## Various Pet Names for Cacti

So at the hotel I stayed at from August 5-6, there was a courtyard area with a cactus garden. Yep, only in Arizona. (Okay, probably not just Arizona.) At the same time of this realization, I was trying to toy around with every single mode on my DSLR in clockwise order from the dial on the top, and the ‘close-up’ mode just so happened to be next on the list. So I played around with some settings, and…

I have no idea what the exact species of the above cactus is. If anyone specializes in cactus nomenclature, now is your time to shine.

The above close-up is part of a monster cactus, which is apparent when zoomed out all the way.

Unfortunately that is all I can give you regarding the fascinating cactus family of plants for now. Maybe I’ll stumble upon another cactus garden soon?

## A Road Trip, Day One

Yesterday (well, since I’m writing this post a little past midnight I should say two days ago) I landed in Pheonix, Arizona and the following morning my family (3) plus my mom’s friend and her two sons (+3=6) went on the first leg of our road trip across the Southwestern United States.

Leg 1 | August 6, 2016: Pheonix, Arizona > Grand Canyon West Rim

The car ride that should have taken four hours took more like six hours because of some stops along the way for lunch and supplies. Eventually, around four in the afternoon, we finally saw some semblance of the Grand Canyon (although it wasn’t yet).

Finally, after a lot of twists and turns, we got there. The six of us paid a total of around \$400 for entry, we boarded a bus, went on the skywalk (where we weren’t allowed to bring our cameras), got our photos taken very haphazardly by a “pro,” got our pictures back and all of them were shit, and walked out of the skywalk with quizzical, slightly disappointed looks on our faces. But the views were amazing. Off to the side were places where you could take some more daring photos, if you were brave enough to risk not falling off the cliffs to a painful death.

The clouds overhead were pretty and gave a unique feel to these shots that I’m sure were taken thousands of times from these exact angles every year.

The view was exhilarating. I wanted so desperately to be one of the black crows (more on that later) and just fly out into the natural sculpture of meandering river and be free. I wanted to be as close to the ledge as possible without actually being stupid, which prompted the following shot:

I’ll end this post here and start a new one because I don’t want to make these too long, but coming up: hunting for some crows and random close-ups of cacti.

PHOTOGRAPHY IS FUN!!

## First pictures

So previously I mentioned I got a new camera. I took a couple pictures at the train station (and one bonus picture in the city) and just toyed around a little bit.

Next to learn: what are all these fancy settings in my DSLR like aperture and shutter. how to manual. also long exposure photos seem really pretty and i want to try that sometime. also what the hell is post processing

## Elementary school math, again

Remember when you were five years old and you learned that one plus one is two? Remember when you were just about to enter middle school or sometime around there and you learned about finding x? And how you had to add some stupid constant on both sides or divide through by a variable and you were like, “What’s so goddamn important about this x anyway?”

And then you went to high school and finding x through simple addition and subtraction gradually got easier, but then you learned harder ways to find x for seemingly no reason. And if you got far enough in high school math you learned about some seemingly black magic where x was zero but apparently not really zero because of something called a limit?? What the heck?? And what were these fancy elongated S symbols?

And then you went even further than that, you went so far that you stumbled onto abstract algebra and realized that almost everything that you learned when you were little could be thought of as operations equipped on the set of integers $\mathbb{Z}$ with the algebraic structure of a group or ring.

Let me explain.

By algebraic structure I just mean that the integers (which math people like to call $\mathbb{Z}$) have some super obvious but also really nice properties that we’ve taken for granted in elementary school, because not all sets are created equal and $\mathbb{Z}$ just happens to be super privileged.

For example, there’s this property that for any two integers $a$ and $b$, adding them up will give you another integer. Really obvious, right? In fancy speak we say that the integers are closed under addition, or: $\forall a,b\in\mathbb{Z}, a+b\in\mathbb{Z}$.

Then, there’s the property of associativity which should be obvious to us since we’ve seen it jammed down our throats so many times in grade school. Formally, $\forall a,b,c\in\mathbb{Z}, (a+b)+c=a+(b+c).$

Next, there’s the property that there exists an additive identity 0 such that every integer plus 0 equals itself. Again, really obvious. In fancytalk, $\forall x\in\mathbb{Z}, \exists 0\in\mathbb{Z}$ such that $0+x=x$ and $x+0=x$.

Finally, there is the property that there exists an additive inverse -a such that every integer plus its additive inverse equals the identity, 0. This is obvious: -a+a=0, or a+(-a)=0, any way you slice it.

So these four properties being satisfied, as obvious as that is, defines a special object in mathematics called a group. A group is a set G equipped with some operation such as addition, multiplication, composition, rotation, or what have you; as long as that set with that operation satisfies these four group axioms. So when we try and solve for x in one-step equations like in seventh grade, what we were really doing was working in the additive group of integers. I won’t get into it now, but if we can also throw multiplication and the distributive property in the recipe of a group, we get an even spicier object called a ring.

So why define this horribly formal thing? Why do we need to call this the “additive group of integers?” Why can’t we just do our damn addition problems? Well, for starters, it’s freaking cool. The first time I saw this stuff my mind was blown. We just grow up assuming that this is just the way it is, and we add 5 on both sides to cancel the -5 in x-5 just because that’s how it’s done. We just follow a strict recipe for solving equations without knowing why. Well, this is the why.

Second, group theory can be applied to pretty much anything and everything that might have such a structure. The next basic additive group that we could talk about are the additive group of integers modulo n, otherwise known as cyclic groups because the integers just keep recycling around and around like numbers on a clock. The circle of fifths, an analog clock, the days of the week, anything that has a cycle can be isomorphic to the cyclic group of integers.

What the heck does isomorphic mean? The formal definition is kind of long to explain from scratch, but intuitively it means that two groups are isomorphic to each other if one group seems like another group in disguise. For example, the days of the week {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} equipped with the operation of “progressing to the following day” (yes, operations can be defined in an informal way like this!) is isomorphic to the more abstract representation of this group, $\mathbb{Z}/7\mathbb{Z}$, defined as {0, 1, 2, 3, 4, 5, 6} with the operation of addition and the added caveat that 7=0. (Formally speaking, this definition is horribly incorrect, but I’m just trying to put it in a way that makes sense.) For example, 8 is equal to 7+1, which is equal to 0+1, which is equal to 1. So 8 is 1 modulo 7. In the same vein, 25 is 7+7+7+4=0+0+0+4=4, so 25 is 4 modulo 7.

Why are they isomorphic? Because the former group (the days of the week) looks like it could be the same thing as $\mathbb{Z}/7\mathbb{Z}$. We can see this if we assign the element “Sunday” to 0, “Monday” to 1, “Tuesday” to 2, and so on all the way to “Saturday” being 6. With this one-to-one assignment (keyword being one-to-one! or, formally, bijection) we can see that “progressing to the following day” is the same thing as “adding by 1” in $\mathbb{Z}/7\mathbb{Z}$.

If you’re beginning to realize it (and start getting your mind blown), you’ll see that “If today is Friday, what day of the week will it be 4 days from now?” and “What is 5+4 in $\mathbb{Z}/7\mathbb{Z}$?” are the same question. Instead of counting “Saturday, Sunday, Monday, Tuesday!” we can say “5+4 is 9, which is 2 mod 7” and if we wanted we could see that 2 corresponds to Tuesday, as we had assigned earlier. Obviously this is a really easy example, but what if I asked what the day of the week would be 1000 days from now? You can’t count those days one by one ad infinitum.

You might be saying, “Well, only a stupid person would do that. A realistic person would break the 1000 up into multiples of 7, with the knowledge that every 7 days it’ll be the same day of the week. Then we get 7(142)+6=1000, so we’d just need to count 6 days instead of 1000.”

Dude. You just did the same exact thing as computing 1000 modulo 7.

Of course, cyclic groups are still pretty basic. There are a huge variety of groups that are insane in their own right when explored. But I’ll leave that up to you to explore, if this blog post sparked your curiosity even a little bit.

## I am a double agent (kind of)

Subtitle: I have so many things I want to do with my life and there’s no time to do any of them in time because I’m leaving for college in less than a month.

Why the title? I’ve been looking for a good proofwriting course that is pretty easy to get into so that I could have an easier time learning some higher-level math courses if the time comes. That search is finally over thanks to many of my friends that decided to go to Caltech instead of MIT (I will miss them so much and it makes me a little sad every time I think about it). Caltech’s curriculum is theoretical from the start, and it shows since they have an into math class that freshmen can take over the summer called Math 0.

The full class apparently comes equipped with video lectures and the ability to send assignments to graders that give super helpful, detailed replies and annotations. But I was only able to get my hands on the PDF notes and problems, which honestly is more than enough seeing as I’m having a lot of fun figuring out the problems on my own and I’m halfway done. If anyone’s interested in the PDFs, Caltech or some administration would probably shoot me, so just give me a private message or email me (found in my About page).

So another thing that I’ve really put off but I really want to do is review 18.02, something I touched on last blog post. Denis Auroux is honestly such a wonderful human being and I love listening to his lectures and his amazing accent and his perfect handwriting and speed-erasing. I’m really sad he’s not at MIT anymore (he teaches at Berkeley now). ASEing out of 18.02, a.k.a. getting full credit for it after passing an equivalent final exam, would mean I could pursue math a semester early like differential equations or linear algebra instead of reviewing what I already know (or maybe even take an introductory engineering course or other elective). I’ve gone through most of 18.02’s material once, and I took a course on the differential half of multivariable calculus last summer, so I’m hoping I can just pass by winging it at this point.

Other weird things that I committed myself to:  a real estate salesperson license. It shouldn’t be that hard, just a lot of time to learn the material so I can pass the exam. And I have to slog through a 75 hour timed course online before I take that. The good thing is I technically have until December until the online course times out, so I’ll be doing that when I have truly reached the pinnacle of boredom.

Speaking of boredom, some other goals and activities I’ve been trying to do when I have free time this summer: get a 99 on Oldschool. I have no idea why I started playing again (I checked in-game that I first made my account over ten years ago) but one thing led to another and me and three other school friends memed our way back into Gielinor. I’m talking we would Skype almost all day for a period of several weeks just grinding out the early game, fueled by our desire to simply keep up with each other. Now there is much less unity, but me and my very close friend still play from time to time. Since I’m at 87 fletching right now and it’s profitable (if only by a small amount) to fletch magic longbows until 99, I’m just going to string those anytime I have a few minutes of free time. Those minutes add up.

Another videogame-related activity that I seem to have lost all desire for: League. The only time I play is usually very late at night if friends are on. When I get to college, I’m assuming that I will get my ass handed to me with all the coursework, so that coupled with the fact that the game doesn’t really appeal to me much anymore means that I’ve weaned myself off pretty effectively. It’s still fun to watch LCS, though, don’t get me wrong. (Congrats to Doublelift for getting 1000 season kills!)

And finally, I got a DSLR. My first one, actually; it’s the Canon Rebel T6i. I literally just got it and haven’t even read the manual yet, and I’ve just taken a couple practice shots, but I feel like I’m already starting to fall in love. That coupled with two new whiteboards that I bought (a smaller one for my house and a huge one for my dorm room) would probably make for a revitalization in Youtube videos.

And I also want to learn Java further, but at this point when I finally get to doing that I’ll probably be learning it for my major anyway.

THERE’S NEVER ENOUGH TIME.

Okay, I’m going to end this blog post here because I really want to tackle this Math 0 problem: Let $a,b\in\textbf{R}$. Show that if $a+b$ is rational, then $a$ is irrational or $b$ is rational.