Infinity is something not really understood by many but still used as if it were understood. The two most prevailing ideas are 1) it is a really big number, and/or 2) it’s a concept, not a number.
With regards to 1, this isn’t well defined. What is a number? Is it something you count on your fingers and toes? Then 0.5 isn’t a number. Is it something you define a range, say, all the numbers between 0 and 1, and then just extend positively and negatively to include things like “everything between -5 and +5” and so on? This is a bit better, but then you have issues like “well what is a number divided by 0?”
Ultimately, a number itself is a very ambiguous term, and it’s not a very well defined one within mathematics (or even by the average person. Go ask someone what a number is—if you’re clever, you’ll always have a “number” that seems to not be included in the definition). You might as well try and precisely define “red” as a color we see—you can’t, and you can’t even prove my red is your red, even if their light wavelengths are the same.
The best we have in the mathematical world is a set of various names of various types of numbers. Natural numbers are your counting numbers, 1, 2, 3, etc. Whole numbers also include the number 0 (this is probably the most ill-defined set for a few ugly reasons). Integers also include all the negative numbers. Rational numbers include any two integers that can be divided (anything divided by anything except 0), which basically says “decimals that stop or have a repeating pattern). Irrational numbers are only those numbers with decimals that have no real pattern or repetition. And real numbers include both rational and irrational and are basically the “parent” of everything. Its sibling is the complex numbers which include the grossly named “imaginary” numbers, or the even roots of negative numbers (like the square root of -1, the definition of the imaginary number i ).
With regards to 2, well, you’re on to something, but not really. You might as well hold up two apples and say “this apple is more apple-y than that apple.” What are you saying? Nothing intelligent, I promise you.
So where does infinity fit into all this? Well, we have two other types of numbers: cardinal numbers and extended real numbers. Extended real numbers are just a “convenience,” wherein we simply take the real numbers and just take them all the way to infinity. This is the idea of infinity most people are aware of. And there are clearly defined rules:
Infinity + Infinity = Infinity
Infinity * Infinity = Infinity
0 – Infinity = -Infinity
Infinity * -Infinity = -Infinity
Infinity / number not 0 = Infinity
Infinity + or – number = Infinity
-Infinity + or – number = -Infinity
However, there are two caveats when allowing such an extension:
Anything divided by infinity isn’t defined (an engineer will tell you this is 0. An engineer doesn’t understand numbers any more than a janitor does, I promise you. The difference is that a janitor knows this, and the engineer believes otherwise). Also, infinity minus infinity isn’t defined (and there’s no way around this).
Engineers came up with the concept, too, that a number divided by 0 is infinity (or a negative number divided by 0 is negative infinity). See the previous parenthetical for what engineers know about numbers. With regards to this, we know that 1/X as X approaches 0 converges to infinity (or negative infinity if it’s -1/0, or either if it’s any other number divided by 0). But limits and numbers aren’t really the same—limits give numbers, but they aren’t numbers. A bank gives you money, but it isn’t money, if you need an analogy.
The other type of number is the cardinal number, and cardinal numbers are perhaps the most important numbers for the real person (even if they don’t realize it), and in nearly 100% of situations, they are identical to normal numbers. Cardinal numbers measure things. They aren’t just mere numbers—they are actively doing something, namely measuring something.
How much liquid is in your cup? How much square feet is your home? How much more do you prefer to this show over that show? You may give a number (or an abstraction for that final question), but you’re measuring something, and this is cardinality.
Extended real numbers only have two infinities: a positive one and a negative one, and if you come up with a positive infinity and I come up with a different one, they are the exact same.
Cardinal numbers are different. How many counting numbers are there? 1,2,3,4,… Infinity, right? Well, now what if I count by 3s? 3,6,9,12,… Infinity of those, too! But they clearly aren’t the same infinity—one appears to be smaller than the other because it’s excluding some numbers (indeed, counting by 3s excludes an infinite number of numbers and is still infinity itself, which gives itself erroneously to the possibility that Infinity – Infinity = Infinity).
Take it a step further. How many numbers are there just between 0 and 1? Or 0 and 0.0001? An infinite number! Now what about ALL the numbers ever! Is this the largest infinity? (No, actually, there are infinitely many infinitely larger infinities than the infinity of ALL the numbers EVER. Isn’t that neat?).
We do know some things, though. We certainly know that the counting infinites can be ordered. I know that 3,6,9,12,...’s infinity is smaller than 1,2,3,4,5….’s infinity. I don’t know by how much (other than by infinity, which is useless), but “logic” and “common sense" make sense here. And I also know that all the numbers’ infinity is larger than these two.
The first two have infinites known as “alephs.” The 1,2,3,4,5,… infinity is known as “aleph naught,” which is a strange looking N (the aleph) and a little 0 under it (the naught). We have other alephs, like aleph-1, aleph-2, etc. for other counting things. The infinity that includes decimals, say, all the numbers between 0 and 1 (or any two numbers X and Y), we call this c. The letter c, specifically lowercase.
And you might think it crazy, but we know that all cs are equal. There are an equally infinite number of numbers between 0 and 1 as there are between 2 and 17, or -1000 and 47287827827871870. Here common sense eludes us, but this is a mathematical thing.
As for the big infinity for all numbers ever, we conveniently just call that one infinity with the normal sideways-8 symbol. This is the not the largest infinity, but it is the one most people are familiar with, so it makes sense to give it the same symbol as we give the extended real number infinity.
Now, with cardinal numbers, you can’t operate on them. I can’t add two measures. Do you think it makes sense to add 3 liters of water and 4 square feet? No, it doesn’t. You might say “Well, why don’t I add 3 liters of water and 6 liters of water?” But you’re not adding measurements anymore! You’re actually just adding the quantities 3 and 6, and then you’re “keeping” the measurement that they both share! “Why don’t we just add the quantities of 3 liters and 4 square feet and keep the measurement they share?” They don’t share a measurement, and you’re not allowed to assign a measurement arbitrarily (or do you really think you can just hold up 4 liters of water and say “this is 4 miles of water” and expect anyone to believe you?).
In fact, ANY time you’re adding measurements together, you’re actually just adding the numbers inside the measurements and then slapping the measurement to the answer. In math, this is 4 feet + 7 feet = (4 + 7) feet = 11 feet. This makes more sense than you might think!
Go back to kindergarten or first grade, when you learned to add and subtract. You have 3 apples, Timmy gives you 2 more. What did you do? You counted to 3, then you counted 2 more. You added the quantities first, and only after you were done did you bother putting on the measurement (here, “apples&rdquo on it.
So you can’t operate on measurements: no adding, subtracting, dividing, or multiplying. So infinity doesn’t mean much except as a measurement, and this is where the idea of “infinity is a concept, not a number” really comes from.
But infinity is a number.
The real question is, what on Earth is a number?
If you can answer “what is a number” in a very precise, rigorous way, you’re going to actually solve a lot of open problems in mathematics and philosophy that have never been solved before, some of them having nothing to do with numbers or math in the slightest. ...And people have tried very hard to answer this or a very similar question (specifically, “what is an object within a set of objects?&rdquo.
I’m sorry, but I don’t believe in you enough that you’ll actually come up with the answer.
Food is very hard to come by where I live. Not food food, mind you, but a quality and local restaurant. Where I grew up, you had a really diverse pick, though some places you had to drive out a bit farther to. But if you wanted a steakhouse, a seafood and/or oyster shack, a sandwich shop, unique pizza (my birthplace actually has our state’s #1 pizza according to every single list), Thai food, Mexican food, Tex-Mex mix, Japanese (hibachi or sushi or both), Chinese, Vietnamese, Mediterranean, Greek, Indian, Italian, French, Cajun, Irish, up-scale, low-scale, whatever you wanted we had, and we had both a variety of national chains and at least 2 of everything above that’s pure local. So far, the best Mexican food I’ve had comes from there, too.
But around here? We have three terrible Mexican places, a lonely single Chinese buffet and two non-buffet Chinese places, and “up-scale” (overpriced but generic) places. Everything else is a chain or it’s all fried or barbecue.
Admittedly, we do have this one Thai place and one Indian place, and while the Thai place is great, the Indian place is located in a cursed spot where nothing there ceases to exist for more than a year.
But if you’re not going to a chain or a bar, there are about a dozen unique places that all serve the same myriad of things, so nothing really stands out as unique. Long ago, this never bothered me—I more or less ate the same things all the time by choice. Now that I’m much older and I want to try new things, I’m forced to eat the same things all the time. You get tired of lemon pepper trout, chicken with “homemade sauces” (that everyone seems to share), that unique fruit-nut salad, or that uniquely cooked steak (that further reinforces the belief that the only good steak is one grilled at home).
So the natural solution, of course, is to drive out a bit farther and see what the neighboring city has to offer.
My ex and I did just that a few days ago. ...They have a Cracker Barrel. Yes, that’s it. Of anything unique or new that we hadn’t done before, they have a Cracker Barrel.
Admittedly, they do have the only place within 70 miles that serves oysters that aren’t fried (I don’t like oysters, but she does, and the place has some unique seafood options), but we’ve eaten there more than once before. Otherwise, it’s the same thing.
And I realize that this is what we as citizens encourage. I see a lot of really neat things that fast food joints of all places are offering around the world, and how the only fast food joint in America that seems to constantly update its menu is Taco Bell (which isn’t really adding but more permuting ingredients in new forms and calling it new), and I hear about how all these people want these worldly things, but clearly we don’t. We’re not spending our money on new things when they are offered to us.
Now, that’s not entirely our fault. Every large-scale multinational conglomerate of restaurants follows the same business model: expand the menu to attract and then keep a significant portion of the market share, and then cease upgrades to reduce cost. That’s precisely why Taco Bell just permutes ingredients and calls it new; there’s nothing actually new, so there’s no significant extra cost (and when things are new, you find out they don’t really stick around). And of course there are some exceptions to the rule via “branching,” such as expanding the menu with a breakfast, but this doesn’t really fall in the same business plan of reducing costs as making the decision to expand the menu or not does. That’s just trying to capture a market share.
But outside of that realm, yeah, it’s our fault. We’ve been consistently encouraged to desire convenience above all else. This is why we might whine but act largely complacent when it comes to nearly anything. Reality is, most people just aren’t whining; they’re happily spending money on the chicken tenders they love so much (even if “chicken tenders” has a new name, but it’s still referring to just the general disdain for trying out new things when you know you already like this other thing and are comfortable in its reliability).
And when it comes to fast food, I’m fine with that. Fast food should be a convenience and nothing else; I don’t want to get to the menu and have to think too hard about what I want; I more or less already have an idea in mind, and I’m happy to continue that mentality.
But opening up a local restaurant, serving the same things as everyone else around you, and expecting to steal a market share?
Good luck with that one. No wonder most of our restaurants fail after 3 years, and only nearly a dozen have actually started out small and expanded by moving to a bigger location over the past 30 years. We are a college town, so you can always grab those freshmen who haven’t eaten there and are going by word of mouth from their more experienced buddies “oh yeah man this place is the best.” But that only carries so far and definitely is a more short-term strategy than a long-term one.
Oh, well. It’s not too big a deal. Only so big a deal as to write a silly Mindsay entry wherein I just discuss some shower thoughts I had regarding the nonissue as a whole. Our city definitely has very intelligent business owners; that’s how some of our local places have stuck around for so long. But it’s about an 80/20 split on not-so-intelligent/intelligent owners. Most places it’s usually a 70/30.
I’m fortunately in that I only have about $20k in loans, and because I’d taken a previous loan long ago for my little Alabama stint and then paid it all off in full well before any deadline approached, I’m always guaranteed a subsidized loan each time I take a loan out for education purposes (little known fact!).
For those not aware, subsidized loans are fantastic. You don’t accrue interest until either immediately after you graduate or 6 months after you graduate, depending on the loan; the Stafford kicks in immediately after you’re no longer in school. The unsubsidized loans build interest immediately after you withdraw them, but if it’s a federal loan you of course don’t have to start paying until you graduate (just know the interest is still building).
I have 3 loans: 2 unsubsidized and 1 subsidized. I have no intention of paying the subsidized loan anytime soon for precisely what I stated above, but I did get rid of one of my unsubsidized loans removed thanks to this scholarship.
I feel good, and I’m well on my track to making sure I graduate without an ounce of debt and maybe some profit. Between this, clearing out my Amazon wishlist, paying my rent through January, putting a down payment on utilities that’ll last a good while, and probably going to be getting a new mattress (I’ve had mine for about 10 years now and I inherited it from my mom), who says money can’t buy happiness?
Well, at least up to a point, right?
Last night at around 5:30 PM, we recipients of a scholarship were required to meet for a sort of orientation. We introduced each other, learned of a few changes (this mostly was for returning recipients), and got several questions answered. As I’m a graduate student and one of only two in the room of about 30, most of it didn’t pertain to me because the rules for graduate students are a bit different (more lenient) than for undergraduates.
We also listened to a spokesman (someone part of an outreach program) from the National Security Agency who had been working within security there for several years (I learned a lot about the NSA during that talk, though primarily it revolves around the fact that the NSA is divided into three branches, and only one branch is the spooky “spy on everyone” branch).
In its entirety, the meeting lasted until nearly 7:00 PM. Quite a long meeting, but fortunately there won’t be another one for some time.
As for the scholarship itself, I was quite ecstatic to see a great sum of money in my account, and I’ve already paid rent through January, put down payments on my utilities, emptied out my Amazon wishlist (well, not entirely), and my last step is to get rid of one of my loans (I have three, though one I will not touch as it is subsidized, meaning it doesn’t accrue interest until I graduate). I expect to be debt free in three semesters, and then my goal will be to get a new car since my car is probably on its last legs.
Anyhow, for every year I take it (rounded up, so any one semester counts as one year if I just stop after one semester), I owe the government that many years of service. Basically a guaranteed job since the man in charge has a 100% placement rate. However, I found out that I can instead go work at any public state university for the three years I will be accepting the scholarship, so this is a huge plus for me!
I also have to maintain full time status, and classes have begun and are well under way as we are halfway through the second week. I always like to talk about what I'm taking, so let’s do that now! First, the list:
BQA 8333 Statistical Methods for Business
IE 8990 Special Topics in Industrial Engineering: Stochastic Optimization
ST 8253 Regression Analysis
The letters represent what department the courses come from (BQA is the department of business and quantitative analysis, IE is industrial engineering, and ST is statistics). The first letter determines the level of course, with 1-4 being for freshman-senior, 6 being a split level senior/graduate course (6 for graduates, 4 for seniors), 8 for purely graduate level, and 7 for masters theses. 5 and 9 are very special, reserved for undergraduates (5) or graduates (9) who are doing a combination of field work and research (basically, use your work to create a research paper). The last number is how many hours the course is worth, typically 3, though 2 and 4 are common for labs. The 0 on 8990 is always used for special topics, dissertation hours, or the like, as they are up to the professor to designate. The middle two numbers are tricky.
The third number represents a slot in the “sequence,” though these sequences do not always makes sense. Typically 1-4 refer to a Course I, Course II, Course III, and Course IV, but if there are no numbers after the course (like Calculus I, Calculus II, etc.), it may refer to a specific name. So I’m taking courses in a third, fifth, and “end” sequence, wherein a 9 is always reserved for special topics courses. Having said that, I have not taken any previous entries in any of those “sequences,” and what those sequences are I do not know. And they don’t necessarily follow by changing only that one number. For example, 1713 might be the first in a sequence that ends in 3253 (indeed, it is, as the sequence is 1713, 1723, 2733, 2743, and 3253, and I know this one because it’s a standard Calculus I-IV + Differential Equations sequence), so nearly all the numbers changed.
The second number is meaningless to students and is purely for departmental organization purposes. It refers to what “concentration” is. 9 and 0 are special in referring to ambiguous special topics and dissertation hours. But I know for the IE department, 5 represents the “industrial management" classification, and 4 represents the “operations research” classification. I don’t actually know how far they go, and the maximum number (excepting 9) is different for every department—the English department has only 3 concentrations, so its max is 3, but the Math department has 8 concentrations, so it has all the numbers 0-9 (because every department has 0 and 9).
Anywho, I think it’s time to discuss the classes themselves.
Stat Methods for Business: I took this class so I can take a later class in Data Analytics (“big data” to most people). I have a continuously growing statistical background, and this is a very remedial type course as it’s designed with the assumption in mind that the students have never done any statistics at all. And yet, this is so far my most interesting class; we keep things very descriptive, and because it’s a graduate business college course, there is a lot of research paper reading, and so I’m getting exposed to a lot of neat ideas and techniques. As implied, this is my favorite class so far, and I’m really enjoying it. There are 6 of us in there, and so things are kept very informal.
Stochastic Optimization: we learn how to optimize stochastic problems. For example, if we have so many acres dedicated to three crops, we need to minimize the total cost of purchasing seed, maintaining the land, and purchasing more crop if we don’t sell as much as we’re required to (an interesting, but realistic approach to the market), while maximizing profit. ...Only we don’t know how much we’re going to grow, so the output is random. When the output is random, we call this a stochastic process. This is taught by my professor, so I’m enjoying it, but it’s my last class in the day and I find myself awful sleepy. Right now it’s pitifully slow because we’re focusing so much on building blocks, which is a good thing, but I just personally don’t need it.
Regression Analysis: this is a very advanced course in regression, a tool used very frequently (and often incorrectly) with data and statistics. Basically, you have a bunch of points, you fit a curve to them, and you use this curve to predict future trends or observations. All courses I’ve had previously focus only on linear regression and simply how to calculate things. We’re going to do more advanced ideas and understand how they all work, so it’s a very mathematical course where I can take advantage of my background in math. However, I hate this course right now. Our professor is stuck in India and having issues getting back to the country, so we’ve had substitutes since Day 1. The first substitute was AMAZING and I’m going to be seeing what else she teaches to see if I can take more classes from her (I’m going for a Statistics minor, so I need 3 more classes in this department after this one). However, all the others since are awful, and the one we had Tuesday will be the one we have Thursday, and I don’t want to deal with that. The professors are smart, for sure, but they’re clearly not interested in being there, and lack of enthusiasm is really disgusting in a higher level math course and can easily break future interest in it. And that already has happened with some of our students. Few are statistics majors; we have four industrial engineers in there (three others plus me), seven agricultural students in there, only three statistics majors, and a few finance students in there. The agricultural students voiced their loud (and rude) opinions very clearly after Tuesday’s class.
Oh, well. I only have the Regression course that will be having continuous homework and exams; the others only have a final exams, infrequent homework, and a project. I’m expecting this semester to be a much needed mental/intellectual break.
In the meantime, I’m trying to stay positive. Not entirely hard to do; I got lots of neat things I’ve wanted for some time (thanks to the money), I’m in a good, quiet environment, and I’m at peace.
But some of these movies on my Netflix list are pretty terrible, and I can’t help but just sigh one too many times.