A funny derivation

March 19, 2008 — Karthik

Every new scientist must learn early that it is never good taste to designate the sum of two quantities in the form:

1 + 1 = 2 \qquad (1)

Anyone who has made a study of advanced mathematics is aware that:

1 = \mathop{ln} e

1 = \sin^2 x + \cos^2 x

2 = \sum_0^\infty \frac{1}{2^n}

Therefore eq. (1) can be expressed more scientifically as:

\mathop{ln} e + \sin^2 x + \cos^2x = \sum_0^\infty \frac{1}{2^n} \qquad (2)

This may be further simplified by use of the relations:

1 = \cosh y \sqrt{1 - \tanh^2 y}

e = \lim_{z->\infty} ( 1+\frac{1}{z} )^z

Equation (2) may therefore be rewritten as:

\mathop{ln}{\lim_{z->\infty} \left(1+\frac{1}{z}\right)^z} + sin^2 x + cos^2 x = \sum_0^\infty \frac{\cosh y \sqrt{1 - \tanh^2 y}}{2^n} \qquad (3)

At this point it should be obvious that eq. (3) is much clearer and more easily understood than eq. (1). Other methods of a similar nature could be used to clarify eq. (1), but these are easily divined once the reader grasps the underlying principles.

Posted in General, Humour. 8 Comments »

totient?

February 2, 2008 — Karthik

Just thought of starting up a blog on math. I expect to put hear real arbit stuff on math that interests me and other math enthusiasts too!

Why totient? Basically wanted the title to be some glorious mathematical symbol. Then thought of functions. The totient function certainly seemed the most exotic one. Mustn’t everything beautiful in math have something to do with Euler or Gauss? I chose Euler for the moment, but I am sure to have interesting posts on all the other geniuses like Gauss, Fermat, Galois or Ramanujan, to name a very few, and maybe even Knuth(!) or Wiles.

But the question remains — “Did Euler ever envision applications in cryptography for his seemingly innocuous function?”.

Now to the introduction of the totient function:

In number theory, the totient \phi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n.

Posted in General, Number theory. Leave a Comment »