Decimals of π in 10TET

The digits of π have been translated into music a number of times. Sometimes the digits are translated to the pitches of a diatonic scale, perhaps accompanied by chords. The random appearance of the sequence of digits is reflected in the aimless meandering of such melodies. But wouldn't it be more appropriate, in some sense, to represent π in base 12 and map the numbers to the chromatic scale? After all, there is nothing in π that indicates that it should be sung or played in a major or minor tonality. Of course the mapping of integers to the twelve chromatic pitches is just about as arbitrary as any other mapping, it is a decision one has to take. However, it is easier to use the usual base 10 representation and to map it to a 10-TET tuning with 10 chromatic pitch steps in one octave.

Here is an etude that does precisely that, with two voices in tempo relation 1 : π. The sounds are synthesized with algorithms that also incorporate the number π. In the fast voice, the sounds are made with FM synthesis where two modulators with ratio 1 : π modulate a carrier. The slow voice is a waveshaped mixture of three partials in ratio 1 : π : π².

Despite the random appearance of the digits of π, it is not even known whether π is a normal number or not. Let us recall the definition: a normal number in base b has an equal proportion of all the digits 0, 1, ..., b-1 occuring in it, and equal probability of any possible sequence of two digits, three digits and so on. ("Digit" is usually reserved for the base 10 number system, so you may prefer to call them "letters" or "symbols".) A number that is normal in any base is simply called normal.

Some specific normal numbers have been constructed, but even though it is known that almost all numbers are normal, the proof that a number is normal is often elusive. Rational numbers are not normal in any base since they all end in a periodic sequence, such as 22/7 = 3.142857. However, there are irrational, non-normal numbers, some of which are quite exotic in the way they are constructed.

A new kind of square root

The postmodernism generator has been translated to mathematics. Now there is a program called Mathgen that outputs nonsensical papers on the advances of mathematics, complete with theorems and references. It has certain idiosynchrasies that makes it easy to recognize its papers. Authors are often drawn from among the most famous names of mathematics, but usually getting the first initial wrong. Theorems and conjectures are generously attributed to pairs of collegues across history, often using centuries old personalities as authors of brand new theories. Who has ever heard of the Conway-d'Alembert conjecture? Well, now we have.

The tone is exactly as condescending as one might fear: 'Clearly' such and such result follows; 'as every student knows …', and what follows is invariably clear as mud. Proofs are safely omitted because they are 'obvious'.

All this remarkable research, those 'little known results', are published safely beyond accessibility in Transactions of the Kenyan Mathematical Society, South Korean Journal of Integral Category Theory, Iranian Journal of Homological PDE, and the like. Surely most of these publications cannot be found at your local library anytime soon.

It is not hard to generate plain gibberish by TeX. Begin with listing a few elementary symbols and operators:

const char *alpha = "\\alpha";

const char *beta = "\\beta";

...

const char *r_arrow = "\\rightarrow";

const char *sqrt = "\\sqrt";

const char *sup = "^ ";

const char *sub = "_ ";

Put all the symbols in an array, so they can be easily accessed and picked at random. Concatenate several of the symbols into a string and print it. With some luck, the symbol sequence will not break the TeX syntax. This doesn't happen by itself, so next one might like to do something more structured. Elementary functions (program routines, that is) that generate small expressions like x ∈ ℂ² or f : ℝ → ℝ are not hard to write.

This is a sample of the babbling that results from the mere concatenation of a few symbols and numbers without regard for syntactical rules:

Difficulties arise when operators are used, because these expect arguments. An expression should not end with, say, an empty square root with no argument, as the above formula seems to do. In fact it ends with \sqrt{%0^{\sum}}, but this is apparently beyond the wits of TeX.


The notorious mathgen paper Independent, Negative, Canonically Turing Arrows of Equations and Problems in Applied Formal PDE by M. Rathke contains a larger sortiment of abstruse mathematical symbols in hilarious combinations. Or what about the frequent use of various powers of zero? Already the first formula contains expressions such as 0^-4, 0⁵, and other meaningless entities such as tan(∞^-1). A judicious use of elaborate idempotent expressions may even accidentally result in a true statement despite the funny appearance.

The derivative of products

An elementary proof 

Knowing some important formulas by heart can be very useful, but if one knows how to derive them, it is no longer necessary to remember the formula. From reading math textbooks (many — or most of them?), one can gain the false impression that the process of deriving a formula follows the same sequence as the proof.

Here is an elementary proof that nevertheless involves some not so obvious steps. 

Suppose that

then the formula for the derivative is 

but how do we prove this? Although the proof is straightforward, it is perhaps difficult to remember all the tricks that are required and when to apply them. Here is a standard proof. First, apply the definition of derivative to the product of the two functions:

The next step is the crucial operation, at once trivial and far from obvious. We are going to both subtract and add u(x)v(x+h) and rewrite the ratio as

Now, who would think of adding two terms that sum to 0 into such an expression? This is an idea that doesn't make much sense at this point. Indeed, one needs to look a few steps ahead and see what it is going to be needed for. What follows are just some simple factorizations of terms.

Break out some terms to get 

then take limits and replace derivatives, and we are done:

Here, the simple formula seems much easier to memorize than all the steps of the proof. (In fact, you may impress your friends far more if you memorize Hugo Ball's poem Karawane than if you learn to recite the steps of this proof.)

It is highly misleading when formulas such as the above are just plainly stated and then concisely proven. This is most likely not how the formulas were originally discovered. Rather, one would observe a few instances of derivatives of multiplied functions and conjecture a formula. Then, starting from the formula as well as the definition of derivative, one would work backwards and find all the arithmetic manipulations that make the proof work.

Instead of learning a fixed set of steps that are used in particular proofs, one would probably learn a bag of tricks that can be applied in various situations. Then, out of this bag one can grab various operations that can be tried out, until something is found that leads the proof in a promising direction.