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1. MADE BY :-
SUNDEEP MALIK.
Ⅷ A.

2.  Each of 1,2,3,4,etc is a natural number.
 The first and the smallest natural number is 1 whereas
the last and the largest natural number cannot be
obtained.
 Consecutive natural numbers differ by 1.

3.  Each of 0,1,2,3,4,etc is a whole number.
 The first and the smallest whole number is 0
whereas the last and the largest whole number
cannot be obtained.
 Consecutive whole numbers differ by 1.
 Except 0 every whole number is a natural number.

4.  Each of 0,1,2,-1,-2,-3, etc is an integer.
 Every integer is either:-
a) A negative of a natural number , such as -1,
-2,-3,-4, etc.
b) 0.
c) A natural number i.e. 1,2,3,4, etc.
 The smallest and the largest integers cannot be
obtained.
 Consecutive integers differ by 1.

5.  It is a number which can be expressed as ab where a and b
both are integers and b is not equal to 0.
 Every fraction such as 3/8,5/7, etc is a rational number.
 Since, 8=8/1, -3=-3/1 etc, therefore each integer is a rational
number. For the same reason, each natural number and each
whole number is a rational number.
 0 can be written as 0/5,0/-3,0/9, etc, therefore 0 is also a
rational number .
 Every decimal number can be expressed as a fraction; so it is
also a rational number. e.g. 0.7= 7/10, 2.5=25/10=5/2.etc.

6.  The numbers which cannot be expressed as a/b;
where a∈ I, b ∈ I, and b is not equal to 0; is called
an irrational number.
 Each of √2,3-√7,√5, etc., is an irrational number.
 Each non-terminating and non-recurring number is
an irrational number.

7.  Real numbers and irrational numbers, taken
together, are called real numbers.
 Every real number is either rational or irrational.

10. 9) TWIN PRIME:- Prime numbers differing by two are said to
be twin prime numbers. E.g.:- 5 and 7, 11 and 13, etc.
10)PRIME TRIPLET:- The set of three consecutive prime
numbers with a difference of 2 is called the prime triplet. Note
that there is only 1 set {3,5,7} of the prime triplet.
11)CO-PRIMES:- The pair of numbers, which are not divisible
by a common number other than 1, are called co-primes. E.g.:-
10 and 21 are co-primes, since these numbers cannot be divided
by the same number other than 1. Similarly, 7 and 16, 8 and 25.

11. 1) DIVISION BY 2,4 AND 8 :-
 A number is divisible by 2, if its last digit is divisible by 2.
E.g.:- 78, 96,192 etc.
 A number is divisible by 4, if the number formed by its last 2
digits is divisible by 4. E.g.:- 172,392,500,29320.
 A number is divisible by 8, if the number formed by its last
three digits is divisible by 8. E.g.:- 33176,436000 etc.

12. 2) DIVISION BY 3 AND 9:-
 A number is divisible by 3, if the sum of its digits is divisible
by 3. E.g.:- 27,192,54924 etc.
 A number is divisible by 9, if the sum of its digits is divisible
by 9. E.g.:- 198, 27,99 etc.

13. 3) DIVISION BY 6:-
 If a number is divisible by 3 and 2, it is divisible by 6.
4) DIVISION BY 5 AND 10:-
 A number is divisible by 5, if its last digit is 5 or 0.
E.g.:- 345, 240 etc.
 A number is divisible by 10, if its last digit is 0. E.g.:-
310,4000 etc.

14. 5) DIVISION BY 11 :-
 Find the sum of the digits in the even places of the given
number and the sum of the digits in its odd places. If the
difference between the two sums is 0 or a multiple of 11, then
the given number is divisible by 11.
 E.g.:- Consider the number 72512. The sum of its digits at
even places = 2+1=3 and the sum of its digits at its digits at odd
places = 7+5+2=14. The difference between the 2 sums = 14-
3= 11; which is a multiple of 11. Therefore, 72512 is divided by
11. Similarly, 957, 1496, 68772 etc is divisible by 11.

15. 1) FIRST PROPERTY : CLOSURE :-
If a number a and b belong to a certain type of numbers such
that :-
 a+b belongs to the same type of numbers, the numbers are said
to be closed for addition.
 a-b belongs to the same type of numbers, the numbers are said
to be closed for subtraction.
 a×b belongs to the same type of numbers, the numbers are said
to be closed for multiplication.
 a÷b belongs to the same type of numbers, the numbers are said
to be closed for division.

16. Now we shall discuss the closure property, as stated in the last slide, for
all four fundamental operations on different types of numbers :-
 Natural numbers (N) :-
 closed for addition and subtraction.
 not closed for subtraction and division.
 Whole numbers (W) :-
 are closed for addition and multiplication.
 are not closed for subtraction and division.
 Integers (I) :-
 closed for addition, subtraction and multiplication but not for division.
 Rational numbers (Q) :-
 closed for addition, subtraction and multiplication but not closed for
division.

17. 2) SECOND PROPERTY :
COMMUTATIVITY :-
E.g.:-
 a+b = b+a, i.e. 3+5 = 5+3.
 a-b = b-a, i.e. 3-5 ≠ 5-3.
 a×b = b×a, i.e. 3×5 = 5×3.
 a÷b = b÷a, i.e. 3÷5 ≠ 5÷3.
For these facts, we say :-
 addition is commutative.
 subtraction is not commutative.
 multiplication is commutative.
 division is not commutative.

18. 3) THIRD PROPERTY :- ASSOCIATIVITY :-
E.g.:-
 a+(b+c) = (a+b)+c.
 a-(b-c) ≠ (a-b)-c.
 a×(b×c) = (a×b)×c.
 a÷(b÷c) ≠ (a÷b)÷c.
For these facts, we say :-
 addition is associative.
 subtraction is not associative.
 multiplication is associative.
 division is not associative.

19. 4) FOURTH PROPERTY :-
DISTRIBUTIVITY :-
For any three numbers a, b and c, the following
results are always true :-
 a×(b+c) = a×b + a×c. Multiplication is distributive
over addition.
 a×(b-c) = a×b - a×c. Multiplication is distributive
over subtraction.

20. 5) FIFTH PROPERTY :- IDENTITY :-
 The role of 0 :-
If 0 is added to any number or any number is added to 0, the
value of the number remains unchanged. E.g.:- 5+0 = 5, 124+0
= 124.
For this property of 0, it is called the identity for addition.
 The role of 1:-
If one is multiplied with any number or any number is
multiplied with 1, the value of the number remains unchanged.
E.g.:- a×1=1, 1×a=1.
For this property of 1, it is called the identity for multiplication.