Tag math

Me — Back to Audiobooks!

Kasey-Rayn 3 Comments

I’ve recently started going through audiobooks again—before that I had been listening to Jukebox the Ghost almost exclusively, and about 7 weeks later their charm is only just starting to wear off on me. So, I thought I would tune that down while I catch up on books.

Well, back at my old job at Target this worked pretty well. I worked night shifts, generally, and being cart attendant meant I was outside in the quiet dark a lot. So, two or three nights a week I’d listen to 4 or 5 hours of an audiobook and I made good time.

But now, I’m working full time and listening to audiobooks for pretty much all of it. At 1.4x speed, so, well, that’s about 50 hours of content a week, or 4 typical novels. Problem is, I only had about 6 books to catch up on, so here we are. I’ve still have 2 to go, and by the time I’m through with Friday I expect I’ll only have half a book left.

Now, in this circumstance my preference is certainly audiobooks, but if I kept at it at this rate, that would be over $200 a month of new books. Now, don’t get me wrong, if I was overflowing with money, I would love to do that. At 4 books a week, that’s over 200 per year, which, if I keep up this rate for all of my working career (which I certainly hope I don’t), would be about 8,000 books, and on the clock to boot. Nevermind the $80,000 that audio library would cost me.

No, no. Podcasts are what it will have to be for the foreseeable future. Which is fine, I have no shortage of these, either. I have Hello Internet, Welcome to Nightvale, and if I find them as interesting as a brother does, My Brother, My Brother, and I as well as the half a dozen other D&D campaigns he follows. I’m sure all of those podcasts combined is well over 1,000 hours, or, 18 weeks if I can make good time.

This is part of the reason why I love my job. It’s sort of complex—there’s a lot of nuance that goes into it, but for the most part lately it’s just been a lot of things that take a lot of time, I’m not bouncing around everywhere. I’m not talking to customers and I don’t need to talk to coworkers all that much, and so it’s a prime environment to listen to stuff.

I love that fact because it means I can multitask in the most efficient of ways: gaining money while also working on what I consider to be self-improvement. Maybe I’m a little crazy in thinking that podcasts are about improving the self, not a mere replacement for music, but that’s what I think of it as. Maybe less so the D&D related podcasts, but you get the idea.

In other news, the fact that I’m going to be listening to so many new things in the near future means more meaningful Review posts! I’ve just finished the first book of the Belgariad as well as the first book of Dan Wells’ Mirador series. By the time Tuesday rolls around, I expect to have at least another review candidate! Maybe the next few Thursdays will be more Reviews rather than Me posts.

 

Me — Quick Maths

Kasey-Rayn 2 Comments

I love stats and data, as you probably know by now, but I’m starting to realize just how intrinsic it’s become to my personality. I do quick math in my head just as idle thoughts.

And before you think that I’m saying this to sound smart, it really is very simple, stupid math. Yesterday my work just got an order of a bunch of postcards, and I had to transport them. So in the few minutes it took me to do that, I did the math: 17 boxes, each box has 5 stacks of 100 postcards each. 8500 postcards. Sometimes I’ll do the math differently just to see which way is fastest. Disregarding the 100 because it’s just adding zeroes, is it faster to multiply 10×5 and 7×5 and add them together? What about multiplying 3 stacks of 5 by five, with 2 remainder? (In other words, 3x[5×5] + 2×5).

My production manager has started asking me “What’s A times B?” and I’ll do the math real quick while I’m doing whatever.

The weird thing is that I don’t consider myself to like math. The class I hated the most in high school was physics, because I would plug in all the numbers into the equations and I would still get the wrong answer. It’s worth pointing out that I didn’t have the best teacher, but nonetheless. I also probably would have hated calculus even more, but I never took it.

The trouble with higher maths, for me at least, is that it becomes too abstract too quickly, and the visualizations and the datas start to turn into meaningless numbers. I don’t like doing pointless things, so if I don’t know what foiling polynomials does, what am I really learning?

I think math is at its best when it helps you better understand things that you couldn’t have figured out with standard observations. I can know that most established authors are far older than I am, but I can’t appreciate that until I gather data on all my favorite writers and calculate the average age of when they were first published (32.9 years old, by the way). And that math is easy! You just add up all the numbers and divide by how many numbers there were! Now I can do something with that information—like breathe, because by that standard I’ve got quite some time to figure myself out and get published.

I’m always confused when people don’t share my love for data. It’s just cool to see and understand the world better through objective means, how can you not appreciate that? As somebody whose entire goal in life hinges on my capacity to know and understand, data gives a very simple and tangible way of doing so.

A post hit my Reddit feed (from r/dataisbeautiful, as it were) of somebody’s heart rate as their significant other left the country. They calculated what moments correlated with which spikes, and as I’m looking at it I’m nodding my head, thinking yeah, I totally understand that feeling of seeing somebody for the last time. That rush of “Oh, no”, is your heartrate spiking to an intense degree, so just reading this and comparing it to the rest of the graph is really interesting to me.

How can anyone not love data?

Learning! — The Gregorian Calendar

Kasey-Rayn 3 Comments

Today I won’t be teaching you a thing you actually will ever need to know: just a concept I find quite interesting, and that is the issue of tropical years versus calendar years. So buckle up because its time for a little bit of a lesson on both history and math. (And because large numbers are involved I’m going to write things numerically rather than spell out the words as I usually do. For your convenience.)

As I probably don’t have to explain, one year is not 365 days. In this instance, when I say “one year”, I mean the time it takes for the Earth to make one complete orbit around the sun (i.e. First day of spring to first day of spring). This doesn’t take 365 days: it takes about 365.2421891 days.

As a result of this, making a calendar turns out to be pretty tough. With a calendar that only contains 365 days, you’ll start being further and further behind, marking the first of spring days before it actually occurs. (1 day behind every 4 years, to be precise).

In 46 BCE, Julius Caesar normalized the calendar by adding the leap year rule, which I’m sure you’re familiar with. So every 4 years, we add an extra day, and this largely solved the problem. With a calendar year being 365.25 days, the first of spring will remain the first of spring for a long time. With this system, it’ll take 128 years to be 1 day ahead!

Except, hold on, 128 years isn’t actually very long. Longer than pretty much anyone has been alive, sure, but about 1500 years later, the Julius calendar was an entire 10 days ahead!

This introduced the Gregorian calendar, named after Pope Gregory XIII. What’s the difference between the Julius calendar and this one, you ask? Well, it’s largely the same, but it takes out 3 leap days every 4 centuries. Specifically, the rule is this. If the year is divisible by 4, there is a leap year. Unless the year is divisible by 100 (ex. 1800,1900), in which case you do not add a leap year. Double unless that year is also divisible by 400 (ex. 2000), where you do add a leap year. Basically, we have a leap year every 4 years unless its the dawn of a new century. In most cases.

As a result of this calendar being implemented, we had to shave off some days on the calendar. October 5th-14th of 1582 never happened in most countries. Except a few countries don’t like the Pope (England), so they didn’t adopt the Gregorian calendar until 1752. This meant that for England and its colonies, September 3-13th of 1752 never happened.

The Gregorian calendar is what we still use to this day. With the new rules, its so accurate, we will only be 1 day off after every 3,216 years. So, while you won’t get the leap day you may or may not expect in the year 2100, you can rest easy knowing that the first of spring by our standard was the first of spring so many years ago.

As it turns out, the rotation of the Earth and its orbit around the Sun (along with some small but measurable factors) make calendars pretty complicated. In the year 2000, we calculated the tropical year to be 365.2421897 days long. Only 10 years later, that same number was calculated to be 365.2421891! Pretty close, but still noticeably different. It’s why mathematicians haven’t solved this problem after so many millennia: the number we’re trying to hit is one that is in constant fluctuation.

In any case, here are the two videos where I got this information. The first video, by StandUpMaths, focuses primarily on the calendar issue, whereas the second, VSauce, talks about how time works in general. VSauce in particular I find extremely interesting, and I’d highly recommend if you enjoy learning things like this!

 

Learning! — More Factoring

Kasey-Rayn 5 Comments

I’m going to give a disclaimer here and say that this was really hard to write out purely in word form. If you’re a visual learner, this will probably confuse you more than anything. It’s not easy to explain mathematical processes without visual aid, so while I tried my best here, I’m not actually sure how effectively I accomplished that. (I’ll also be breaking my typical rule of writing out numbers in sentences just to make things a little easier to read.

Often, in algebra, we have to both factor and expand equations of increasingly complexity. This post will assume you know how to use the “foil” method when expanding equations such as (x + 4)(x – 2). We won’t be talking about expanding today. Instead we’ll look at factoring binomial equations like x² +2x – 8 back into something like (x + 4)(x – 2).

Let’s start simple with x² + 5x + 6. The goal here is to foil backwards. We have to find two factors of +6 that add up to +5. The factors of 6 include 1&6, and 2&3. (Remember we have to look at the factors in pairs. We can’t take 2 and 6 because we’ll be left with 12, which isn’t what we want). 2 and 3 add up to 5, so there we go: (x + 2)(x + 3) =  x² + 5x + 6. This is because when foiling it, you’ll end up with x*x + 2x + 3x + 6, and when we combine like terms, this means we can simplify it down to x² + 5x + 6.

Let’s throw in some subtraction to make things a bit more difficult. How about x² – 3x – 28? Again, we look at all the factors of -28 here. In this case we have a few more: 1&28, 2&14, and 4&7. Now, we know one of these numbers has to be negative, because we’re looking for a -28 here. But we can pick which number we want to be negative, so it’s not too daunting. Our target is -3. This means that when adding these two factors together, we’ll be left with a negative number, so the negative will have to be the larger of the two factors. This leaves us with -7 + 4. After that, we can piece it together and get (x – 7)(x + 4) = x² – 3x – 28.

For a leap of faith, one last challenge: adding numbers to x². Let’s factor 3x² – 7x – 20. In a case like this, we have to look at the factors of both 3x² and 20. Obviously, the only two factors of 3 are 1&3. The factors of 20 are 1&20, 2&10, and 4&5. So how do we get all these numbers to add up to -7? This time around order becomes important. Now we have to multiply the factors by each other to get to -7, its not as simple as adding them up anymore.

So, multiplying 1&3 by a pair of 1&20, 2&10, or 4&5, we have to hit -7. One of the numbers in the factors of 20 will also be negative, and again, we want it to be larger than the positive factor. This time, we can solve by trial and error.

Let’s start with 1&3 * 1&20. 3*20 leaves 60, which is way too high for our target. 3*1 leaves 3, and 20*1 leaves 20. This leaves us with either -17 or +17 when we add them together, so the factors of 1&20 are out.

How about 1&3 * 2&10? 3*10 leaves thirty which is still way too big, so we have to multiply 1*10 and 3*2. This leaves us with 10 and 6, and whichever factor of 20 we make negative will leave us with ±4 (± means positive or negative, if you didn’t know).. Much closer, but we need a seven.

So, 1&3 * 4&5 could work. 1*4 and 3*5 gives us 4 and 15, leaving ±11. 3*4 and 1*5 leaves 12 and 5, which adds up to ±7! In order to get -7 specifically, we needed 12 to be negative. Since 3 can’t be negative in this case, this means that we have a -4.

Now, how do we use that information? Let’s back up to 3x² – 7x – 20. We now know that the factors of -20 are -4 and +5, but is the answer (3x + 5)(x – 4) or is it (x + 5)(3x – 4)? Well, for that we look back to the multiplication. In order go get -7, we needed to multiply 3 and 4, which means they cannot be in the same grouping of parenthesis. This means that the answer is (3x + 5)(x – 4) = 3x² – 7x – 20.

There are a lot of special rules and easier ways to learn these techniques than by reading how to do them, though. As I said last week, here is the link to the site that helped me relearn all of this. Though I’m not a math major or anything, I’m generally pretty good at math and am open to help anyone that needs it. There is also a foiling calculator that can solve the problems for you, but if you genuinely don’t know how to do it, use the calculator to check your work, not give you answers. Getting answers to something you don’t understand is the worst thing you can do because it will tell teachers you’re either ready for the next level or you cheated. That said, the calculator is very useful for checking whether or not you got the right answer, and if you already know how to work out the problems, it saves a lot of time on homework.

Learning! — Simple Factoring

Kasey-Rayn 5 Comments

It’s been some years since I’ve taken a math class, so I admit I did need to relearn a little bit, but factoring is actually really simple once you get the hang of it. It’s super important in pretty much any math class you take, so if you never understand it, life will be hard for you. So, if no teacher or tutor explained any of this in a way it made sense, allow me to give it a shot.

The baseline here is that factoring is taking out and simplifying complex equations. You can chop them up and organize them in a way that looks more orderly, so eventually you’ll organize “z³ – z² – 9z” into “(z – 3)(z + 3)(z – 1)”.

But lets start more simple. A factor is a whole number that, when multiplied with another number, makes a new one. So, 2 and 3 are factors of 6, because when you multiply them together, they equal 6. So if something multiplies into another thing, it is a factor of that thing. It’s worth noting that every number that can be multiplied into a larger number is a factor of it, so 1,2,3,4,6,8, and 12 are all factors of 24.

Which brings us to letters. “Letters don’t belong in math!” you proclaim, and you’d be right. Letters themselves hold almost no value in mathematics. But the letter itself is meaningless. “X” is a variable. The number can change because its “variable”: its a placeholder for a number we don’t know. It holds no linguistic value, because we aren’t referring to language here. We could just as easily use a drawing of a hippopotamus in place of the letter X. In math, letters aren’t letters at all, but convenient, understandable symbols. when we say A = L * W, it serves as both a sentence and a mathematical expression for “area equals length times width”. I could say “T = & * $” and get the same answer when I plug in the same numbers. We just use letters, or variables, because it’s easier to convey what each specific symbol means when we involve other people.

So, all variables mean in this context is “we don’t know what this number is”. The Area of that rectangle could be a lot of different things, but if you just say A, we know what you’re talking about even if we can’t quantify it with a number.

So, if we say “3x”, we’ll take that to mean “3 times whatever number X happens to be”. Without knowing what X is, we’ll have to leave the equation at that. We can treat “3x” as one number, though, because when it’s all said and done they would be mixed together.

So if we try to factor “4x + 8”, we are going to try to take out the common elements of both (or all, if there’s more than two) numbers. When we’re factoring, we want to simplify the equation as much as possible. So we can take out a 2 from both 4x and 8. We put that outside of some parenthesis, and we divide 2 from everything because that’s what we took out. This leaves us with 2(2x + 4).

But we’re not done. We can take out another 2 here, because they are still factors of both numbers inside the parenthesis. We’ll divide that from everything inside, and then multiply it to whatever is on the outside, leaving us with 4(x + 2). (If this doesn’t make any sense, sorry, it’s hard to describe numerical processes with words!)

If you’ll notice, we could have simply removed a four from both numbers in the beginning and saved ourselves the extra step, but we get the same answer in the end.

Now that we’ve got the basics, let’s throw in an extra step. Let’s bring in exponents. You probably know that anything squared (ex: 3²) is a number multiplied by itself. We can do that with variables, too! Let’s try factoring 6x² + 18x. We can easily take out the largest number, 6, and be left with 6(x² + 3x). You may think we’d be done, but we’re not. Once you start getting into more complex math, we’ll need to start factoring variables, too. If x² means x*x, then x² divided by x will just leave us with one x. So, from 6(x² + 3x), we can divide X from inside the parenthesis. x² divided by x will leave one x left, and dividing x by itself will just leave 1. Since 3*1 will just be 3, we can shorten this equation all the way down to 6x(x + 3).

“Hold on,” you say. “We’re still super far from being able to factor z³ – z² – 9z” into (z – 3)(z + 3)(z – 1).” And you’d of course be right, but in the interest of keeping things simple, we’ll stop here and continue on next week. Or, if you need to know how to do that now, here is the link to the website that helped refresh my memory.