# 6th Class Mathematics Number System and its Operations NUMBER SYSTEM

NUMBER SYSTEM

Category : 6th Class

Learning Objective

• To know about the types of Numbers.
• (Natual numbers whole numbers. even. odd, prime, composite.co-prime, twin primes. perfect numbers, integers etc.)
• To understand the divisibility of whole numbers.
• To learn how to round off a number nearest to ten, hundred and thousand.
• To know how to compare numbers.
• To learn how to calculate factors, multiples, prime factors H.C.F. (highest common factor) and L.C.M (lowest common factor) of numbers
• To understand the representation of integers (positive and negative) on number line.
• To learn the order of integers, absolute value of integer.
• To team how to add and subtract integers with and without number line.
• To learn BODMAS rule and uses of brackets.

NATURAL NUMBERS

(i) Counting numbers 1,2,3,4, 5, ....are called Natural numbers.

(ii) The set of natural numbers is denoted by N i.e.,

N = {1, 2, 3, 4, 5,.........}

(iii) 1 is the smallest natural number.

(iv) There is no largest natural number, i.e. the set of natural numbers is infinite.

(v) Any natural number can be obtained by adding ' 1' to its previous natural number.

WHOLE NUMBERS

(i) All natural numbers together with zero are called whole numbers, as 0, 1, 2, 3, 4,... are whole numbers.

(ii) The set of whole numbers is denoted by W, i.e.

W = {0, 1, 2, 3, 4, 5.....}

(iii) $W=N\cup \{0\},$ where N is the set of natural numbers.

(iv) 0 is the smallest whole number.

(v) There is no largest whole number i.e. the number of the elements in the set of whole numbers is infinite.

(vi) Every natural number is a whole number.

i.e. $N\subseteq W$ i.e. N is a subset of W.

(vii) 0 is a whole number, but not a natural $0\in W$ but $0\notin N$

EVEN NUMBERS

(i)   Whole numbers which are exactly divisible by 2 are called even numbers.

(ii) The set of even numbers is denoted by 'E', such that

E= {0, 2, 4, 6, 8.....}.

(iii) The set E is an infinite set.

ODD NUMBERS

(i)  Natural numbers which are not exactly divisible by 2 are called odd numbers.

(ii) The set of odd numbers is denoted by '0' such that

O = { 1, 3, 5, 7, 9.....}

(iii) The set 0 is an infinite set.

PRIME NUMBERS

(i)  Natural numbers having exact two distinct factors i.e., 1 and the number itself are called prime numbers.

Example: 2, 3, 5, 7, 11, 13, 17, 19,... are prime numbers.

(ii) The set of prime numbers is infinite.

(iii) 2 is the smallest prime number.

COMPOSITE NUMBERS

(i)  Natural numbers having more than two factors are called composite numbers.

Example: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 ... are composite numbers.

(ii) Number 1 is neither prime nor composite number.

(iii) All even numbers except 2 are composite numbers.

(iv) Every natural number except 1 is either prime or composite number.

(v) There are infinite composite numbers.

CO-PRIME NUMBERS OR RELATIVELY PRIME NUMBERS

Two natural numbers are said to be co-prime numbers or relatively prime numbers if they have only 1 as common factor.

Example: (8, 9), (15, 16), (26, 33) etc. are co-prime numbers.

Co-prime numbers may not themselves be prime numbers.

As 8 and 9 are co-prime numbers, but neither 8 nor 9 is a prime number.

TWIN PRIMES

Pairs of prime numbers which have only one composite number between them are called twin primes.

Example: 3, 5; 5, 1; 11, 13; 17, 19; 29, 31; 41, 43; 59, 61 and 71, 73 etc. are twin primes.

FACTORS AND MULTIPLES

(i)  A factor of a number is that number which completely divides the number without leaving the remainder.

For example: Factors of 18 are: 1, 2, 3, 6, 9 and 18.

(ii) A multiple of a number is a number which is obtained by multiplying it by a natural number.

For example: Multiples of 4 are:

$4\times 1=4,$ $4\times 2=8,\,4\times 3=12,$ $4\times 4=16$ etc.

(iii) Those factors (or multiples) which are common among the factors (or multiples) of two or more numbers are known as common factors (or multiples)

For example: 18 and 36

Factors of

Factor of

Common factors of 18 and 36 are: 1, 2, 3, 6, 9 and 18 Multiples of 4

etc.

Multiples of 3 are

3, 6, 9,  15, 18, 21,  27, 30, 33,  etc.

Common multiples of 3 and 4 are 12, 24, 36 etc.

PERFECT NUMBERS

If the sum of all the factors of a numbers is twice the number, then the number is known as a perfect number.

For example: Factors of 6 are 1, 2, 3 and 6.

Also, 1 + 2 + 3 + 6 = 12

$=2\times 6$

So, 6 is a perfect number

Factors of 8 are 1, 2, 4 and 8.

Also, 1 + 2 + 4 + 8 = 16 = 2 x 8.

Sum of all the factors of 8 = 2 x 8

So, 8 is a perfect number.

PRIME FACTORIZATION

Expressing a number as a product of prime factors is called a prime factorization of a number.

For example: Prime factorization of 24 is

24 =2 x 2 x 2 x 3

Prime factorization of 56 is

= 2 x 2 x 2 x 7.

DIVISIBILITY OF WHOLE NUMBERS

Divisibility Test for whole numbers

• A number is divisible by 2 if the unit place digit in it is an even number.
• A number is divisible by 3 if the sum of its digits is multiple of 3.
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• A number is divisible by 5 if it ends in 0 or 5.
• A number is divisible by 6, if it is divisible by 2 and 3 both.
• A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 10 if it ends in zero,
• A number is divisible by 11 if the difference of the sums of alternative digits is zero or a multiple of 11.
• A number is divisible by 12 if it is divisible by both 3 and 4.

For example: 1236 is divisible by 2,3,4,6 and 12.

1255 is divisible by 5

8955 is divisible by 9

134574 is divisible by 11

((sum of odd digits = 1 + 4 + 7 = 12, sum of even digits = 3 + 5 + 4 = 12)

difference = 12 – 12 = 0

$\therefore$ divisible by 11)

PLACE VALUE

Place value of a digit in a given number is the value of the digit because of the place or the position of the digit in the number.

For example: 6503 place value of 3 = 3.

Place value of 0 = 0, Place value of 5 = 500

Place value of 6 = 6000

ROUNDING OFF A NUMBER

ROUNDING OFF A NUMBER TO THE NEAREST TEN

(i) If the one's digit is less than 5, replace it by 0.

For example: 23 is rounded off as 20

284 is rounded off as 280

(ii) If the one's digit is equal to or more than 5, increase the ten's digit by 1 and replace one's digit by 0.

For example: 249 is rounded off as 250.

ROUNDING OFF A NUMBER TO THE NEAREST HUNDRED

(i) If the ten's digit is less than 5, replace the tens and ones digits by 0 only.

For example: 627 is rounded off as 600.

2434 is rounded off as 2400.

(ii) If the ten's digit is equal to or more than 5, increase hundreds digit by 1 and replace each digit on its right by 0.

For example: 6587 is rounded off as 6600

6557 is rounded off as 6500

ROUNDING OFF A NUMBER TO THE NEAREST THOUSAND

(i) If the hundreds digit is less than 5, replace the hundreds, tens and ones digit by 0.

For example: 1482 is rounded off as 1000

4341 is rounded off as 4000

(ii) If the hundreds digit is equal to or more than 5, increase thousands digit by 1 and replace each digit on its right by 0.

For example:     1877 is rounded off as 2000

5936 is rounded off as 6000

COMPARISON OF NUMBERS

WHEN THE NUMBER OF DIGITS ARE DIFFERENT

(i) Check the number of digits in each number.

(ii) The number having more number of digits is the greater one than the number having less number of digits.

For example:     12456 > 2345

1534 < 12345

WHEN THE NUMBER OF DIGITS ARE THE SAME

(i) Compare the left most digit of each number.

(ii) The number with the greater left most digit is greater

For example: 4378 > 2378

(iii) In case, the left most digit is same, compare the next digit of both numbers. The number with the greater digit is greater.

For example: 5765 > 5467

ROMAN NUMERALS

A numeral is a word that we write down to stand for a number.

 I 1 XX 20 II 2 XXX 30 III 3 XL 40 IV 4 L 50 V 5 LX 60 VI 6 LXX 70 VII 7 LXXX 80 VIII 8 XC 90 IX 9 C 100 X 10 D 500

Repetition of a symbol means addition.

For example: III

i.e. 1 + 1 + 1 = 3,

XX,

i.e., 10 + 10 = 20

Smaller number written to the right means addition.

For example: In LXX

L Stands for 50

X Stands for 10

LXX stands for 50 + 10 + 10 = 70

A smaller number written to the left it means subtraction

For example: In XC

X Stands for 10

C Stands for 100

XC stands 100 – 10 = 90

H.C.F AND L.C.M.

STEPS FOR FINDING HCF

(i) Factorise the numbers into product of primes expressed in exponential form.

(ii) Select the lowest of the power of common primes

(iii) The product of primes with lowest powers is HCF.

STEPS FOR FINDING LCM

(i) Faetorise the numbers into product of primes expressed in exponential form.

(ii) Select the highest power of a prime present in all or some of the numbers.

(iii) The product of primes with highest power is LCM.

For example: Find HCF and LCM of 45. 75 and 125

$45=3\times 3\times 5={{3}^{2}}\times 5$

$75=3\times 5\times 5=3\times {{5}^{2}}$

$125=5\times 5\times 5={{5}^{3}}$

HCF $(45,\,75,\,125)={{5}^{1}}=5$

(Product of lowest power common prime)

LCM (45, 75, 125) = 32 x 53

(Product of highest power of primes)

$=9\times 125=1125$

INTEGERS

The set of integers is denoted as

{.....-4, -3, -1, 0, 1, 2, 3....}

POSITIVE INTEGERS

The numbers 1, 2, 3, 4, 5, 6,. .....i.e. natural numbers are called positive integers.

Positive integers are also written as +1, +2, +3,...... however, the plus sign (+) is usually omitted and understood.

NEGATIVE INTEGERS

The numbers -1, -2,-3, -4, - 5, - 6,........are called negative integers. The number 0 is simply an integer. It is neither positive nor negative.

REPRESENTATION OF INTEGERS ON THE NUMBER LINE

To represent integers on number line, we draw a line and a point O in the middle of it. Now, we set off equal distances on the right hand side as well as on the left hand side of O. On the right side of O, we label the points of subdivision as 1, 2, 3, 4, 5, etc., and the point O as 0. On the left side of point O, we label the points of subdivision by -1, -2, -3, -4. -5, etc.

ORDERING OF INTEGERS

On a number line an integer to the right is always greater than the integer to the left.

For example:

-3 > -5, Since -3 is to the right of-5.

4 > 1, Since 4 is to the right of 1

SOME USEFUL POINTS

Every positive integer is greater than every negative integer, since every positive integer is to the right side of every negative integer.

Zero is less than every positive integer, since 0 is to the left side of every positive integer.

Zero is greater than every negative integer, since 0 is to the right of every negative integer. The greater the number, the lesser is its opposite.

ABSOLUTE VALUE OF AN INTEGER

The absolute value of an integer is the numerical value of the integer regardless of its sign.

The absolute value of an integer a is denoted by |a|.

For example: Absolute value of 7 i.e. |7| = 7

Absolute value of—7 i.e. |—7| = 7

ADDITION OR SUBTRACTION OF INTEGERS ON NUMBER LINE

In order to add two integers on a number line, we proceed as:

Seep (i)   Draw a number line and mark integers on it.

Seep (ii) Represent the first number on the number line.

Step (iii) Move as many steps as the second number to the

(i) right of the first number, if the second number is positive.

(ii) left of the first number, if the second number is negative.

Step (iv) Obtain the number representing the point. This number represents the required sum of the given integers.

(i)  6+4

To add 4 to 6, move 4 steps right to the 6 we reached to 10.

i.e. 6 + 4 = 10

To add 5 to 7, move 5 steps right to the -7, we reached to -2

(ii) -  7 + 5 To add 5 to 7, move 5 steps right to the ? 7, we reacted to ? 2

(iii) - 4-3 To substract 3 from -4, move 3 steps left to the -4, we reached to -7

ADDITION OF INTEGERS WITHOUT USING THE NUMBER LINE

To add two positive integers or two negative integers, we add their values regardless of their sign (absolute value) and give the sum their common sign.

For example: 7 + 4 = 11

$-7+(-4)\,=-7-4=-11$

To add a positive and a negative integer, we determine the difference of their values regardless of their sign and give the sign of integer with greater absolute value to the result.

For example:     8 + (- 10) = -2

10 + (-8) = 2

Numbers such as 3 and -3, 2 and -2, when added to each other give the sum zero. They are called additive inverse of each other.

SUBTRACTION OF INTEGERS

If a and b are two integers, then to subtract b from a, we change the sign of b and add it to a, i.e.,

a - b = a + (-b)

For example:

(i)  Subtract 7 from 12 = 12 -7 - 12 + (-7) = 5

(ii) Subtract 9 from 5 = 5 – 9 = 5 + (-9) = - 4

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