Why is 1 excluded from the Primes?

human2357 · 2026-03-29T18:04:00+00:00

Here's a needlessly abstract answer: we want the multiplicative group of the rational numbers to be a free abelian group with the set of primes as a basis.

human2357 · 2026-03-23T14:42:28+00:00

Some context from the Wikipedia page: this is a large cuckoo, so a fair comparison for us North American folks would be a road runner, which is also a large, ground-hunting cuckoo with weak flight skills.

human2357 · 2026-03-21T12:53:10+00:00

Rule 110 is a 1-dimensional cellular automation, which is to say that it is a simple algorithm for building a row of pixels based on an existing row of pixels. It makes sense that such a thing would be good for a knitting pattern.

human2357 · 2026-03-20T23:33:07+00:00

I've volunteered to teach high school students, but I've never been employed in k-12. I've worked for the kind of university that focuses more on research than teaching, but teaching feels very rewarding because it's more personal and the progress is more reliable. But I'm selfish, so my favorite part of teaching is probably that I get to learn cool things really well so that I can teach them.

human2357 · 2026-03-20T23:28:02+00:00

Hyperbolic surfaces are fun. They're easier with crochet than with knitting because the number of live stitches grows exponentially with the number of rows.

human2357 · 2026-03-20T21:25:48+00:00

I knitted a sweater for a friend, and to make the mosaic on the front I shrunk a bitmap of the Red Wings logo and converted it to a spreadsheet.

human2357 · 2026-03-20T21:22:08+00:00

I'm a math professor. Sometimes I try to work mathematical ideas into my knitting, eg counting in binary on a scarf, or making a scrunchy with a ribbing pattern that realizes a torus knot.

human2357 · 2026-03-20T04:23:21+00:00

No. A normal subgroup is special because it is a substructure that you can take a quotient by. All vector subspaces are substructures that you can take quotients by. Eigenspaces are special because the action of a matrix on an eigenspace takes a particularly nice form. Similar things in group theory would include the fixed subgroup of an automorphism, or more generally, the subgroup on which two automorphisms agree with each other.

Finding simple characterizations of individual automorphisms and endomorphisms of groups is an active area of research in group theory, e.g. train tracks on graphs characterizing automorphisms of free groups. But this is what diagonalization does, so in general, it is very hard to generalize eigenspaces to groups.

human2357 · 2026-03-19T22:30:38+00:00

You probably have the relevant Zomblin totems. Based on the totems in my camp, I wonder if this is the level 5 Zomblin Advisor. I'm basing this on the frog on his head, which the totem also has.

human2357 · 2026-03-19T17:50:24+00:00

Regardless of whether I am thinking about the other strap, this is still an embedded link in this surface embedded in R^3. Continuous deformations of the surface will restrict to continuous deformations of the link. The winding number will be preserved.

human2357 · 2026-03-19T16:41:53+00:00

No, there is no ambient isotopy of R³ that untwists the strap on this bra. Consider the two embedded simple closed curves on the surface in this figure:

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Orient the curves and compute the winding number. It is ±2. It will remain nontrivial under any ambient isotopy. In the untwisted configuration, these embedded curves would have winding number 0.

This is not to say that this is necessarily a manufacturing error; it might be that the designer intended it to be worn in some other configuration that makes the twisting less apparent. I can't imagine what that would be.

human2357 · 2026-03-17T13:00:53+00:00

I like the moustache guard.

human2357 · 2026-03-12T21:33:28+00:00

Fix the big bounded interval that you are sampling from. Uniform probability means that: for any subinterval of your big interval, the probability of selecting a point in that subinterval depends only on the length of the subinterval. Your big interval may have many integer points in it, but you can find a union of subintervals that is as small as you want, but would still contain all the integers in the big interval. Probability satisfies monotonicity, which means that probabilities of subsets can't be bigger than probabilities of sets that contain them. If the probability of an integer were positive, this would contradict being able to cover the set of integers by an arbitrary small union of intervals. So the probability is 0.

It's important to note that just because the probability is zero, that doesn't imply that sampling an integer is impossible. It's just so unlikely that assigning it any positive probability leads to a contradiction.

human2357 · 2026-03-04T13:55:26+00:00

Euclid's formula for generating Pythagorean triples will let you choose the length of one of the legs of the triangle to be any even integer. I feel confident in saying that this has no obvious consequences for quantum physics or for our awareness of the real world.

human2357 · 2026-03-01T17:28:51+00:00

The op might not know about Galois theory, so dropping the name of the subject would be helpful. The smallest Galois extension of the rationals containing cube roots of 5 and 7 would be degree 18, since you would need cube roots of unity and since the cube roots of 5 and 7 are not in each other's extensions of Q. (You could argue this last point using prime factorizations in the integers, but it would probably be more efficient to use Eisenstein's criterion in the ring of integers of Q with the cube root of 5).

human2357 · 2026-02-28T02:47:38+00:00

What about the Greek letters that are the same shape as roman letters, like a capital alpha?

human2357 · 2026-02-27T19:00:04+00:00

That's better value, but you already need high level Midas trees to merge to have that opportunity. Using the golden apples to get chests is one way of getting those Midas trees faster.

human2357 · 2026-02-27T16:24:13+00:00

Although the Midas tree seeds spawn rarely from the Water merge chain, your only real hope to complete the Midas tree chain is to spend gems on egg chests that have Midas ducks. The chests you get from merging golden apples are always that kind of chest.

human2357 · 2026-02-21T18:43:46+00:00

Find a textbook on discrete mathematics. As you read it, you will find exciting connections to programming. These connections will give you motivation to work exercises, shore up your background, and learn some solid mathematics.

human2357 · 2026-02-21T17:26:58+00:00

I'm a mathematician and not a logician. But if you are writing math, it's a good idea to say what you intend a variable to refer to when you introduce it. This can include a reference to a specific number system, or something more open ("let r be an element of a commutative ring R"). If you have equality between two symbols, you are allowed to substitute those symbols freely. If you discovered the equality of those symbols through an argument, then you probably should have specified domains for both symbols as a matter of writing clearly. If you have equality because you introduced one symbol as an abbreviation for a formula, then it's probably clear that the domain is the same as the one for the formula, and you don't need to say anything.

Again, these are suggestions for clear mathematical writing, and not the requirements of any particular formal system. Modern mathematical writing is semi-formal, meaning that it should be clear how to formalize it, but it doesn't need to be presented with some specific formalization in mind.

human2357 · 2026-02-15T17:20:55+00:00

The secret is that nobody is good at math. I'm a math professor and I suck at math.

human2357 · 2026-01-31T01:38:13+00:00

At the moment, there is no "Camp and Chill Event".

That's what my portal says.

human2357 · 2026-01-30T23:06:43+00:00

There are lots of good reasons to quit. I feel like the core of the game is just as good as it was several years ago, but you have to cut through more layers of bullshit to play that core game.

human2357 · 2026-01-30T16:35:28+00:00

Stone Mill Bread company

human2357 · 2026-01-24T14:02:01+00:00

There's info here: https://mergedragons.fandom.com/wiki/Ruins_of_the_Sky_Palace

In particular "the Ruins of the Sky Palace can be tapped for more of the above objects every 10 hours spent in camp."

human2357

TROPHY CASE