[anupam will return]

Okay peeps,

We are yet to have a thread on higher level maths.
In this thread we'll discuss :
1)differential and Integral calculus.
2)probability & Statistics.
3)Limits, Continuity and Discontinuity.
4)Convergence and Divergence of series.
5)Matrices & Determinants.
6)Pigeon Hole and other similar theorams.
7)Co-ordinate Geometry ( parabolas,hyperbolas,ellipses,pair of st.lines,etc)

and many more..

To start with .. one very simple question which was asked once in an IIM-C Interview..

if we take any big number and keep on taking its square root continuously (by continuosly I mean taking the square root of the result of the 1st computation and so on..)
,why do we end up with 1 ?
The same is true for any small number also.. take 0.2 and keep on taking its square root infinitely.. we end up with 1. I want to know the reason for this..

[anandv]

Obviuosly after some decimal places..the computer/calculator gives up the significant decimal digits....

Like it approximates 1.000000000000000000000000000000001 to 1

The vice versa is not true for the problem..Like 1 raised to any real power will always be 1

[anandv]

Prove that 0!=1

[Govi]

Originally Posted by anandv

Prove that 0!=1

From the defination of a factorial we can write for a number n

n! = n(n-1)!

n=1 gives

1!=1(1-1)!

1! = 1 (0!)

0! = 1

[The Raven]

Originally Posted by anupam will return

Okay peeps,

We are yet to have a thread on higher level maths.
In this thread we'll discuss :
1)differential and Integral calculus.
2)probability & Statistics.
3)Limits, Continuity and Discontinuity.
4)Convergence and Divergence of series.
5)Matrices & Determinants.
6)Pigeon Hole and other similar theorams.
7)Co-ordinate Geometry ( parabolas,hyperbolas,ellipses,pair of st.lines,etc)

and many more..

To start with .. one very simple question which was asked once in an IIM-C Interview..

if we take any big number and keep on taking its square root continuously (by continuosly I mean taking the square root of the result of the 1st computation and so on..)
,why do we end up with 1 ?
The same is true for any small number also.. take 0.2 and keep on taking its square root infinitely.. we end up with 1. I want to know the reason for this..

If we see mathematically effectively what is asked is lim n-> infinity a^(1/x)
This is equal to 1 as per the formula.

ut let me try to approach this logically. For a number bigger than 1, every time we take a square root , we have to get a smaller number. Assume smallest number possible is less than 1. This means the square root of a number greater than 1 ia less than 1, which is not possible. So smallest is 1. So every square root has to tend to 1.

By converse logic of suare root of a number less than 1 can not be greater than 1, that also tends to 1.

Does this sound convincing??

[anandv]

Originally Posted by Govi

From the defination of a factorial we can write for a number n

n! = n(n-1)!

n=1 gives

1!=1(1-1)!

1! = 1 (0!)

0! = 1

Not a very elegant way to answer !
But still you saved the day

Any more elegant answers please!

[chautauqua]

how about this..
nCr=n!/(n-r)!.r!
put r=n...
gives 0!=1

[anupam will return]

Good show there ...

What about.. "why is the mod of any number (excepting 0 ) always +ve ?"

[Govi]

Originally Posted by anandv

Not a very elegant way to answer !
But still you saved the day

Any more elegant answers please!

how about this n! is also the number of ways of arranging exactly n things and there is only one way to arrange zero things. Thus 0! = 1

[Mahip]

Originally Posted by anandv

Not a very elegant way to answer !
But still you saved the day

Any more elegant answers please!

The number of ways of choosing k things from a collection of n things is n!/k!(n-k)!.

For example, the number of handshakes that occur when everybody in a group of 5 people shakes hands can be computed using n = 5 (five people) and k = 2 (2 people per handshake) in this formula. (So the answer is 5!/(2! 3!) = 10).

Suppose that there are 2 people and "everybody shakes hands with everybody else." Obviously there is only one handshake.
But what happens if we put n = 2 (2 people) and k = 2 (2 people per handshake) in the formula? We get 2! / (2! 0!).
This is 2/(2 x), where x is the value of 0!.
The fraction reduces to 1/x, which must equal 1 since there is only 1 handshake.
The only value of 0! that makes sense here is 0! = 1.
And so we can elegently define 0! = 1.