a performance dialogue

Thursday, December 6, 2007

Repeating decimals

Here's something that I find rather amusing about fractions.

Write each fraction as a decimal:
1/1 =1.000000... Repeats in 1
1/2 =0.500000... Repeats in 1
1/3 =0.333333... Repeats in 1
1/4 =0.2500000... Repeats in 1
1/5 =0.2000000... Repeats in 1
1/6 =0.1666666... Repeats in 1
1/7 =0.142857142857142857... Repeats in 6
1/8 =0.1250000000... Repeats in 1
1/9 =0.111111111... Repeats in 1
1/10=0.1000000... Repeats in 1
1/11=0.09090909090909.... Repeats in 2
1/12=0.0833333333333... Repeats in 1
1/13=0.076923076923076923... Repeats in 6
1/14=0.0714285714285714285... Repeats in 6
1/15=0.0666666666666... Repeats in 1
1/16=0.0625000000000... Repeats in 1
1/17=0.05882352941176470588235294117647... Repeats in 16
1/18=0.055555555555... Repeats in 1
1/19=0.052631578947368421052631578947368421... Repeats in 18

The point is that if p is a prime then
1/p expressed as a decimal will have a repeating pattern of
length N where N divides p-1.

[More generally if a number n=p(1)xp(2)...xp(k) then 1/n written as a decimal will have a repeating
pattern of length N where N divides (p(1)-1)(p(2)-1)...(p(k)-1). See for example 1/14.]

The interesting question is: which primes p do you get expansion where the repeating pattern of length N is equal to (p-1), the maximum length of a repeating pattern?

In the examples above this happens for 7, 17 and our prime 19. One reason I like the 19th step. Some of the primes after this for which you get this maximal repeating pattern are: 23, 29, 47, 59, 61, 97, 109.

Gauss made the conjecture that there are infinitely many primes for which this is true.

This is still unknown. We know that if the extended Riemann Hypothesis is true, then there are infinitely many such primes.

It relates to something called Artin's conjecture. Artin was a mathematician who did his mathematics in the 20s and 30s. I rather liked the following quote I found from Artin:
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasions the feeling of elation in realising that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications."

I was quite struck by this statement that mathematics was non-linear. This sense of suddenly seeing what was going on is quite true. One might proceed through a linear, logical argument. But then there is this moment when you suddenly see what's going on. Like someone switching on a light and revealing the architecture of the room. Does music have that moment? A non-linear experience amongst the very linear nature of listening to a piece of music?


kate said...

I can relate to the light being switched on experience had it alot learning new things generally especially ballroom dancing when your learning the steps its logical and you learn the pattern but theres a moment when the light comes on and you see the dance as a whole rather than one move after another. Also get this with books have read stuff that I can't really understand until certain moments in life happen then you get a realisation a sort of over view of what the author was trying to tell you.Unfortunately my light hasn't come on in the fraction room I don't understand how it can be amusing yet?

Dorothy said...

It is entirely in the nature of music that the linear becomes architectural. Rothstein penetrates this really well I think in his book 'Emblems of the Mind'. Many music primers teach kids 'notes', introducing one at a time, but much more effective is to teach shapes.

Many composers will say they 'hear' a piece of music all at once. But we have this experience as listeners or performers also, just as Kate is describing with dance, there is a point when you come to understand the shape of the piece as a whole.