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
3

381
(3+8+1=12, and 12÷3 = 4) Yes

217
(2+1+7=10, and 10÷3 = 3 1/3) No

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
= 2) Yes

308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No

7

If you
double the last digit and subtract it from the rest of the number and the
answer is:

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by 7

(Note: you can apply this rule to
that answer again if you want)

672
(Double 2 is 4, 67-4=63, and 63÷7=9) Yes

905
(Double 5 is 10, 90-10=80, and 80÷7=11 3/7) No

8

The last three digits are divisible
by 8

109816 (816÷8=102) Yes

216302 (302÷8=37 3/4) No

9

The sum of the digits is divisible by
9

(Note: you can apply this rule to that answer again if you want)

1629
(1+6+2+9=18, and again, 1+8=9) Yes

2013
(2+0+1+3=6) No

10

The number ends in 0

220 is

221 is not

11

If you
sum every second digit and then subtract all other digits and the answer is:

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<![endif]>0, or

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by 11

1364 ((3+4) – (1+6) = 0) Yes

3729 ((7+9) – (3+2) = 11)
Yes

25176 ((5+7) – (2+1+6) = 3)
No

12

The number is divisible by both 3 and
4

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  .

 

 

 

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