Written mathema…

Written mathematics is a very analytic language in the sense of <a href=”http://en.wikipedia.org/wiki/Isolating_language“>isolating language</a> except we want to say that the grapheme-per-word ratio is small. Greek letters are overloaded. Nobody takes this seriously

That is, if written mathematics were a completely regular language.

It bothers me that no one is taking these jokes seriously: http://sitinajihah.blogspot.com/2008_08_01_archive.html

But I don’t think the cure is a new programme of language for mathematics — that usually fails, at least if the language is in the standard sensory gamut of written mathematics: few pictures, lots of squiggles. People are yammering about higher dimensional algebra when the “two dimensional symbolic manipulation language” contains scores of unwritten rules which are mostly stored in people’s muscle memory these days.

What’s bizarre is that the written visual language of formal mathematics is enormously context dependent.  Feynman was irritated by “log” being written in three characters because he felt they were separate variables.

For instance:

q(z) = \sum_{n=0}^{\infty} \frac{\sin(z^{n})}{}

is not a complete word. Later tonight I’m going to try to calculate the grapheme/word ratio of written mathematics and compare it with natural languages.

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fun little radical cancellation thingy

Starting with \sqrt[3]{a} +1 = \sqrt[2]{a}:

a + 3\sqrt[3]{a^{2}} + 3\sqrt[3]{a} + 1 = \sqrt[2]{a^{3}}

Substituting \sqrt[2]{a}-1 for \sqrt[3]{a}

I have no idea why the rendering is broken for this line, will fix later:

a+3[\sqrt[2]{a}-1]^{2} + 3[\sqrt[2]{a}-1] + 1 = \sqrt[2]{a^{3}}

a + 3[a-2\sqrt[2]{a} +1] + 3\sqrt[2]{a} -3 +1 = \sqrt[2]{a^{3}}

a + 3a - 6\sqrt[2]{a} +3 + 3\sqrt[2]{a} -3 + 1 = \sqrt[2]{a^{3}}

4a - 3\sqrt[2]{a} + 1 = \sqrt[2]{a^{3}}

4a +1 = \sqrt[2]{a^{3}} + 3\sqrt[2]{a}

What’s amazing here is that if we square both sides, we completely

eradicate all the radicals on the right side (eradicalize!)

viz \sqrt[2]{a^3}\sqrt[2]{a} = \sqrt[2]{a^{4}} = a^{2}

16a^{2} +8a + 1 = a^{3} + 6a^{2} +9a

And you get a single polynomial from this:

a^{3} -10a^{2} +a -1

throwing that into wolfram alpha, we find that

a = \frac{1}{3}\left(10+\sqrt[3]{\frac{1937-33\sqrt[2]{93}}{2}}+\sqrt[3]{\frac{1937+33\sqrt[2]{93}}{2}}\right)