NUMBER
Digit-Digit
is the most basic element of mathematics. A digit in mathematics is similar to
an alphabet in English language.
The number system which we follow
consists of 10 symbols or digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All the
numbers, however large or small they may be, are formed by these digits.
Number-A
number is a word or symbol that represents an amount or quantity.
Numeral-The
representation of a number by a group of digits is called a numeral.
Or,
A numeral is a symbol to
represent a number.
Or,
It is a digit or a group of
digits which we write to represent a number.
e.g. - Five thousand
eight hundred forty two.
The
numeral is 5842.
Place-Every
number is made of digits. Every digit in a number has a certain position called
place.
e.g. - 538904
Lakhs T Th Th hundreds tens ones
Place Value-The
place-value of a digit depends on its place in the number. Starting from the
right, the value of each successive place is 10 times its previous place.
e.g.-
Ones = 1
Tens = 10´ones = 10´1 = 10
Hundreds = 10´tens = 10´10 = 100
Thousands = 10´hundreds = 10´100 = 100
Ten thousands = 10´thousands = 10´1000 = 1000
…… ……
….. ….. ….. …… …..
…… ……
….. …… ……. ……. ……
Face Value-The
face value of a digit in the value of the digit itself and does not change.
e.g. - 4676
the face value of 7 is 7
Place Value and Face Value-
Every digit of a number
has two values – The face value and the Place Value. The face value of a digit
is the value of the digit itself and does not change, while the place value of the
digit changes according its position in the number.
|
Number
|
Digit
|
Face
Value
|
Place
Value
|
|
67,923
|
6
7
9
2
3
|
6
7
9
2
3
|
60,000
7,000
900
20
3
|
There
are two commonly used method to express a number –
i.
The Indian System
ii.
The International System
In both systems, a number is split
up into groups called periods.
In the Indian system, staring
from the right, the groups are called ones or units, thousands, lakhs, crores,
arabs etc. The ones are split into hundreds, tens and units.
Places
Value
Units
(U) 1 10
Tens
(T) 10 101
Hundreds
(H) 100 102
Thousands
(Th) 1000 103
Ten
Thousands (T-Th) 10000 104
Lakh
(L) 100000 105
Ten
Lakh (TL) 1000000 106
Crore
(C) 10000000 107
Ten
Crore (TC) 100000000 108
Arab
(A) 1000000000 109
Ten
Arab (TA) 10000000000 1010
Kharab
(Kh) 100000000000 1011
Ten
Kharab (T-Kh) 1000000000000 1012
Note:
- To write a number in the Indian system, beginning from the right, a comma is
put after first three digits and then after every two digits.
e.g. - 27, 39, 825 – Twenty seven lakh,
thirty nine thousand, eight hundred twenty five.
In the International system of
numeration, starting from the right, the groups or periods are called ones,
thousands, millions, billions, trillions, etc.
Place Value
Ones
(O) 1 10
Tens
(T) 10 101
Hundreds
(H) 100 102
Thousands
(Th) 1000 103
Ten
Thousands (T-Th) 10000 104
Hundred
Thousand (H-Th) 100000 105
Millions
(M) 1000000 106
Ten
Millions (TM) 10000000 107
Hundred
Millions (HM) 100000000 108
Billions
(B) 1000000000 109
Ten
Billions (TB) 10000000000 1010
Hundred
Billions (TB) 100000000000 1011
Trillions
(T) 1000000000000 1012
Ten
Trillions (TT) 10000000000000 1013
Hundred
Trillions (HT) 100000000000000 1014
Quadrillions
(Q) 1000000000000000 1015
Ten
Quadrillions (TQ) 10000000000000000 1016
Hundred
Quadrillions (HQ) 100000000000000000 1017
Note:
- To write a number in the International system, beginning from the right.
Commas are put after every three digits.
e.g. - The number is 42745013829
42,745,013,829 –
Forty two billion, seven hundred forty five million, thirteen thousand, eight
hundred Twenty nine.
Number-A number is a word or
symbol that represents an amount or quantity.
Types of Numbers-
(a) Natural Numbers- The
numbers which is used for counting are called natural numbers.
Two consecutive natural
numbers differ by 1.
N = {1, 2, 3, 4, 5, 6, …}
(b) Whole numbers- All
natural numbers together with 0 form a set of all whole numbers.
W
= {0, 1, 2, 3, 4, 5, …}
(c) Integers- The set
of naturals numbers, 0, and the negatives of natural numbers from the set of integers.
Zero is neither negative nor positive.
Z= {…, -3, -2, -1, 0, 1, 2, 3, …}.
Ø The
absolute value of an integer a is its numerical value regardless of its sign
and is denoted by
(d) Even numbers- The
number which is exactly divisible by 2 is called an even number.
Ø Two consecutive even
numbers is differ by 2.
Ø An even number is
represented by 2n when n
ϵ N.
(e) Odd numbers- The
number which is not exactly divisible by 2 is called odd number.
Ø Two consecutive odd
numbers also differ by 2.
Ø An odd number is
represented by 2n-1 where n ϵ N or 2n+ 1, where n ϵ W.
(f) Prime numbers- The
number which has only two factors (1, and the number itself) is called a prime
number.
e.g. - 2, 3, 5, 11, 13,
17, etc.
Ø 1 is not a prime number
as it has only one factor which is itself.
(i)
Co-primes- A pair of two natural
numbers having no common factor, other than 1, is called a pair of primes.
e.g. - (3, 5), (4, 5), (5, 6), (7, 9),
(6, 7) etc.
Ø Two prime numbers are
always co-primes.
Ø Two co-primes need be
prime numbers.
E.g. - 6, 7 are co-primes, while
6 is not a prime number.
9, 10 are co-primes, while
none of 9 and 10 is a prime number.
(ii)
Twin primes- Two prime numbers which
have a difference of 2 between them are called twin primes.
e.g. - (3, 5), (5, 7), (11, 13)
etc.
(iii)
Prime triplet- A set of three
consecutive prime numbers which have a difference of 2 between them are called
prime triplet.
e.g. - (3, 5, 7).
Ø The numbers 2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89
and 97 are the prime numbers between 1 and 100.
“Every prime number except 2 is odd
but every odd number need not be prime”.
(g) Composite numbers- The number which has more than two factors is called a
composite number.
e.g. - 4, 6, 8, 9, 10, 12, 14, etc.
Ø 1 is neither prime number
nor a composite number.
Ø 2 is the smallest prime
number and also the only even prime number.
(h) Perfect number- If the sum of all the
factors of a number is two times the number, then the number is called a
perfect number.
e.g. - Factors of 6 are
1, 2, 3, and 6.
1 + 2 + 3 + 6 =
12 = 2
6.
Hence, 6 is a
perfect number.
(i) Division algorithm- For any two given positive integers
and
there exist unique whole numbers
and
such that
Here, we call
as dividend,
as divisor
as quotient and
as remainder.
Formulae-
Rules of divisibility – There are some rules which help us to
find whether a given number is divisible by another quickly, without indulging
in the process of actual division. These rules are called the rules of
divisibility.
Divisibility
by 2 – A
number is divisible by 2, if the digit at the ones place of the number is even i.e. 0, 2, 4, 6 or
8.
e.g. - 152; 3,470; 53,
17,954 etc.
Divisibility
by 5 – A
number is divisible by 5, if the digit at the ones place is either 0 or 5.
e.g. - 485; 5,970; 12,
63,435 etc.
Divisibility
by 10 – A
number is divisible by 10, if the digit at the ones place is 0.
e.g. - 570; 63,590; 13,
69,150 etc.
Divisibility
by 4 – A
number is divisible by 4, if the number formed by its last two digits is
divisible by 4 or
the last two digits are
both zeroes.
e.g. – 15,324; 6, 37,900.
Divisibility
by 8 – A
number is divisible by 8, if the number formed by its last three digits is
divisible by 8 or
the last three digits are all zeroes.
e.g. – 53,144; 8, 36,000 etc.
Divisibility
by 3 – A
number is divisible by 3, if the sum of its digits is divisible by 3.
e.g. – 477; 2,898; 58,797 etc.
Divisibility
by 9 – A
number is divisible by 9, if the sum of its digits is divisible by 9.
e.g. – 477; 2,898; 58, 797 etc.
Divisibility
by 11 – A
number is divisible by 11, if the difference between the sum of its digits at
odd places
(from the right hand side) and sum of its
digits at even places is either 0 (zero) or is divisible by 11.
4 + 2 + 9 = 15 (sum of even
digits)
26 – 15 = 11.
11 is divisible by 11.
Thus, 64, 62,896 is divisible by 11.
Divisibility by 6 – A number is divisible by 6, if it is divisible by both
2 and 3.
1,518
1 + 5 + 1 + 8 = 15 (divisible by 3)
Thus, 1518 is
divisible by 6.
Divisibility by 12 – A number is divisible by 12, if it is divisible by both
3 and 4.
e.g. – 7,668
7
+ 6 + 6 + 8 = 27 (divisible by 3)
7,668
68
4 = 17 (divisible
by 4)
Thus, 7,668 is divisible by
12.
Divisibility by 7 – A number is divisible by 7, if the difference of the twice of the
digit at its ones place and
the number formed by
the remaining digits either 0 (zero) or a multiple of 7.
e.g. – 1,652
165 –
(2
)
= 165 – 4= 161
17,024
1702 – 8 = 1694
Divisibility by 14 – A number is divisible by 14, if it is divisible by both 2 and 7.
e.g. – 1,652; 17,024.
Divisibility by 15 – A number is divisible by 15, if it is divisible by both 3 and
5.
e.g. – 5,970; 1263435.
Divisibility by 16 – A number is divisible by 15, if the number formed by the last
four digits is divisible by
16.
e.g. – 51,792; 1792.
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