Ask HN: Teach me something new

There are two possible extensions of the rational numbers: the reals and the p-adic numbers. This is because of Ostrowski’s theorem which states that any norm defined for rationals, can either be the normal absolute value or the p-adic norm. For p being a prime, n and d being integers relatively prime to p and e being an integer, a rational can be written as r=p^e*n/d. And p-adic norm is p^(-e). Then in 3-adic numbers you can write 1/2 = 2 + 3 + 3^2 + 3^3 + 3^4 + …

which actually converges because the norm has the negative of the exponent.

pari/gp has a native p-adic calculator, where you can type 1/2 + O(3^5) and get the first p-adic digits.

I don’t know why this impresses me, but it is probably because there are just two ways of extending the rationals, and the p-adic numbers are like the lesser known brothers of reals. I feel, I should have known this for ages. And no, I don’t know if there are known applications, outside number theory, but still, its cute.

Almost everything around you has been designed by another person, or teams of people.

I've described this on HN before; forgive me if you've already seen it.

I have chronic migraine. It sounds worse than it is; I rarely suffer attacks anymore, and when I do, they're usually mostly harmless, as in the case I'll describe below.

Contrary to popular belief, migraine is not a headache. It's something a little more like an epileptic seizure. Intense headaches are one common symptom, but there are a lot of others.

One common symptom of migraine is called *scintillating scotoma*. A scotoma is a blind or blank spot in the visual field. A scintillating scotoma caused by migraine is an area of the visual field that is temporarily replaced by a vivid visual aura. Its appearance is commonly the first symptom of a migraine attack, and is often followed by more unpleasant symptoms.

I've seen scintillating scotoma many times over the years. A few years ago I was reading and enjoying a good book, and my scotoma appeared. It was a humdinger: roughly triangular prisms of white light with utterly black zebra stripes moving along them kaleidoscope-style. It took up the lower left middle of my visual field, pulsing and radiating and turning, covering the book in my hand.

I was disappointed. I didn't want to stop reading. I was enjoying the book. So I decided to keep reading until the scotoma made it impossible to continue.

It never did.

It became so vivid that I couldn't see my hand at all, but I still had no trouble reading the book. I even started reading it aloud without any difficulty.

After a few minutes the scotoma faded. In that instance, it was not followed by any other symptoms (that's happened more and more frequently over the years since probably my forties).

I could now see the book again, and could confirm that what I had been reading was indeed what was on the page.

I thought about how to explain my experience. The best hypothesis I've come up with so far is that the neurological process of seeing and the neurological process of being consciously aware of what I'm seeing are not the same thing. They're independent processes. The scotoma prevented me from subjectively experiencing seeing the book, but did not prevent me from actually seeing it, nor from correctly interpreting what I was seeing.

This experience (and one two other odd experiences) has led me to adopt the working hypothesis that many of our cognitive experiences are more complicated than we tend to assume, and that they're often made up of several more or less independent processes. We usually benefit if related processes pretty much work together, so they pretty much do. Because they do, we experience them all together as a single experience, but that's an illusion that unravels if circumstances screw up their synchronization.

I hope someone finds that as interesting as I do.

To be clear, this is Ctrl+Shift+T . I know you had T in capitals, but keyboard keys are normally written that way