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NUMBER SYSTEM
INTRODUCTION
A numeral system (or system of numeration) is a writing system for
expressing numbers; that is, a mathematical notation for representing
numbers of a given set, using digits or other symbols in a consistent
manner.
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NUMBER SYSTEM
RATIONAL NUMBERS
IRRATIONAL NUMBERS
A rational number is a
number that can be written
as a ratio of two integers.
𝑷
( , q ≠0)… Ex... 4 , 𝟏𝟔
Irrational number is any real
number that cannot be
expressed as a ratio
,
𝒒
𝟐
𝟓
of integers. Ex…. 𝟐 ,𝝅..
..
NUMBER SYSTEM
COMPOSITE NUMBER
PRIME NUMBER
All natural numbers
which are divisible by
only 1 and itself. Ex.
2, 3,5,7……….
All the natural numbers
which are not a prime
number except 1.
Ex- 4, 6,8,9…….
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INTEGERS
–
6
–
5
–
4
–
3
–
2
–
1
0
+
1
+
3
+
2
– Ve Integers
+
5
+
4
+Ve Integers
n(n+1)
 The sum of first ‘n’ natural numbers =
𝟐
 The sum of first ‘n’ even numbers = n( n+1)
2
 The sum of first ‘n’ odd numbers = n
 The sum of square of first ‘n’ natural numbers =
 The sum of cube of first ‘n’ natural numbers =
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n(n+1)(2n+1)
𝟔
n(n+1) 𝟐
𝟐
+
6
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PROPERTIES OF PRIME NUMBERS & COMPOSITE NUMBERS
 Only ‘2’ is even prime number.
 There is only one set of three prime numbers with a gap of 2
between two prime numbers and that set is 3, 5 , 7.
 Lowest composite number is 4.
st
 1 odd composite number is 9
 There are total 25 prime numbers up to 100.
 There are total 46 prime numbers up to 200.
PRIME NUMBERS (Between 1 - 100)
Prime Numbers 1 to 100
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
78
79
83
89
SUM OF PRIME NUMBERS:
 Sum of 1 to 50 prime numbers = 328
 Sum of 1 to 100 prime numbers = 1060
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97
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Some examples:
Ex: Find the product of all the prime numbers lies
between 80 to 90?
Prime numbers between 80 to 90 are 83 & 89
So Product = 83 × 89 = 7387
Ex: The sum of the squares of three consecutive
natural number is 194 then find the sum of these
numbers?
Suppose number x , x+1 ,x+2
x2 + (x+1)2 + (x+2)2 = 194
x = 7 , -9
so number are 7 , 8 , 9
sum = 7+ 8 + 9 = 24
Ex: Find the number of prime factors in
307 × 225 × 3412 × 125 ?
= 307 × 225 × 3412 × 125 = 27+5+12+5+5 × 37+5 × 57 ×
115 × 1712
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= 234 × 312 × 57 × 115 × 1712
the required number of prime factors
= 34 + 12 +7 +5 + 12 = 70
Ex: There are three prime number, and the product of
the first two of them is 551 and product of last
two of them is 1073, then find the sum of these
number?
Suppose number is a , b, c
a × b = 551
similarly
b × c = 1073
b × c = 29 × 37
so ,
a × b = 19 × 29
therefore sum will be = 19 + 29
+37 = 85
𝐧
𝐧
𝐧
 Number of zero’s = 𝟓+ 𝟓𝟐+ 𝟓𝟑+ − − − − −

numerator ≥ 𝒅𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓
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Ex: Find the number of zeroes at the end of !1000
Sol: Number of zeroes =
1000 1000
1000
+
+
5
52
53
= 200 + 40 + 8 + 1
= 249
+
1000
54
Ex: How many zeroes will be there counting from the
right side when will solve the following
expression 5 × 10 × 15 × 20 ------ × 50
Sol:
There will be 8 pairs of (5×2) so 8 zero
Ex: Find the number of zero’s in the given expression
10 × 20 × 30 × 40 × ……….× 1000
Sol: 10 × 1 × 10 × 2 × 10 × 3 × ……. × 10 × 100
= 10100 × 1 × 2 × 3 × 4 × 5 × ………..× 100
= 100 + 24
= 124 no. of zero’s
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104 + 90 = 194 (e)
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