property of Natural Number

(1) Coprime Number : 

The two natural numbers whose H.C.F Is one(1), is known as coprime number .

Example - 

(1,2) , (2,3) , (24,25) etc .

(2) Tween Prime Numbers : 

The difference between two natural numbers is two , is called Tween Prime number . examples

(3,5) , (5,7) , (7,9) ,(17,19) e.t.c

Questions : 

(a) . The  alternative natural numbers are  always Composite number (True/ False).

(b) . The alternative Composite numbers are always composite number (True/False) .

(C) . The alternative odd natural numbers are Composite number (True/ False) . 

(d) . Multiply of three natural numbers is  divisible by six (True/False) . 

Answers :

(a) . True 

We know alternate natural numbers are 

2,3,4,5,6,7,8,9,10,11,12,

by defination (2,3),(3,4),(4,5),(6,7) are composite numbers because their H.C.F is 1.

(b) . True 

alternative composite numbers are (2,3),(3,4) we see that these are composite .

(C) . False 

Alternative odd Composite numbers are 

3,5,7,9,11,e.t.c

Now 

(3,5),(5,7)   we see that the H.C.F between 3&5 are not one hence these are not a composite numbers .

(d) . let the three alternative numbers are 3,4,5  and the resultant of their multiplication is 60(3×4×5)

Now by question sixty should be divisible by 6 

60÷6 = 10 .

Similarly 

21,22,23 & their multiplication is 10626

10626 ÷ 6 = 1771 .

 Here we see that the multiplication of three alternative numbers are divisible by 6  so the answer will be " TRUE" .

TYPE OF NATURAL NUMBER

Natural Numbers are divided into three type - (1) Prime Number           (2) composite Number, (3)Neither Prime nor Composite Number.               

(1) Prime Number : The natural number which are divisible by one(1) and himself only is none as Prime number. Example of prime numbers

2,3,5,7,11,23,17,19,13,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,93,97 etc.

If we divide these types of number by any other number except 1 and itself we see that they can not be divided exactly .

(2). Composite Number : The natural number which has minimum three factors or more than two factors is known as Composite Number. Examples of composite number are -

4 = 1×2×2

6 = 1×2×3 

8 = 2×2×2

etc.

NOTE : 

(1) Most smallest composite number is 4.

(2) Most smallest even Composite number is 4 (1×2×2) .

(3) most smallest odd composite number is   9 (1×3×3) .

(4) 1×1×2  , 1×1×3 are the Composite number ?

The answer Will be definitely not . These types of number are not consider as Composite number . 

(3) . Neither Prime nor Composite Number : 

The natural number "1" is only a number in math which is neither Prime nor Composite number .

Do you know 29 september is celebrated as a World Heart Day .

About Heart day :  Heart day was introduce or founded in 2000 for spraying about heart disease . Each year it introduce different - different theme about heart disease . This year the theme of world day is Heart healthy environments .

2.  Today 28  September We are celebrating  as National Daughter's Day . Every year on fourth Sunday of September month in our country we are celebrate National Daughters day .

Good morning

Examples of irrational & Rational number :

A.

Which numbers are rational number ?

a. √2+√2 

b. 2√2

c. 2

d. √2×√2 

Answer : c&d

c. 

We know 2 is written as 2/1 which is in the form of rational number .

d. When two same irrational numbers are multiply with each other the resultant number is always a rational number thus 

√2×√2 = 2 which is a rational number.

a. When two irrational numbers  add to each other than resultant will also be an irrational number.

thus √2+√2 = √2+√2 is an irrational number.

b. When an irrational number is multiply with an rational number the resultant will be an irrational number.

2 is a rational number while √2 is an irrational number hence 

2√2= √2×√2×√2 = √8 which is an irrational number.

Question 2 : 

Which of in these two numbers  are rational & irrational  number and why?

(a). 3.14

(b). π 

Answer : 

a.  is a rational number because it is write as 

3.14 = 22/7 which is a rational number .

b. π is an irrational number because it is not write as a rational number 

π = 3.141592-------

.

Questions 3 : 

Which one of irrational value is large ?

a. √(4^2+5^2) 

here 4^2 = 16 (here sign "^" comes for power of any number like 4^2 means 4 raise to the power 2 similarly 6^3 = 216 etc ).

b. √(11+12^2)

c. √(6+6)

d. √(4^8)

Answer : b

b. √(11+12^2) = √(155)

 

a. √(4^2+5^2) = √(41) 

C. √(6+6) = √(12)

d. √(4^8) = 256 which is a rational number

In options a,b,c we see that root of 15 {(√15)}

is greater than another two option .

Solve the Questions:

1. Which is Natural number?

a. 9.76

b. 8/9

c. 5003

d. 2π

2. Find the lowest Natural number ?

3. 5^6/6^5 is an?

a. Natural number

b. Rational number 

c. Irrational number

d. None of these 

Rational Number:

When a number is written in form

                    P/q

where p,q is any natural number & 

q not equal to zero .

Irrational Number:

An irrational number are a real number that can not be written as a fraction.

It means not rational number .

Example of irrational number -

1. 

π is an irrational number because 

It's value is π = 3.1415926535 -----------

It mean that it is not written in the form of fraction .

2.

√2,√3,are the other example

3.

Base of natural logarithm (ln) e:

Often called Euler number is an irrational number

The value of e = 2.718281828459045235360287471------------

The e constant is divined as the limit 

e  = limit (1+1/x)^x

       x tends to infinite

The value of e is also divined as 1+1/1!+1/2!+1/3!+-------

Where ! Is the sign of factorial

Example of Rational number:

1.

1/3 ,2/6,3/4,16/68 are the rational number because all of these can be presented in the form of fraction .

Questions 1: difine which is an irrational number -

a. 3.14.                     b. √26

c. .33.                        d. .05 

Answer : b

Because it is not written in the form of fractional number

a. 3.14 is a rational number because it can be written as 22/7 which is a fractional form .

c. .33 = 10/3 .

d. .05 = 5/100 = 1/20 .

Natural Number :

All the counting numbers are called natural

number.

{1,2,3,---------n}

Whole Number :

When natural numbers are started with zero(0) then these numbers are called whole number.

{0,1,2,3,-----------n}

Point to be remember: 

1. All the natural numbers are whole number but all the whole number are not natural number .

2. Natural Numbers are group of - 

(a) Rational Number

(b) Irrational Number

(c)Whole Number

(d) Intizer Number .

3. Natural Number are represented as symbols N.

Question.

1. Difine which number is natural number

a.3\2

(b) .34

C. 34 

d. 0

Answer - C (34)

Solutions:

Option a : it is a rational number because it is a written in form of ratio or in decimal form (1.5).

Option b:

It is written in decimal number (because 3 & 4 comes after point).

Option c: 

It is a natural number because it is a combination of natural number as unit and ten did it (10×3+1×4 = 34 ).

Option d: 

Zero is a whole number.

Congratulations to all of you for di scovery of the most small galaxy M60 - UCD1 which has discovered by a team of Indian researchers .M60 - UCD1  IS place at 5.4Crore Light year . We live in Milkway and in Milkiway we can see 4 thousand stars from our eye but in M60 we can see 10 lakh stars from our eye.
light year - Distance travel by light in vaccum in one year is known as light year .
1 light year = 9.4605284×10^15M in S.I unit
Light year is used as an Astromical unit .
One light year is equal to one Julian year
Julian year : Julian year is a measurement of time difined as exactly 365.25 days of 86,400 seconds each .Julian year is the average length of year in the Julian calander in western societies in previous centuries.

In ganita he define sine table, Second order Sine Difference &Sine Mathematical Series ,quadratic Equation ,Compound Interest ,Ratio , Solution of various linear Equation ,Arithmetic and Alegebric topic included.

Today we know about the great mathematician Arayabhatta . He is born in 476,Ashmaka or Kusumapura India He is an astronomer also .He flourished in kusupura near Patliputra(Patna) then the capital of Gupta dynasty. He is particularly popular in south India at that time . Written in verse couplets this work deals with Mathematics and Astronomy. The work is characteristically divided into three sections - 
1. Ganita(Mathematics)
2. Kala - Kriya (Time&work)
3.Gola(Sphere)
In Ganita he developed the first ten decimal places and gives algorithm for obtaining square and cubic Root utilizing the decimal number system .The value of π is 1st of all given by him (62832÷20000) and developed 
that the value of π is goes up to infinite times,developed property of similar right angle triangle and property of two intersecting circles utilizing the Pythagorean theorem.

Hello 

You know all of you that 16th September organized as a international day of preservation of ozone layer ,which play an important roll for all human beings and save us from harmful sun light ,so let's go to save ozone layer. 

I am proud to be an Indian 

You are welcome in mathmtx