCLUDES ONLY THE HIGHEST VALUE OF EXTREME BUT NOT THE LOWEST EXTREME VALUE
[ )INCLUDES ONLY LOWEST EXTREME VALUE BUT NOT INCLUDES THE HIGHEST EXTREME VALUE.
FACTORS AND MULTIPLES
WHEN TWO OR MORE INTEGERS ARE MULTIPLIED GOTETHER THEN THE RESULTANT VALUE IS CALLED THEIR PRODUCT :
3 X 4 =12 , 2 X 7 X 5 = 70, 3 X 5 X 11 = 165
Where 12, 70 and 165 are called the products.
But we see that 12 = 4 x 3 it means 4 and 3 are the factors of 12.
And 70 = 2 x 5 x 7 where 2, 5, 7 are known as factors of 70.
Again 12 is called as the multiple of 3 or multiple of 4.
Similarly 70 is called as the multiple or 2 or 5 or 7 or 10 or 35 or 14.
Thus a number which divides a given number exactly is called factor ( or divisor ) of that given number and the given number is called a multiple of that number.
Now , 15 is exactly divisible by 1, 3, 5, 15, so 1, 3, 5, 15 are called as the factors of 15 while 15 is called as the multiple of these factors. Where 1, 15 are improper factors and 3, 5 are called Proper factors of 15.
Therefore 1 and itself (the number ) are called the improper factor of the given number.
So the factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 35 = 1, 5, 7, 35
Similarly the multiples of 2 are 2, 4 , 6, 8, 10, 12,
Multiples of 7 are 7, 14, 21, 28, 35…….
Multiples of 10 are 10, 20 30 40…….
1 is a factor of every number.
Every number is a factor of itself.
Every number, except 1 has atleast two factors viz. 1 and itself.
Every factor of a number is less than or equal to that number.
Every multiple of a number is greater than or equal to itself.
Every number has infinite number of its multiples.
Every number is a multiple of itself.
PRIME FACTORISATION : If a number is expressed as the product of prime numbers ( factors ) then the factorisaton of the number is called its prime factorization.
For example
(i) 72 = 2 x2 x 2 x 3 x 3
(ii) 420 = 2 x 2 x 3 x 5 x 7
