Online Learning Summit

NUMBER SYSTEM

MAT111

Number, Real, complex, Rational, Irrational, Fraction, Integer, Classification

Factor, Divisor and Multiple, Find prime factorization

& all factor of a number

LCM, HCF/GCD, Find LCM & HCF/GCD of Integers or Fractions

Complex number, Modulus, Argument and its various forms

TOPIC-1

Number :

For example: The weight of a man is 153 kg

A number is a numeral which represents a quantity and also used for

counting, measurement & labelling.

Real

Number :

A number is said to be real if the square of it is non-negative otherwise imaginary.

( √ ^{2 )}

²

   

¿ 2

Non-negative

( √ ^{− 2 )}

²

   

¿ − 2

^Negative

A real number geometrically

represents a point on X-

axis.

−

 

� −

 

� −

 

� �

 

�

 

�

 

�

 

Complex

Number :

A complex number is the linear combination of two real numbers &

with a special sign .

 

A complex number geometrically

represents a point on Complex plane or

Argand plane.

� = � + ��

 

(±

 

� ) ² =− �

where: and real number & 

(

 

. � , � )

  

�

 

� ( ��������� )

 

�

 

(����)

Rational Number:

For

example:

2 3 4

2 ,3 & 4 .

1 1



1

etc

   

0101 Integer / Whole number

15 3 13

1.5 ,1.3 .

10 2 10

etc

  

0202 Terminating decimal number

0303

Recurrence decimal number /Infinite decimal

point number

1 4 9

, & . 3 3 5

etc

0404 Fraction

.

1

.

13 1 4

0.3 ,1.3 .

3 9



3

etc

  

Is 4 is a rational number ?

Numerator Fraction =

Denominator

A number of the form where and are integers.

 

Irrational Number:

For

example:

 

2 , 3 & 2



3

etc

.

0101 Inexpressible as fraction

1.32495879...

2.78997824...

0202

Non-terminating decimal number/Infinite nonrepeating

decimal number

A number which is not expressible as where and are integers.

 

Divisio n

Try it

: What are the factors of the number 6 ?

Factor :

For example: are the factors of .

 

Factors Multiples

Factors of a number are all those numbers which can exactly divide the given number.

Multip le:

For example: The multiples of are

 

Multiples of a number are those numbers which are exactly divisible by the given number.

               

Prime

Number :

Composite Number :

A prime number is a natural number greater than 1, that has exactly two factors 1(one) and itself.

7

 

7 1

 

 

A composite number is a natural number

greater than 1, that has more than 2 factors.

6

 

6 1

 

   

2

 

3

Classificatio

n:

12

 

3 4

 

 

Prime

Factorizatio n

Prime factorization of a number is the

representation of the number by its prime factors.

2

²

   

3

Problem 01:

Find the Prime Factorization of 1600.

Division method:

2 1600 2 800

2 400 2 200

2 100 2 50

5 25 5

Therefore ,the prime factorization of 1600 is

6 2

2 . 5



Tree

Diagram:

5 1600

2 800

200 400

100 50

25 5

2

2 2

2

2

Multiplication Method :

2

3 4

6 6 2

1600 2 800 2 2 400 2 2 200 2 2 100 2 2 50

2 5 5 2 .5

       

     

   

Formula :

Try it.

How many factors of 520?

� =�

^�

�

^�

�

^�

 

( � + 1 )( � + 1)( � + 1 )

 

N u m b e r o f f a c to rs

2

¹

   

3

¹

6

 

6 1

 

   

2

 

3

( 1 + 1 ) ( 1 + 1 ) = 4

 

Problem 02: Find the all factors of 1600.

2 1600 2 800

2 400 2 200

2 100 2 50

5 25 5

Therefore ,the prime factorization of 1600 is

6 2

2 . 5



So ,the total number of factors of 1600 is

1600 1 1600 2 800 4 400 5 320 8 200 10 160 16 100 20 80 25 64 32 50 40 40

 

 

 

 

 

 

 

 

 

 

 

The factors of 1600 are

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800 & 1600.

( 6 + 1 )( 2+ 1 ) =7 ⋅ 3 =21

 

Calculation for all factors

Exercise

: 1. Find the prime factorization of 1240 using three different methods.

2. Find the all factors of 1240 using tree diagram.

3. Find the all prime factors of 1240.

4. Find the all composite factors of 1240.

LCM = Least Common Multiple

2  2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , ...

4  4 , 8 , 12 , 16 , 20 , 24 , 28 , 36, ...

(2, 4) 4

LC M 

HCF / GCD = Highest Common Factor / Greatest Common Divisor

2  1 , 2

4  1 , 2 , 4

(2, 4) 2

G C D 

LCM is a smallest number which is divisible by that numbers.

GCD is a biggest number

that divides those numbers.

Calculation of LCM and GCD by using Prime Factorization.

Problem 03: Find the LCM & GCD of the numbers 2 & 4.

Solution:Given numbers are  

Therefore, the prime factorization of are

 

LCM = Common/Uncommon factors GCD = Common factors

 

LCM ( 2,4 )= 2

²

= 4 ∧GCD ( 2,4 )=2

¹

= 2

 

Problem 04 : Find the LCM and HCF of

 

Soluti on:

Given numbers are .

 

2 2

2 3 4

50 2.25 2.5.5 2.5

140 2.70 2.2.35 2 .5.7

240 2.120 2.2.60 2 .2.30 2 .2.15 2 .3.5

  

  

    

4 2

(50,140, 240) 2 .5 .3.7 8400

 LCM  

& H C F (5 0,140, 240 )  2.5  10.

LCM & HCF of Fractions

LCM of Numerators LCM of Fractions=

HCFof Denominators



HCFof Numerators HCFof Fractions=

LCM of Denominators



2 8 16 Find the LCM and HCF of , and .

3 9 81 Calculation for

Numerators

Calculation for Denominators

LCM (2, 8,16 )=2 4 =16

 

HCF ( 2, 8,16 ) = 2

¹

=2

 

LCM ( 3, 9,81)=3 4 =81

 

HCF ( 3, 9,81 ) =3

¹

= 3

 

 

 

LCM of 2,8,16

2 8 16 16

LCM of , and =

3 9 81 HCF of 3,9,81  3

 

 

HCF of 2,8,16

2 8 16 2

of , and =

3 9 81 of 3,9,81 81

HCF LCM 

Exercise :

Find the H.C.F & L.C.M of

 

Find the H.C.F & L.C.M of