Number System
Types of Numbers:
I. Natural Numbers:
Counting numbers are called natural numbers
II. 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.
III. Integers :
Allnaturalnumbers,
0and negatives of counting
numbers i.e., together form the set of integers.
(i) Positive Integers:
is the set of all positive
integers.
(ii) Negative Integers: is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers:
0 is neither positive nor negative.
So,represents
the set of non-negative
integers,
while represents the set of non-positive
integers.
IV. Even Numbers:
A number divisible by 2 is called an even number, e.g.,, etc.
V. Odd Numbers:
A number not divisible by 2 is called an odd
number. e.g., etc.
VI. 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 up to 100 are
Prime numbers 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 . Test whether p is divisible by
any prime number less than k. If yes, then p is not prime.
Otherwise, p is prime. Example:
We have to find whether 191 is a prime number or not. Now, .
Prime numbers less than 14 are
191 is not divisible by any of them. So, 191 is a prime number
VII. Composite Numbers:
Numbers greater than
1 which are not prime, are known as composite numbers, e.g.,
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.
Theorem (Euclid’s Division Lemma):
For a pair of given positive integers ‘a’
and ‘b’, there exist unique integers ‘q’ and ‘r’ such that
Explanation:
Thus, for any pair of two positive integers
a and b; the relation
Divisibility Rules:
Divisible by: |
If: |
Examples: |
2 |
The last digit is even (0,2,4,6,8) |
128 is 129 is not |
3 |
The sum of the digits is divisible by |
381 217 |
4 |
The last 2 digits are divisible by 4 |
1312 is (12÷4=3) 7019 is not |
5 |
The last digit is 0 or 5 |
175 is 809 is not |
6 |
The number is divisible by both 2 and 3 |
114 (it is even, and 1+1+4=6 and 6÷3 308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No |
7 |
If you <![if !supportLists]>• <![if !supportLists]>• <![endif]>divisible (Note: you can apply this rule to |
672 905 |
8 |
The last three digits are divisible |
109816 (816÷8=102) Yes 216302 (302÷8=37 3/4) No |
9 |
The sum of the digits is divisible by (Note: you can apply this rule to that answer again if you want) |
1629 2013 |
10 |
The number ends in 0 |
220 is 221 is not |
11 |
If you <![if !supportLists]>• <![if !supportLists]>• <![endif]>divisible |
1364 ((3+4) – (1+6) = 0) Yes 3729 ((7+9) – (3+2) = 11) 25176 ((5+7) – (2+1+6) = 3) |
12 |
The number is divisible by both 3 and |
648 (By 3? 6+4+8=18 and 18÷3=6 Yes. By 4? 48÷4=12 Yes) Yes 524 (By 3? 5+2+4=11, 11÷3= 3 2/3No. Don’t need to check by 4.) No |
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
The nth term of
this A.P. is given by .
The sum
of n terms of this A.P.
Some Important
Results
an= a1+ d
( n – 1 ) .
2. Geometrica 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
In this G.P.
sum of n terms,
when
bn= b1 q n –1 .