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.

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