The nice thing that happened in class today:
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@ersatzmaus @lapis @futurebird
The Romans used an abacus or similar for arithmetic. Or slaves that used other number systems.
Roman Numerals were only used to record numbers.
Read Georges Ifrah
"From One to Zero" revised as "The World's First Number-Systems" (The Universal History of Numbers 1).
The ancients knew about zero, but the big breakthrough was using it for place number system instead of a gap.
The Romans invented concrete, bureaucracy & some war machines. Most else was copied.@raymaccarthy @lapis @futurebird I'm aware, I was using the roman numeral system as an example. I did mention place-value as the key innovation.
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird
They could invent 0+0i, with a zero imagination
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@Unixbigot @futurebird damn warp core runs on taking the local warp constant and dividing by the number of seconds since midnight. It crashes at midnight with an F_DIV_ZERO error.
Kit Bashir (@Unixbigot@aus.social)
âWeâre out of warp, whatâs wrong?â âNothing, it happens every morning at this time. Just reset it. You havenât been getting that on B-shift?â âNo, and how longâholy crap!â âWhat?â âWarp degradation has added three days to our ETA so far. TELL ME if stuff breaks; if we miss the book sale on Rigel Four everybodyâs getting Curium ash for christmas.â #Tootfic #MicroFiction #PowerOnStoryToot
Aus.Social (aus.social)
@becomingwisest @Unixbigot @futurebird
We would've figured this out sooner had there been the usual warp core dump, but ever since StarshipOS 11.0 fricking systemd has hidden them away somewhere stupid.
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The nice thing that happened in class today:
Grade 5 students solve a puzzle where they put cuneiform numbers in order (there is no guidance, just work with the symbols, how do you order them?)
I told them they are like archeologists cracking a code. They did it!
"But where is zero?"
"It wasn't invented yet." I said this seriously. I mean ... it's true.Later that day the same student asked if it was a joke. I got to tell them no! Zero had to be invented. Everything had to be invented!
@futurebird
A good video on the history of 0:
https://youtu.be/ndmwB8F2kxA -
This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird I love how this math theory was first created a decade after NaN already existed on computers https://en.wikipedia.org/wiki/Wheel_theory
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@faithisleaping @futurebird I mean, nonstandard analysis and infinitesimals are a thing, so maybe they're just really forward thinking.
@eleanor
a really great thing that has an absolute conspiracy against it. -
@futurebird @SarraceniaWilds algebra with a "bottom element" is fun also
@somebody
Thats what she said.
@futurebird @SarraceniaWilds -
@futurebird @johncarlosbaez addition table, multiplication table, it doesn't matter, its an abstract operation. But yeah, I do call it "addition", not multiplication, at least when introducing this stuff.
I think I have a reasonably simple angle for introducing the symmetry group of the square, and that's (imperfectly) represented in the repo as it currently exists. You should print out the calculator front-to-back and play with it for a bit.
I have somewhat developed ideas about how to move from the intuitive approach of my mechanical number line for D_4 to implementing the arithmetic of D_4 using pencil-and-paper calculations. Namely, I think the semidirect product, the 2x2 integer matrix approach, and the permutation-based (i.e. subgroup of S_4) approach are particularly notable.
I don't know where I'd place the lesson on automorphisms, as honestly it need not depend on anything other than the intuitive approach. On the other hand, I'd probably want to prioritize at least one or two of the pencil-and-paper approaches to performing addition in D_4.
@futurebird @johncarlosbaez Anyway, my point is that "decoding alien hieroglyphs" is actually a pretty good way to start to get an intuition for what an automorphism is.
For example, if you were given all the alien hieroglyphs that correspond to their integers, and given access only to their addition table, then you'd be able to figure out what 0 is, as any identity must be unique, if it exists. Similarly you'll be able to figure out which two hieroglyphs correspond to ±1, in the sense that you could call one of them "1 up" and "1 down", and from there figure out what "2 up" and "3 down" are, but you'll never be able to decide if "up" correponds to positive and "down" corresponds to negative, or vice-versa.
This intuition is captured by the fact that the group of integers under addition has exactly one non-trivial automorphism: we can negate everything everywhere and things will still work out. (And in fact, this is the only such change that is guaranteed to work perfectly in all cases.)
Of course, if you then gain access to the alien's multiplication table, you can multiply "1 up" by "1 up", and that answer will correspond to the positive direction. Thus we can fully decode the alien's integers, which corresponds to the fact that the ring of integers exhibits only the identity automorphism: when multiplication is involved, we can't just flip everything's sign and expect things to work out.
This intuition is a bit hard to operationalize, though, as the addition tables are infinitely large. In reality, if the alien heiroglyphs are truly capable of expressing arbitrarily large members of an infinite set, such as the rational numbers, the notation must involve some regularity. That regularity can provide insight into the alien's interpretation of their rationals in ways that don't correspond to what could be learned from their operation tables alone.
The automorphisms of the group of rationals under addition correspond to multiplying by a non-zero rational number, capturing the intuition that you'll never be able to definitively decode the scale of the alien's unit of measurement from the addition table alone. But if you see something like 1/10000, you can guess it's probably not the unit, versus something much simpler like 1/1.
However, the field of rationals exhibit only the trivial automorphism, meaning that you could fully decode alien rationals from their addition and multiplication table.
Switching to a finite system avoids these complications, and also is capable of providing much more interesting examples of automorphism groups than your more widely-appreicated number systems can.
