The comfortable routine I have settled into has been disturbed… in a good way… I think.
If I can preserve some of the routine today, I may be able to write up another post about trusses. This one will even be a real truss – a Howe roof truss.
Or, it looks like I have the mathematics for reading the parameters of sinusoids from their discrete Fourier transform… and, as a follow-up post, a nice real-world example. Both look fairly easy to finish off.
In the meantime, my alter ego the kid has finished reading – not working! – Oystein Ore’s “Number Theory and Its History”. It is an introductory level book… very smoothly written… available as a Dover paperback.
Although it had some unfamiliar mathematics in it, what I got out of it was that the 8-gallon puzzle goes back to medieval times.
Also in the meantime, my alter ego the graduate student has begun focusing on “unique factorization”. In fact, he’s been focusing on number systems which are not “unique factorization domains” (UFD). Instead of looking at unique factorization, then, he’s been looking at its failure.
What’s all this?
We should all know that any integer can be written essentially uniquely as a product of prime numbers. The word “essentially” is there because the order of the prime numbers in the factorization is not unique, and because we could replace pairs of them by their negatives. Another way to phrase it is that the product is unique up to order and units – and in this case, “units” are ±1.
This is an easy property to take for granted. Back in the 1840s someone thought he had a proof of Fermat’s Last Theorem – this mistake was in assuming that the special numbers he was working with were a UFD.
To me, the most fascinating thing is that some number systems do not have unique factorization. In addition, however, we discover that it is necessary to distinguish 2 properties of prime integers… because the 2 properties do not coincide in a number system which is not a UFD.
The following 2 properties of prime integers are equivalent, and either might be taken as the definition:
- A number is prime if its only factors are itself and 1 (up to sign).
- A number p is prime if p | ab (“p divides ab”) implies either p|a or p|b.
Once we want to move beyond the integers, we find it necessary to define instead
- A number is irreducible if its only factors are itself and “units” (numbers of magnitude 1).
It turns out that any prime is irreducible in any number system – but if unique factorization fails, then there are irreducibles which are not prime.
Then the property of unique factorization reads
- any integer (in the number system in question) can be written uniquely (up to order and multiplication by units) as a product of irreducibles.
One simple example of a collection of numbers that is not a UFD is those of the form
, where a and b are integers.
We have, you see
which would be unremarkable, except that 2 and and are all irreducible – so we see that 4 cannot be written uniquely as a product of irreducibles.
All of that is well and good… but….
I think I’m going to a lecture in a couple of weeks… about the Higgs particle… and I really want to know the Higgs mechanism – how the Higgs particle gives mass to all other particles.
I understand it in broad outline… if we have “spontaneous symmetry breaking” and invariance under a global phase change, then we create what is called a Goldstone boson. If, instead of invariance under a global phase change, we have invariance under a local phase change, then an initially massless particle combines with the Higgs field to acquire mass.
But the broad outline isn’t enough… I could get seriously distracted by this…. Actually, of course, I already have been… for most of yesterday evening.
We’ll see what happens.
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