Monthly Archives: July 2011

A misconception concerning maths

A common misconception about maths is that it is originally meant for practical usage in the real world. Nothing could be farther from the truth. Maths, in its true sense is an art, a real mathematician is far removed from any attractions the practical importance of what he does has. What he does, is exactly what other artists do: He has fun with maths! He plays. What does he play with? Musicians play with music, colorists play with colors, the mathematician plays with ideas. What kind of ideas? Not the ones inspired by daily practical life, most of them are far too boring! He leaves all that to the physicist. He instead plays with ideas created by his own imagination. Simple. He creates a reality of his own and starts playing in it.

For example, in his reality, he construes two creatures called 0 and 1 and imagines how similar creatures mate (aka add up): If he wants 1+1 to be 0 in his reality, he just goes ahead and imagines it. Thats it. Thats what maths is. And the reason the mathematician does maths is also not borne out of any “desire for scientific advancement of mankind”. The reason is simply that he does maths because he enjoys doing it. Like other art forms give new ways to human expression, maths gives one too. And like other artists enjoy this freedom of expression he does too.

By the way I agree certainly that there are practical uses of maths, but these result as a by-product and are not the original intention of doing maths by real mathematicians.

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Euler’s famous formula

A hand waving proof of Euler’s formula e^{i\theta}=\cos\theta +i\sin\theta starts from the following Taylor series:

\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots

e^{x}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots

Plugging in i\theta in this series and separating the real and imaginary parts yields the formula.

Here is a sketch for a more formal proof:

1. Consider the initial value problem \frac{dw}{dz}=w; w(0)=1. We solve it by formally setting

w=a_0+a_1z+a_2z^2+\cdots

\frac{dw}{dz}=a_1+2a_2z+\cdots

Evidently to satisfy the differential equation we must have a_{n-1}=na_n and the initial condition gives a_0=1. It follows by induction that a_n=1/n!.

2. The solution w is now redesignated as e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots.

3. We note that this power series has an infinite radius of convergence as (n!)^{1/n}\rightarrow\infty. By the existence and uniqueness theorem of differential equations, it is certain that this is the only solution of the given IVP.

4. The trigonometric functions are now defined by \cos z=\frac{e^{iz}+e^{-iz}}{2} and \sin z=\frac{e^{iz}-e^{-iz}}{2i} and it immediately follows that e^{iz}=\cos z+i\sin z. We thus have Euler’s formula.

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