intuition | Unjustified Margins

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$e^{\pi i} + 1 = 0$

$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

$\int_0^1 f(x) \, \mathrm{d} x$

$\displaystyle e = \lim_{n\to\infty}\left(1+ \frac{1}{n} \right)^n$

$\displaystyle e = \sum_{n=0}^\infty\frac{1}{n!}$

$e = 2.718281828\ldots$

The three values of “e” immediately above lead to a question. e is the base of natural logarithms. It is a “transcendentally irrational number”. It is an irrational number which is not the root of a simple polynomial equation like — $\sqrt {2} = 1.41421356 \ldots$ — which is just the solution to $x^2 = 2$ after all.

An interesting observation can be made. “e” is an irrational number but both of the presented equations which give rise to it are made up entirely of rational numbers. Why this is so is something every student of introductory calculus has to learn. Such students are often very discouraged because the ideas involved are counter-intuitive. Nobody sees the need to tell them the finest minds of the human race needed about 2000 years to get the answer. And the answer really is not all that “intuitively satisfying”.

This brings up questions of what “intuition” is and the boundaries (or margins) within which it needs to be confined if we are to avoid confusion. It also brings us to considerations of those attempts to understand reality which direct themselves too closely to the “intuitive”. We can also consider the idea of “identity elements” in mathematics. Identity elements are those values which do not change a result when a specific operation is carried out. The most common examples are 0 and 1.

0 is the identity element for addition and subtraction.

Adding or subtracting zero to anything does not change it.

1 is the identity element for multiplication and division. Dividing or multiplying anything by 1 does not change it.

And if we consider the Euler equation encountered earlier:

$e^{\pi i} + 1 = 0$

Many regard it as the “most beautiful” equation so far found. It combines addition, multiplication, exponentiation, and imaginary numbers along with two of the most important irrational numbers in an elegant form. “e” however has another interesting property.

When raised to the power of “x”, represents the “rate of change of e raised to the power of x”.

$\displaystyle \frac {de^x}{dx} = e^x$

Differentiation leaves $e^x$ unchanged.

Many articles and books have been written to explore the properties of “e” and the Euler equation.

Another digression can involve the recognition of “identity elements” which leave various values unchanged and how they can be related to dimensionality and boundaries themselves. A boundary can be argued as a place of “balance”, a place where there is “equivalence” or “identity” for a particular quality or quantity and on either side of the boundary a preference for one or another value can be found. This would also be a point of either stability or metastability. It would be in some sense “continuous” along an attribute or dimension. Another kind of boundary could be seen as discontinuous. It would have one set of attributes on one side and a different set on the other. Parallels between the physics of condensates and the physics of exclusion principles are seen here.

$V_{n}(R)=\frac{\pi\frac{n}{2}}{ \Gamma( \frac{n}{2} +1)}R^{n}$

$\Gamma(n) = (n-1)!$

Matrices are also going to be of use in this site:

This brief excursion began just as a way of testing how LATEX works in WordPress.

$\sqrt{-1} \:\: 2^3 \:\sum \: \Large \pi$

This post can now end.

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