All about the numbers

When a particular subject lights something up inside you, it’s worth taking notice. For me, one of those is the numbers – specifically the whole numbers, or integers. Thus was I from childhood drawn to mathematics, and later to Greek philosophy via Pythagoras. The former gave the outer mechanics of numbers, the latter suggested that numbers had a more mystical and imprecise meaning, leading to later interests in subjects such as numerology, and to astrology, where the numbers lurk in the background.

So I was a sucker for these two books which approach the numbers in completely different ways:

  • Music by the Numbers by Eli Maor
  • The Archetype of Number and its Reflections in Contemporary Cosmology, by Alain Negre

music by the numbersFor people such as me, Eli Maor has written an engaging book about the relationship between music and mathematics. The development of musical scales from Pythagoras to the early 20th century is an interesting story, reasonably well explained, from Pythagoras’s whole number ratios through the equal tempered scale exemplified in the work of JS Bach to the experiments of Stravinsky and Schoenberg.

The fascination still seems to lie in those magical simple ratios of musical resonance: the octave 2:1, the fifth 3:2 and the fourth 4:3, from which are derived the Pythagorean Scale, which is nearly ‘right’, but in the end not adequate for use in orchestras with different sort of instruments, as Maor explains. Always the whole numbers are beautifully simple, but prove too limited to describe the real world, hence the subsequent invention of all the panoply of mathematics, irrational numbers, imaginary numbers, the calculus and on and on.

And in the end, always and tantalisingly, the maths cannot fully describe the real world, which we know thanks to the insights of Kurt Gödel.

archetype of the numberAlain Negre’s book is about number as archetype – the qualitative aspect of number, which was revived in the 20th century by psychologist Carl Jung and physicist Wolfgang Pauli. All begins with 1,2,3, and 4 – just as with the Pythagorean scale. The qualities of these 4 basic numbers are explored and particularly related to the work of Jung, and to the triplicities and quadruplicities of astrology.

There are rather incomprehensible (to me) chapters relating the numbers 3 and 4 to current theories on the evolution of the cosmos – rather speculative, I think. Negre goes on to suggest that the astrological zodiac with the 12 signs is another projection of these number archetypes, including discussion of the axis crosses and the oppositional polarities in a chart of the 12 signs.

So the book is both familiar to me, in an astrological sense, and almost incomprehensible when relating to modern cosmology, which must be partly due to my own failure to keep up with this field. In fact, I had a similar reaction to an earlier work some years ago Number and Time by Marie-Louise von Franz. It feels like there is something important there, but the author has not quite managed to express it in a way that is easily comprehensible to me (of course this may be a commentary on me, rather than on the author’s work).

So yes, number still has that magical pull, but these books didn’t greatly enlightened me. Nor did they blunt that fascination with the numbers.

Music by the Numbers is much the more readable.

Fermat’s Last Theorem

fermat coverI was a sucker for this book, having been fascinated by the history of mathematics from an early age. As Simon Singh’s book Fermat’s Last Theorem explains, the origins of this theorem came from the early days of mathematics, with Pythagoras in Ancient Greece. Everyone knows Pythagoras’ theorem that the sum of the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, e.g:

32 + 42 = 52

In fact, it was eventually demonstrated that there are an infinite number of triples of integers x,y,z for which

x2 + y2 = z2

Mathematicians puzzled for centuries as to whether a similar equation might be possible with higher powers of any integers, i.e. cubes, power of 4, 5, 6,…

xn + yn = zn

Pierre de Fermat was a supreme mathematician of the 17th century, who worked largely alone, rather than with colleagues. When his work was subsequently examined he was found to have made major advances to mathematics in a number of areas. In particular there was a famous note written in a margin that he had found ‘a truly marvellous proof’ that there could be no instance where such an equation was possible, yet there was insufficient space in the margin to explain it. This became a challenge to all the top mathematicians since then.

Simon Singh takes us through much of the history of mathematics in recounting the development of efforts to solve what had become known as Fermat’s Last Theorem. And a fascinating tale he tells, with potted histories of the involvement of many leading mathematicians over the centuries – including the story of the 21-year-old Frenchman Evariste Galois, who jotted down what proved to be key insights during the night before he was shot and killed in a duel early the next morning.

Finally there came the assault by Cambridge mathematician Andrew Wiles, working for some years in a solitary fashion similar to that adopted by Fermat himself. Finally, in June 1993 Wiles outlined to a packed meeting of leading mathematicians a proposed argument that demonstrated that Fermat’s Last Theorem was true. But this was only a prelude to drama, as a fault was discovered in the logic of his proof. It was not until October 1994 that Wiles and a colleague finally laid rest to centuries of speculation and completed their proof of Fermat’s Last Theorem.

Simon Singh makes the development of deep ideas in mathematics in some way accessible to us, even though we could never understand the detail. Few people do!

Featured image of Pierre de Fermat from Wikimedia Commons

Fibonacci Grape Pips?

2,1,2,2,3,3,3,2,2,5,1,2,2,2,3,5,2…

I was idly counting the pips in each grape off a bunch from E Leclerc (cf Tesco, Kroger). (It seems that France has not really caught on to the fashion for seedless grapes; most on sale had pips. Yes, they were more tasty.) My idle counting had spotted a potential ‘pattern’ – so far these are all Fibonacci numbers, and it is well known that Fibonacci numbers appear frequently in nature. Could it be…?? Then came the next sequence:

4,3,3,1,2,2,5,2,2,6

Now FOUR is not a Fibonacci number, so appears to be anomalous. Well, science does allow for anomalous results that don’t fit the current theory. Then comes the SIX. But here I notice two tiny black dots in the grape – putative pips that did not develop – which makes 8, another Fibonacci number. Maybe I’d missed a black dot with the 4?

So I can hang on to my theory for a while, until more anomalous data emerges. A rather trivial example of the scientific method in action? Of course, there are far too few results to draw conclusions…

Featured image by Thamizhpparithi Maari, via Wikimedia Commons

1089

Choose a 3 digit number whose first and last digits differ by at least 2.

E.g 853

Reverse this number

I.e. 358

Subtract the smaller from the larger

I.e. 853 – 358 = 495

Add this number to the reverse of this number

I.e. 495 + 594 = 1089

Why? (That’s the fascinating part of maths.)

Thanks to Alex Bellos’s book “Alex’s Adventures in Numberland” for rekindling my interest in numbers.

My demonstration of why 1089 is the answer:

hundreds tens units notes
start a b c a>c+1
invert c b a
subtract a-(c+1) 10+b-(b+1) 10+c-a +10 carry 1 in tens,units
simplify 9
invert 10+c-a 9 a-(c+1)
add 10 8 9 Carry 1 in tens

The Fibonacci Series

In my youth I was always fascinated by numbers. The Fibonacci series is one of the most interesting sequences of numbers, first mentioned by Leonardo Fibonacci (c1170–c1250), the leading mathematician of his era, who popularized the Hindu–Arabic numeral system in the Western World.

The series of integers comes from 1, the symbol of unity, followed by 2, an expression of duality. (Some people prefer to begin with two 1’s, sometimes preceded by a 0; the resulting sequence is essentially the same.)

Each subsequent number is the sum of the previous two, so we have:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…. (and so on)

I was always intrigued by the fact that 144 is the square of 12. Is it the only Fibonacci square, apart from the trivial case of 1 and maybe 0? This was long the subject of conjecture, but it was eventually proved that 144=12×12 is the only square of the sequence. Now 12 is a multiple of 2’s and 3’s, and occurs in many human systems – measurements, money, astrology, and so on. This seems somehow significant.

A similar conjecture applies to number 8 being the only cube of the series, which I believe is also the case but is far too complex a subject for this blog.

The Golden Ratio

The ratio of consecutive numbers of the series converges onto a number called the Golden Ratio, usually symbolised by the Greek letter φ (phi).

φ = 1.61803398874989484820… (and so on)

It turns out that φ is found all over the place when we measure nature and its patterns. For an excellent, but rather mathematical, overview see this fascinating post on the golden ratio on the blog ‘Rationalising the Universe’.

The Golden Ratio was seen as very important in the art of the Renaissance, and of course turns up in the work of Leonardo da Vinci mentioned in the preceding post.

In one of my particular spheres of interest, the golden ratio turns up as a key ratio in the psychological perspective on the house system in the astrological psychology of Bruno and Louise Huber. So it would appear that maybe the ratio pervades not only the outer ‘objective’ physical world, but also the inner ‘subjective’ world where it relates to space and time.

Another intriguing fact is that Golden Ratio is very similar to the ratio of kilometers to miles, e.g. 8 kilometers is approximately 5 miles. This is entirely coincidental. [Or is it?]