Leap years wouldn't effect them, since they are invention of humans to reconcile the fact the actual year doesn't have an integer amount of days and yet we kind of want an integer amount of days to have a workable calendar. i.e. Each year is actually 365.25 days. But a regular calendar year is 365 days, so every 4 years our calendar starts to diverge from Earth's actual rotation so we add an extra day every 4 years to resync it.

Like the other reply mentioned, leap years probably wouldn't effect it, since the bugs are on year-based cycles and not 365-day-based cycles.

Yeah it would be cool to visit one of those places at the right time. As I mention in the episode, these particular broods have notable traits about their size, which is why the 221 year cycle is extra notable.

@@ComboClass There may not be much to see, at least not much more than a normal emergence (which is already plenty). The overlaps are generally pretty small, so even when they are forecasted, they don't always happen in the same place and they also don't necessarily happen at the same time. Each brood is synchronized based on temperature, but not necessarily 2 different broods that are in different cycles, so the emergence may happen twice in a small area rather than having a double emergence in one spot. This one I think was in a small corner of Eastern Illinois and the next one is in North Carolina if I remember correctly.

Double emergences can also happen occasionally unexpectedly when a brood doesn't emerge when predicted for whatever reason.

oh shit grade -2 is finally ending??? damn i remember grade -1 ending a year ago. i cant believe we are nearly at the end of the next one, and damn its taken so long to get here

@@tristantheoofer2 This grade is taking a little longer than a year since I wanted to do 36 main episodes like last time, but a lot of these episodes were longer (among other factors). Also, I decided I wouldn't cut my hair/beard at all during Grade -2 as a personal experiment, so soon I'll finally get to shave haha.

@@ComboClass I think the Catweazle look contributes to the atmosphere of strange and unlikely events happening while you explain the world to us peasants. The gods must be angry with you for casually revealing their secret knowledge.

@@ComboClass yeah ive noticed the hair hasnt been cut at all (and being real i kinda envy it lmao). and i was also thinkin that there would be 36 episodes like last time so that makes sense :P

1/(an aleph number) is not a defined quantity on the real number line. You can't take the reciprocal of an aleph number in the same way as we are used to doing with a real number. We typically assume that every number non-zero (n) has a version (1/n) in its system, but aleph numbers aren't the same. Some branches of math allow that sort of thing, but those systems are different and lose many typical properties of arithmetic we take for granted. When proving things about .999 repeating we are typically discussing real numbers, not systems that allow infinitesimals of that form.

@@ComboClass Here is proof that ℵ0 ∈ N. The cardinality of the set {1} is 1, the cardinality of the set {1, 2} is 2, and the cardinality of the set {1, 2, 3, ..., N} is N. and the cardinality of the set of natural numbers is ℵ0 (aleph-null). Thus, ℵ0 is a natural number. we know that all natural number are real number . Because ℵ0 is a number in the real number system, 1/ℵ0 is a real number as well ( not zero) . Since ℵ0 < ℵ1, it follows that 1/ℵ0 > 1/ℵ1, which implies that the claim of no gap in the real numbers is false. Therefore, there must be a gap between 0.99... and 1; thus, they cannot be equal in the real number system. So, if this video is correct, which is my opinion, it is 100% accurate and the best one I have seen in years. It proves that 0.99... cannot be equal to 1 in the real number system, although that may not be your intention but it did.

Not sure what you mean with that, since 1 is certainly being used as a number here, not just a symbol. When we say things like (1+1)(1+1+1) = 6 here, it is not true that (e+e)(e+e+e) = 6, and if you mean that in general (a+a)(a+a+a)=6a, that's true but is really just describing that (a)(1+1)(1+1+1) = (a)(6) which is just a multiple of the equation using 1's. You can multiply most equations by a random variable, but that doesn't mean that their simpler form using 1's isn't using numbers.

@@ComboClass (Your series is one of the best btw, but I am always a dark critic.) Yes, quite right, you used 1 as a number, I reminded myself of that after I posted, but I let it ride because when you first asked "how many one's ...?" (if I remember right) I saw a fly in the ointment, and wondered if that fly had infected the whole jar. "One" standing alone, is a monad (see Liebniz?) , and not countable as there is nothing to distinguish it as separate. (How do we include THAT in the number series which is, after all, a spectator perspective?) So we cannot then speak of "many one's". If we insist that "one" is countable, then there are consequences, including the one where you said that by replacing 1 with e we would get a different addition result. But on that point (there are others) I'm in stormy seas, the timbers creak. There is another reality (ontology) underpinning mathematics, one that mathematicians rarely look at, but sometimes go there. But as Wittgenstein said, the philosophy of mathematics is not mathematical, and not everyone's cup of tea of course. Still, it can be very interesting. almost mystical. Thanks for responding.

I used proofs by contradiction, yes. The tricky part for a given type of number isn't to know that you want to use that technique, but to find the contradiction.

At 0:17 I'm next to an apple tree

That really fooled me! The leaves on your apple tree are 3-4 times larger than the leaves on mine, and look just like my orange brugmansia's leaves.

We have to choose which things we take for granted when writing a proof, because you don't want to have to re-prove every basic thing (like that 1+1=2, or a+b = b+a) within every proof. It's true that in this episode I somewhat took the "if a square is even so is the square root" as a pre-existing assumption, although I did briefly explain the logic that if any (x) was odd then (x^2) would also have been odd. If you wanted to prove that part within the proof instead of using it as an assumption, you could use the definitions of even and odd number as 2k or 2k+1 for different integers k and do some arithmetic. But any proof will still have "missing steps" unless it includes dozens of other proofs that people consider already proven/obvious.