How a day, a month and a year is defined?

DAY:

A Day is defined as the time taken by earth to complete one rotation around its axis.

Earth takes approximately 23 hour and 56 minutes to complete one rotation around its axis. So one day of earth is taken as 24 hours. 

Now let's see how much time planets in our solar system take to complete one rotation around its axis.

MONTH:

Generally a  Month is defined as time taken by MOON to complete one revolution around earth which is approximated to 30 days.

YEAR:

A Year is defined as the time taken by earth to complete one revolution around Sun.

Earth takes 365.242  days to complete one revolution around sun.

Every year is considered as 365 days and to adjust 0.242 day which is approximately equal to 0.25 day, in every fourth year one day is added which is called Leap Year.

This extra day is added to the month February, which becomes 29 days. 

see, how long it takes by the planets of our solar system to complete one revolution around sun (in earth days)

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PIN Code and what does its each digit mean

 PIN stands for Postal Index Number.

A PIN refers to a six digit code in the Indian Postal System used by India Post.

HISTORY:

India Post is a government-operated postal system that was established in 1854, this is the largest postal service or most widely distributed postal service in the world.

It had some limitations like manual sorting of posts and delivery of posts by eliminating confusion over names of similar places, and incorrect delivery addresses.

To simplify this PIN system was launched on 15th August 1972 by the Ministry of Communication. This system divided India into 9 different zones and each zones into many sub-zones. The first eight zones i,e. digits 1 to 8 are divided geographically and the digit 9 is reserved for Army Postal.

PIN STRUCTURE

Each digit of PIN code is assigned to a specific region/location. 

First digit of the PIN refers to the zone.

• 1- Delhi, Haryana, Punjab, Himachal Pradesh, Jammu & Kashmir, Ladakh, Chandigarh
• 2- Uttar Pradesh, Uttarakhand
• 3- Rajasthan, Gujarat, Daman and Diu, Dadra and Nagar Haveli
• 4- Maharastra, Goa, Madhya Pradesh, Chhatisgarh
• 5- Telengana, Andhra Pradesh, Karnataka
• 6- Tamilnadu, Kerala, Puducherry, Kerala
• 7- West Bengal, Odisha, Nagaland, Arunachal Pradesh, Tripura, Mizoram, Tripura, Meghalaya, Assam, Sikkim, Andaman & Nichobar Island.
• 8- Bihar, Jharlhand
• 9- Army Post Office(APO), Field Post Office(FPO)

Army Postal Service is a government operated postal service for the delivery of mail of army.

Field Post Office is set up at the time of war or military exercise.

Second digit indicate the sub-zone.

Second digit along with the first digit indiate a sub-zone or a circle.

• 11 - Delhi
• 12,13 - Haryana
• 14,15 - Punjab
• 16 - Chhatisgarh
• 17 - Himachal Pradesh
• 18,19 - Jammu & Kashmir, Ladakh
• 20 to 28 - Uttar Pradesh, Uttarakhand
• 30 to 34 - Rajasthan

• 36 to 39 - Gujurat
• 40 to 44 - Maharstra
• 45 to 49 - Chhatisgarh, Madhya Pradesh
• 50 to 53 - Andhra Pradesh, Telengana
• 56 to 59 - Karnataka
• 60 to 64 - Tamil Nadu
• 67 to 69 - Kerala
• 70 to 74 - West Bengal
• 75 to 77 - Odisha
• 78 - Assam
• 79 - North Eastern State
• 80 to 85 - Bihar, Jharkhand
• 90 to 99 - Army Postal Service

Third digit refers to a geographical region (excluding in the case of functional zone of Army)  called sorting District i,e. headquartered at the main post office of the largest city of the region.

Fourth digit represent the route on which a post office is located. The digit 0 is for offices which are nearer to the sorting district.

Fifth and Sixth digit together represent a specific Post Office starting from 01.

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Co-prime numbers

Set of  numbers that have only common divisor as 1 are called co-prime numbers.

Co-prime numbers are also known as Mutually prime number or Relatively prime numbers.

How to Check weather two numbers are Prime or not?

Consider any two numbers, if no positive integer divide both the numbers except 1 then the two numbers are called Relatively Prime or Co-prime.

Example1: are 15 and 22 co-prime?

Answer: 

Divisors of 15: 1, 3, 5, 15

Divisors of 22: 1,2,11,22

Since 15 and 22 don't have any common divisor they are co-prime.

Example2: are 20 and 34 co-prime?

Answer: 

Divisors of 20 : 1,2,4,5,10,20

Divisors of 34 : 1,2,17,34

These two numbers have 2 as the common divisors so they are not co-prime.

FACTS:

• Every number is co-prime with 1.
• Two even number can never be co-prime because every even number is divisible by 2.
• Every Prime number is co-prime to each other. 
• Any two consecutive number are always co-prime. Take any two number like (1,2), (2,3), (3,4)... (501,502).... are always co-prime.

                                                            

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Roman Number System

7 symbols are used to represent the whole number system in Roman Number System.

The seven symbols are given as:

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500

M = 1000

The value of the numeral is the sum of the values of the symbols.

like:

1=I

2=II

3=III

4=IV

5=V

6=VI

7=VII

8=VIII

9=IX

10=X

In this system the symbols are read from left to right.

You must have noticed IV and VI having same symbols represent different value. But why?

There are specific rules to write the roman number system.

• We can't write the same symbol more than three times.

    we can't write 4 as IIII. it's wrong . 4 is written as IV.

• When a symbol appears after a larger or equal symbol, then it's added.

VI= 5+1=6

XI=10+1=11

XX=10+10=20

•When a symbol appears after a smaller symbol, then the smaller symbol is subtracted from the larger symbol.

as

IV=5-1=4

IX=10-1=9 

LCXIV= (100-50) + 10 + (5-1)  = 64

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HCF AND LCM

Factors and Multiples:

If number a divided another number b exactly, we say that a is a factor of b.

In this case, b is called a multiple of a.

Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):

The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.

There are two methods of finding the H.C.F. of a given set of numbers:

1. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

2. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.

   Similarly, the H.C.F. of more than three numbers may be obtained.

Finding the H.C.F. of more than two numbers: 

Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.

Similarly, the H.C.F. of more than three numbers may be obtained.

Least Common Multiple (L.C.M.):

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of finding the L.C.M. of a given set of numbers:

1. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

2. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Product of two numbers = Product of their H.C.F. and L.C.M.

Coprimes : Two numbers are said to be co-primes if their H.C.F. is 1

HCF and LCM of fractions

1. H.C.F =	H.C.F. of Numerators
	L.C.M. of Denominators

2. L.C.M =.	L.C.M. of Numerators
	H.C.F. of Denominators

H.C.F. and L.C.M. of Decimal Fractions:

In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of Fractions:

Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

EXAMPLES

Exp: 1 Find HCF of 6/5, 12/25 and 18/35 .

Sol:

HCF (6/5, 12/25, 18/35) =

HCF(6,12,18) / LCM(5,25,35)

= 6/175 

Here, every fraction is exactly divisible by hcf 6/175. Exactly divisible means the quotient will be an integer, with remainder=0 like

6/5 ÷ 6/175 = (6*175) ÷ (5*6) = quotient= 35, R=0 . We can check with remaining fractions too.

Exp 2: Find LCM of  6/5, 12/25 and 18/35 .

Sol: LCM( 6/5, 12/25, 18/35 ) =

= LCM(6,12,18) / HCF ( 5,25,35)

= 36/5 

Here, 36/5 is exactly divisible by each given fraction. Like 36/5 ÷ 6/5 = (36*5) / (5*6) = quotient = 6, R= 0. We can check with remaining fractions too.

Exp 3: Find LCM and HCF of  1.20 AND 22.5 .

Sol: Given, 1.20 and 22.5

Converting each of the following decimals into like decimals we get;

1.20 and 22.50

Now, expressing each of the numbers without the decimals as the product of primes we get

120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

2250 = 2 × 3 × 3 × 5 × 5 × 5 = 2 × 3² × 5³

Now, H.C.F. of 120 and 2250 = 2 × 3 × 5 = 30
Therefore, the H.C.F. of 1.20 and 22.5 = 0.30 (taking 2 decimal places)

L.C.M. of 120 and 2250 = 2³ × 3² × 5³ = 9000
Therefore, L.C.M. of 1.20 and 22.5 = 90.00 (taking 2 decimal places)

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INAUSPICIOUS NUMBER 13?

        A popular maxim states that the best way to predict the future is to look into the past. History has repeatedly proven the number 13 to be a diabolical entity. sneaking in corners, ready to cause bad omens and accidents, so much so that the fear of this number over the centuries has compounded manifold to the point of causing looks of horrors at the prospect of residing at the "13th floor" of a building.

            The fear of this number has its own personalized jargon known as "Triskaidekaphobia". It is a combination of the Greek words "tris", meaning "three", "kai", meaning "and", "Deka" meaning "ten" and "phobia" meaning "fear".

* Apollo 13 was launched on 11 April and it underwent an explosion on the 13th of April (2 thirteens).

* Zoroastrian tradition predicts chaos in the 13th millennium.

*Another example is the Columbia Space Shuttle. This one went into space on 1/16/2003. Add all the numbers and you will get the number 13. During its re-entry into Earth, it exploded. All the crew members died.

* Many hotels in China and America don't have a 13th floor, After 12th, either they have 12 and a half or 14th. The same goes for the number of houses too.

* Even Microsoft considers the number 13 very unlucky and that is the reason why there is no

version 13 of Microsoft office. Version 12 is Microsoft Office 2007 and the next version Microsoft Office 2010 is actually version 14. Thus, the company skipped the number 13 altogether. 

But just because some superstition is blindly acknowledged, does it make it true? Does this number really deserve this notoriety?

** In ancient Greece, Zeus is considered as the thirteenth and the most powerful God. This thirteenth God seems to be associated with totality, completion, and attainment.

** The ladder to eternity has thirteen steps, on reaching the 13th step, it is assumed that your soul attains spiritual completion.

** 13 is a prime number, which means it cannot be divided by any number other than itself. Hence, Symbolizes qualities of incorruptible nature and purity.

** in one of the most powerful civilizations of history, the Aztecs decided to have 13 days in a week as they considered 13 to him an extremely lucky number. Each day was ruled by one God. And the God who ruled the thirteenth day was associated with mystery, psyche, and magic.

** As we all know 13 is the age of change or transition for every girl or boy. It is the age when children officially become teenagers.

** The US flag has 13 strips, that represent the union of 13 colonies to fight British rule, later these 13 colonies became the first thirteen states of the United States of America.

**The Thai New Year (Songkran Day) is celebrated on the 13th of April. It is considered to be a day of washing away all the bad omens, by splashing water on people, friends, and relatives.

** In Hindu mythology. MahaShivratri is celebrated on the thirteenth night of the Magha month. which is very sacred and a holy night for all the Shiva devotees.

** In the sacred book of the Sikhs, the "Guru Granth Sahib", the word "Waheguru" which means eternal guru appears 13 times.

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Ramanujan Number

1729 is called as the "Ramanujan Number".

This is also called as the TAXI CAB number.

The number which can be expressed as the sum of cubes of two different digits is called as the Taxi Cab Number.

example:- 2, 1729, 87539319 etc.

1729 is the smallest number that can be expressed as the sum of cubes of two numbers in two different ways.

There is an interesting story why these number called so.

Let's see.

Godfrey Hardy was a professor of Mathematics at Cambridge University.

One day he went to meet the great Indian Mathematician 

Srinivasa Ramanujan by a Taxi.

When Ramanujan heard that Hardy had come in a taxi he asked him what the number of the taxi was. Hardy said that it was just a boring number: 1729. Ramanujan replied that 1729 was not a boring number at all: it was a very interesting one. He explained that it was the smallest number that could be expressed by the sum of two cubes in two different ways.

This story is very famous among mathematicians. 1729 is sometimes called the “Hardy-Ramanujan number”.

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