Elementary
Arithmetic-I
Number System
Number A number tells us how many times a unit is contained in a
given quantity.
Numeral A group of figures (digits), representing a number, is
called a numeral.
Face Value and Place Value of the Digits
In a numeral, the face value of a digit is the value.
In a numeral, the place value of a digit changes according to the
change of its place.
e.g. In the numeral 576432, the face value of 6 is 6 and the place value
of 6 is 6000.
Types of Number System
(i) Binary Number System (Base-2) It represents numerical
values using two digits usually ‘0’ and ‘1’. This system is used
internally by computers and electronics.
For binary systems, as we move left to the decimal point number
gets 2 times bigger and as we move right to the decimal every
number gets 2 times smaller.
e.g. 1011101
. = 1 × 2 3 + 0 × 22 + 1 × 21 + 1 × 20 + 1 × 2−1
+ 0 × 2−2 + 1 × 2−3
Decimal 0 1 2 3 4 5
Binary number 0 1 10 11 100 101
(ii) Octal Number System (Base-8) It represents numerical
values using 8 digits from ‘0’ to ‘7’.
, As we move left to the decimal point number gets 8 times bigger
and as we move right to the decimal point number gets 8 times
smaller.
Decimal 0 1 2 3 4 5 6 7 8 9 10
Octal 0 1 2 3 4 5 6 7 10 11 12
(iii) Hexadecimal Number System (Base-16) Every numerical
value in this system is represented by decimal numbers 0 to 9
and letters ( A, B, C, D, E, F ) in place of number 10 to 15. As we
move left to decimal number gets 16 times bigger and as we
move right to the decimal numbers gets smaller by 16.
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
number
(iv) Roman Number System Roman Numerals and their
corresponding Indo-Arabic numerals
Roman numerals I V X L C D M
Indo-Arabic numerals 1 5 10 50 100 500 1000
(v) Decimal Number System Numeric values are represented
by using digits from ‘0’ to ‘9’.
Classification of Numbers in
Decimal Number System
Real Number
Rational Number Irrational Number
Integer Non-Integer Rational Number
Positive Integer Negative Integer Whole Number Natural Number
Natural Numbers Numbers starting from 1, having no fraction
part, which we use in counting the objects, denoted by N.
N = { 1, 2, 3, K }
,Whole Numbers The system of Natural numbers along with
number 0, is called whole number (W ).
W = { 0, 1, 2, 3, K }
Different Types of Natural Number
(i) Even Number A number, which is multiple of 2 is called an
even number.
(ii) Odd Number A number, which is not a multiple of 2 is called
an odd number.
(iii) Prime Number The number which can be divided only by
itself and 1 is called prime number.
e.g. 2, 3, 5, 7, 11, ...
(iv) Composite Number The number which can be divided by a
number other than 1 and the number itself is called composite
number.
(v) Consecutive Number A series of numbers in which each
number is greater by 1 than the number which precedes it.
Method to Determine a Given Number is Prime or Not
Step I Find a new number larger than the approximate square
root of given number.
Step II Test whether the new number is divisible by any prime
number.
Step III If the new number is not divisible by any of the prime
number, then given number is a prime number otherwise
it is composite number.
Division on Numbers (Division Algorithm)
Let ‘a’ and ‘b’ be two integers such that b ≠ 0 on dividing ‘ a ’ by ‘ b’.
Let ‘ q ’ be the quotient and ‘ r ’ the remainder, then the relationship
between a, b, q and r is a = bq + r.
or in general, we have
Dividend = Divisor × Quotient + Remainder
, Test of Divisibility on a Natural Number
(i) Divisibility by 2 A number is divisible by 2, if digit on unit
place is 0, 2, 4, 6, 8.
(ii) Divisibility by 3 If the sum of the digits of a number is
divisible by 3, then the number is divisible by 3.
(iii) Divisibility by 4 If the last two digits of a number is divisible
by 4 or the last two digits are ‘00’, then the number is divisible
by 4.
(iv) Divisibility by 5 A given number is divisible by 5, if 0 or 5
comes at unit place.
(v) Divisibility by 6 If a given number is divisible by 2 and 3,
then it is divisible by 6.
(vi) Divisibility by 7
(a) If a number is formed by repeating a digit six times, the
number is divisible by 7, 11 and 13. e.g. 666666.
(b) If a number is formed by repeating a two-digit number three
times, the number is divisible by 7. e.g. 676767.
(c) If a number is formed by repeating a three-digit number two
times, the number is divisible by 7, 11 and 13. e.g. 453453.
(vii) Divisibility by 8 If the last 3 digits of a number is divisible
by 8 or the numbers ends with ‘000’, then it is divisible by 8.
(viii) Divisibility by 9 If the sum of the digits of a number is
divisible by 9, then the number is divisible by 9.
(ix) Divisibility by 10 If ‘0’ comes at unit place of a number, then
it is divisible by 10.
(x) Divisibility by 11 A given number is divisible by 11, if the
difference between the sum of the digits in odd places and the
sum of the digits in the even places is either 0 or a multiple by 11.
(xi) Divisibility by 12 If a given number is divisible by 4 and 3,
then it is divisible by 12.
(xii) Divisibility by 25 When the number formed by last two
digits is divisible by 25.
(xiii) Divisibility by 27 When the sum of the digit of the number is
divisible by 27.
(xiv) Divisibility by 125 When the number formed by last three
digits is divisible by 125.
Arithmetic-I
Number System
Number A number tells us how many times a unit is contained in a
given quantity.
Numeral A group of figures (digits), representing a number, is
called a numeral.
Face Value and Place Value of the Digits
In a numeral, the face value of a digit is the value.
In a numeral, the place value of a digit changes according to the
change of its place.
e.g. In the numeral 576432, the face value of 6 is 6 and the place value
of 6 is 6000.
Types of Number System
(i) Binary Number System (Base-2) It represents numerical
values using two digits usually ‘0’ and ‘1’. This system is used
internally by computers and electronics.
For binary systems, as we move left to the decimal point number
gets 2 times bigger and as we move right to the decimal every
number gets 2 times smaller.
e.g. 1011101
. = 1 × 2 3 + 0 × 22 + 1 × 21 + 1 × 20 + 1 × 2−1
+ 0 × 2−2 + 1 × 2−3
Decimal 0 1 2 3 4 5
Binary number 0 1 10 11 100 101
(ii) Octal Number System (Base-8) It represents numerical
values using 8 digits from ‘0’ to ‘7’.
, As we move left to the decimal point number gets 8 times bigger
and as we move right to the decimal point number gets 8 times
smaller.
Decimal 0 1 2 3 4 5 6 7 8 9 10
Octal 0 1 2 3 4 5 6 7 10 11 12
(iii) Hexadecimal Number System (Base-16) Every numerical
value in this system is represented by decimal numbers 0 to 9
and letters ( A, B, C, D, E, F ) in place of number 10 to 15. As we
move left to decimal number gets 16 times bigger and as we
move right to the decimal numbers gets smaller by 16.
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
number
(iv) Roman Number System Roman Numerals and their
corresponding Indo-Arabic numerals
Roman numerals I V X L C D M
Indo-Arabic numerals 1 5 10 50 100 500 1000
(v) Decimal Number System Numeric values are represented
by using digits from ‘0’ to ‘9’.
Classification of Numbers in
Decimal Number System
Real Number
Rational Number Irrational Number
Integer Non-Integer Rational Number
Positive Integer Negative Integer Whole Number Natural Number
Natural Numbers Numbers starting from 1, having no fraction
part, which we use in counting the objects, denoted by N.
N = { 1, 2, 3, K }
,Whole Numbers The system of Natural numbers along with
number 0, is called whole number (W ).
W = { 0, 1, 2, 3, K }
Different Types of Natural Number
(i) Even Number A number, which is multiple of 2 is called an
even number.
(ii) Odd Number A number, which is not a multiple of 2 is called
an odd number.
(iii) Prime Number The number which can be divided only by
itself and 1 is called prime number.
e.g. 2, 3, 5, 7, 11, ...
(iv) Composite Number The number which can be divided by a
number other than 1 and the number itself is called composite
number.
(v) Consecutive Number A series of numbers in which each
number is greater by 1 than the number which precedes it.
Method to Determine a Given Number is Prime or Not
Step I Find a new number larger than the approximate square
root of given number.
Step II Test whether the new number is divisible by any prime
number.
Step III If the new number is not divisible by any of the prime
number, then given number is a prime number otherwise
it is composite number.
Division on Numbers (Division Algorithm)
Let ‘a’ and ‘b’ be two integers such that b ≠ 0 on dividing ‘ a ’ by ‘ b’.
Let ‘ q ’ be the quotient and ‘ r ’ the remainder, then the relationship
between a, b, q and r is a = bq + r.
or in general, we have
Dividend = Divisor × Quotient + Remainder
, Test of Divisibility on a Natural Number
(i) Divisibility by 2 A number is divisible by 2, if digit on unit
place is 0, 2, 4, 6, 8.
(ii) Divisibility by 3 If the sum of the digits of a number is
divisible by 3, then the number is divisible by 3.
(iii) Divisibility by 4 If the last two digits of a number is divisible
by 4 or the last two digits are ‘00’, then the number is divisible
by 4.
(iv) Divisibility by 5 A given number is divisible by 5, if 0 or 5
comes at unit place.
(v) Divisibility by 6 If a given number is divisible by 2 and 3,
then it is divisible by 6.
(vi) Divisibility by 7
(a) If a number is formed by repeating a digit six times, the
number is divisible by 7, 11 and 13. e.g. 666666.
(b) If a number is formed by repeating a two-digit number three
times, the number is divisible by 7. e.g. 676767.
(c) If a number is formed by repeating a three-digit number two
times, the number is divisible by 7, 11 and 13. e.g. 453453.
(vii) Divisibility by 8 If the last 3 digits of a number is divisible
by 8 or the numbers ends with ‘000’, then it is divisible by 8.
(viii) Divisibility by 9 If the sum of the digits of a number is
divisible by 9, then the number is divisible by 9.
(ix) Divisibility by 10 If ‘0’ comes at unit place of a number, then
it is divisible by 10.
(x) Divisibility by 11 A given number is divisible by 11, if the
difference between the sum of the digits in odd places and the
sum of the digits in the even places is either 0 or a multiple by 11.
(xi) Divisibility by 12 If a given number is divisible by 4 and 3,
then it is divisible by 12.
(xii) Divisibility by 25 When the number formed by last two
digits is divisible by 25.
(xiii) Divisibility by 27 When the sum of the digit of the number is
divisible by 27.
(xiv) Divisibility by 125 When the number formed by last three
digits is divisible by 125.
