The Magical Number 142857

So, what's the big deal about a number, you would say! Well, not if this number graces the cover of a book called "The Joy of Numbers" by Shakuntala Devi.

The great thing about this number is when multiplied by an integer you have an 86% chance of the digits just kind of shifting around. Look at this:

142857 = 142857 x 1
285714 = 142857 x 2
428571 = 142857 x 3
571428 = 142857 x 4

See the pattern? All the numbers on the left are just the digits 142857, just in different orders, BUT, 142857 x 7 = 999999.... An implication of the fact that 1/7 = 0.142857 repeating. There are tricks to make the trick continue working for much larger multiples... but you can figure those out on your own... does this have application? Definitely not, but it will help you remember the decimal equivalents of fraction multiples of 7.

1/7 = 14.2857%
2/7 = 28.5714%
3/7 = 42.8571%
4/7 = 57.1428% and so on...

It can be very useful in calculations involving a multiple of 7 in the denominator. BTW, remember the irrational number pi? It is approximated as 22/7, which is a rational number. A rational number must be terminating or recurring decimal. And can you now guess the exact value of 22/7? Yes, that's right...its 3.142857 recurring! Of course, pi value is non-recurring and non-terminating...I heard someone worked hard on a program to find its value to trillion places.

Now for some interesting stuff that I found related to this number...and I quote

"The tune "Further and In Between" takes its basis from the cyclical number 142,857. Cyclical numbers are actually the inverse of certain prime numbers. For example, the number 142,857 represents the first six repeating digits of 1/7 in decimal form (0. 142857142857142857...). Also, a cyclical number with n digits, when multiplied by any whole number from 1 through n, will generate a product consisting of the same digits, which in every case except for when the multiple is 1, will be in a different sequence. Mahanthappa took various forms of his chosen number (142,857; 428,571; etc.), strung them end-to-end, and assigned a musical pitch to each digit. Allowing some leeway in melodic direction—tweaking pitch as needed—and working in some funk-based, start-and-stop rhythms, he constructs a jagged, leaping melody as visceral as it is intellectual."

You may read the whole article here about Rudresh Mahanthappa who brought maths into the music room.

Concepts on remainder theorem

Fermet’s theorem

If ‘a’ and ‘p’ are two numbers such that p is prime number then

( a) ^p-1 divided by p will always leave remainder 1

For example 16^38 when divided by 7 then by ferments 16^6n divided by 7 will give remainder 1

further 16^2 divided by 7 will give remainder 4.

Euler’s theorem

When ‘a’ and ‘p’ are coprime and p is not prime number than a ^E divided by ‘p’ will leave remainder 1 , given

E = p( 1- 1/x) (1- 1/y)

Where x and y are prime factors of number, note we are not concerned with powers of x and y.

For example E for hundred will be
100 = 2^2 * 5^2

E = 100(1-1/2) ( 1- 1/5) = 40

So 3^ 80 / 100 will give remainder 1 by Euler’s theorem

ANNOUNCEMENT FROM IMI

hello,

IMI has announced their prerogative. Read futher to know more.......

The Delhi-based International Management Institute (IMI) will stop giving any weightage to an applicant's CAT score once she is selected for the interview stage starting this admission season, IMI Director Prof CS Venkata Ratnam. The weightage will be more for work-ex.

Relative Pronouns Explained

There seems to be a perpetual confusion in usage of relative pronouns among students. If one finds pronouns a tough ball game then relative pronoun seems totally out of the league. I guess, after going through the following post on usage of relative pronouns, life could be little smooth than it was earlier, in context to relative pronouns.

A relative pronoun links two clauses into a single complex clause.Now look at the following examples:

1. I met Akil who had just returned from Kerela.
2. I have found the book which I had lost.
3. Here is the bag that you lent me.

‘ who’ joins or connects two statements as well as does the work of a pronoun. ‘who’ relates to some noun ( here,akil) which is called its antecedent.

Similarly ‘which’ and ‘that’ are also relative pronouns.

Difference between ‘conjunction’ and a ‘relative pronoun’:

Conjunction just joins two clauses or sentences but relative pronouns joins two clauses and is used in place of a noun.

This is a house.
Jack built the house.

This is the house that Jack built.

Different pronouns are used for human and non human antecedents.

Relative prounouns can of 2 types:

(i) Simple relative pronouns: which, who, whom, that
(ii) Compound relative pronouns: whichever, whoever, whomever.


Simple relative pronouns

WHO Vs WHOM:

Who/ whom is used for human antecedents.

Who is used in subjective case.e.g.

This is the boy who works hard.
‘who’ is used for the subject( boy here).

Whom is used in objective case.e.g.

This is the boy whom all praise.
‘whom’ is used in the place of objective case in the sentence, note that the object in the sentence is ‘the boy’.

Relative pronoun is generally marked in its form. This survives only in who, which has a possessive case form whose and an objective case form whom.

Which and that have no possessive forms.

When a preposition is used the pronoun must be either whom or which and never that.


WHICH Vs THAT:

Which and that both takes non human antecedents and hence a frequent point of dispute.

That is used for restrictive clause and on the other hand which is used for non restricitve clause.

The actual job of identifying restrictive and non restrictive clauses.

A non restricitve clause is normally preceeded by a comma where as restrictive clause is not.e.g.

The builder,who erects very fine houses,will make a large profit.
( this is an example of non restrictive clause, a specific builder is referred here.)

The builder who erects very fine houses will make a large profit.
(this is an example of restricitve clause, refers to any builder who builds such houses will make profits.)

Compound relative pronouns

1.You may invite whomever you like to the party.
The relative pronoun ‘whomever’ is the direct object of the verb ‘may invite’.

2.Whoever broke the window will have to replace it.
‘whoever’ functions as the subject of the verb ‘ broke’.

3.I will read whichever manuscript arrives first.
‘whichever’ modifies the noun ‘amnuscript’ which functions as the direct object.


Read the above and relative pronouns would be definitely smooth sailing.

MAT on 2nd September

Check out the notification here

SNAP & IBSAT both on Dec. 16

SNAP, the single test for admissions into any of the Symbiosis group of institutes would be conducted on Dec. 16th, 2007 @ 2:00 p.m.-4:00 p.m.

For more details click here

For a detailed analysis of SNAP 2006(last year's SNAP) along with the estimated cut-offs etc. click here

IBSAT, the exam for entrance to any of the ICFAI group of schools is also scheduled on the same day albeit earlier in the day @ 9:30a.m.- 12 noon

For more details click here