Math, the Way it Should Be (Unit 3: Imaginary Numbers, Revisited)

It’s been a long time since the last math lesson.  I have been busy with other interests of mine, though I definitely haven’t forgotten about this whole math thing.  If you’re not sure what I’m talking about you can visit the first ever math post, the first post in the first unit, the second unit, or the last math post.  I have decided to do a minor detour in the curriculum to revisit imaginary numbers which I kind of brushed over in a previous lesson.  I do recommend reviewing that lesson if you’re unfamiliar with imaginary numbers.  I didn’t include this exploration of this topic in another unit because it doesn’t really fit in with the areas of mathematics that we’ll be headed into later (until you get into the really weird stuff) so we probably won’t be seeing them for a long, long while, if at all.  And since I can’t really cover this entire topic in one lesson and don’t want to spend more than that, I’ll just hit some of the highlights and things that I think are interesting about them.

One thing you’ll want to know about imaginary and complex numbers is about their modulus.  The modulus is akin to the absolute value of a real number.  Essentially, it gives you the distance the number is from 0.  That is, 4 is 4 away from zero, so $|4| = 4$.  In the same way, -4 is 4 away from zero, so $|-4| = 4$. If you think of a complex number as two separate vectors on the complex plane, you can think of the modulus as the length of the vector that they create when added together.  In other words, you use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle.  For example, if you have the complex number \(z = ai + b\) and want to find it’s modulus, you would take the real part ($b$) and the imaginary part (\(a\)), square them, multiply them together, and then take the square root.  This means that

$|z| = \sqrt{ a^2 + b^2 }$.

The modulus of complex numbers has an interesting property.  If you think about a triangle with two sides made up of the vectors representing complex numbers in the complex plane (visual coming to help sort that out!), you will be able to see that the length of the third side cannot be greater than the sum of the length of the other two sides!  Visually, this looks like this:

Mathematically, it is known as the triangle equality and is written like this:

$|x| +|y| \ge |x + y|$

This is a helpful tool when you’re looking to prove certain things about complex numbers.

Another interesting feature of complex numbers is the ability to use them to transform exponential expressions (which will be showing up often, later) into expressions involving trigonometric functions.  This is for reasons that I do not want to prove here, but let me throw some math down on the page:

\(e^{ai} = cos(a) + i*sin(a)\)

Here it is good to do a “sanity check” and substitute 0 into the equation for \(a\).  We know that any number raised to the 0 power is equal to 1.  We also know that \(cos(0) = 1\) and \(sin(0) = 0\) so this equation is consistent.  If you substitute \(\pi\) into the equation for \(a\), you will find a very important result:

\(e^{i\pi} + 1 = 0\)

This so-called Euler Equation relates pretty much all of the most important mathematical numbers.  It’s pretty elegant and that’s probably why people like it so much.

Homework:

Take a break.  I have assigned far less homework than I originally expected but that’s okay.  I’m not collecting it and in all likelihood, you’re not doing it.  I don’t know how many math posts I’ll be doing in September (if I had to guess, I’d say none) because of a challenge I’ll be undertaking (more on that tomorrow!).

-Lane

The Storm Went Elsewhere

Hurricane Irene was supposed to bring Hell and High Water all up and down the east coast, including where I live in Stamford, CT. And, I suppose, it didn't really let down on its threats. Only, it didn't live up to them at my apartment.

Irene made landfall where I am at about 9:00 Sunday morning. I know, because I read the papers, it didn't even wake me up. When I did wake up, the only damage I could see from my window was one small branch on the ground and one strip of siding off a house up the street. Otherwise, nothing. Power was still on, internet still worked, and my car was fine. I was actually a bit disappointed, not that my car was fine, but that it seemed to be such an over-hyped storm. Stamford had a curfew up; nobody was supposed to be outside. We were in full lockdown. The MTA, for the first time in history, shut all of its operations down in preparation. Everything we heard was buckle up, it's gonna be a bumpy ride. Then, at my place nothing...

It took a little while for me to start getting a good sense of what had actually happened elsewhere. The first glimpse I got was when I decided to head out and drive around town a bit to see the damage. I headed up out of downtown into the suburban area and started to see some medium branches down, nothing huge, but definitely damage. I tried to get on the Merritt Parkway, but it was closed. I started thinking: more serious yet. Finally, I tried cutting across town through a back road-no go. There was a huge tree across the whole road. So, definitely some damage there. But, really I thought, it's not all that bad; a tree or two down, some power out, whatever. I've seen thunderstorms do worse.

Then info started to come through the news about other places...Oh Boy! What a world of difference that was. The commuter line I take to NYC everyday was out with trees tangled in the overhead power lines. Other branches will be out for months due to washouts, some serious damage around. Finally, news started to trickle in about my home state of Vermont. VT was getting hammered. Not very high winds or anything like that. Just lots and lots of rain. Southern Vermont in particular was pounded. Roads were out all over the place. News from Lane came in: I-91 was closed due to flooding. Some towns were cut off completely, all roads in and out of them-washed out. The state even lost a few of its beautiful and historic covered bridges. Many are calling it the worst flooding in a century; even worse than this past spring, which saw Lake Champlain at historic flood heights for a month and a half.

So, I guess Irene did bring plenty with her. I was fortunate enough to be missed entirely on her way through; others were not so lucky. I am, however, confident that Vermont will be fine. It's citizens are used to tough weather, and they will rebuild and get on with life as they always do. I wish them the best of luck in rebuilding.

~Alex

More Wedding Plans!

The blog is behind for a number of reasons.  The first is that I was super busy over the weekend and didn’t get a chance to write any posts until yesterday.  Although Hurricane Irene had not prevented my from having electricity, she had knocked out my internet, so making blog posts available took a bit longer than I would have liked.

Back to the business of the weekend!  The primary reason for the pilgrimage back to Vermont that my fiancé and I made over the weekend was to meet with the priest in her parish.  He usually likes to meet with couples in order to reserve the church for their wedding.  It was actually a very helpful meeting because he let us know some of the things that we need to do in order to get the ball rolling.  There are a couple things we need to take care of.  Because I am a baptized, non-Catholic Christian, we will be entering into what is known as a “mixed marriage”.  Because of this, we need to get special permission from the bishop in Burlington.  I also need a certificate, proving I have been baptized.  Additionally, because Brianna’s uncle will be officiating (conveniently, he is a priest) and is from outside of Vermont, he will need to get in contact with both the diocese of Vermont to be allowed to perform the ceremony in a religious capacity and the county to be allowed to perform the ceremony in a civil capacity.  Luckily we’re thinking ahead so this shouldn’t be a big deal.

On to the fun stuff:  Brianna bought her dress after trying it on again and getting her dad’s approval.  We also ordered our wedding bands.  I went with a simple titanium band (secretly I know that it’s made of titanimite, a rare allow of titanium and dynamite) and Brianna will be getting two bands made to go on either side of her engagement ring.  We were fortunate to be able to meet with the custom designer and she came up with a number of beautiful designs.  We’ve also started talking about flowers and photographers!  Luckily we’ll be headed back to Vermont a couple more times to get even more of the planning out of the way.

The next steps that we need to get out of the way are to get the pre-marital counseling done and to start tackling the guest list.  The guest list is a little daunting because we will have to balance the people that we want to attend, the people that we are somewhat obligated to invite, the people that our parents want to attend as well as logistical and quasi-political concerns.  I would prefer to keep the number as reasonable as possible but with two large families and fair number of close family friends I doubt that we will be able to have less than 200 people in attendance.  The unfortunate reality of a wedding is also that we’ll never be able to invite everyone we want to come and everyone who wants to come.  There will be some tough decisions, to be sure, but in the end I know that it will all work out.

I am more excited than ever about getting married to Bri.  I can’t wait for the wedding and it’s less than 9 months away!  There’s a lot to do but we have plenty of time and are well ahead of schedule.  It’s hard to believe, but I think it will be more awesome than the following picture:

-Lane

Training, Week of 8/28

This week was another “dumpster” week, with a total of only 26.2 miles in 3:12:44 (7:23/mile average).  I started the week off really well with a couple longer runs (one of which was a trip to the Reservoirs in West Hartford).  I then headed back to Vermont on Thursday night with my fiancé (skipping my regularly scheduled track workout).  I wasn’t feeling motivated to run on Friday, so I just ran 4.5 miles and then I was super busy on Saturday and it was raining on Sunday and the excuses just kept building and I ended up running far less than I should have.  I am a bit disappointed in myself but I’m primed to start a fresh week and put in some good miles.  I think I am going to focus more on running farther in the afternoon (rather than trying to add morning runs which has been a huge failure so far).

On the racing front, I have signed up for another corporate 5k and I have been put on a “stacked” team so that hopefully the company that I work for can compete against the local running shoe store.  It’s a little funny that it’s such a heated competition based on the sizes; my company has (literally) thousands of employees in the area and the shoe store’s employees number in the tens.  I suppose that they tend to hire people from two very different demographics!  Once again, I’m looking forward to helping my team and you never know, now that the college kids have gone back to school, I might even get a prize!


-Lane

The Storm Cometh...

Come on Irene...

Growing up in Vermont I have dealt with extreme weather. Vermont can be hot in the summer (not blazing southwest hot, I know, but hot enough to cause problems), cold in the winter (-20F cold enough to qualify as extreme?), snowy as all get-out. We even get earthquakes and tornadoes at times (again, they don't hold a candle to those in California, or the mid-west, but still). There are some extreme weather we don't get though. Wildfires don't typically occur, neither do sandstorms or hurricanes. In fact, the northeast in general does not typically experience any of the three.

So, with the approach of Hurricane Irene to the greater New York City area, I'm not really sure what to expect. Most people around here aren't. In the last 50 years, only 5 hurricanes have tracked within 75 miles of NYC. Irene is predicted to pass directly over Queens. I'm at a loss to even quite comprehend, let alone explain, the problems this may cause. For example, the Metro Transit Authority (the organization responsible for running all the NYC subways, buses, Long Island Rail Road, and Metro-North Railroad) has announced that it will be stopping service for the duration of the storm starting at noon on Saturday. Additionally, they have said that it will take a number of hours to restart once the storm has passed. Thus, the public transit may not be running fully until midday on Monday. So, this all boils down to the commuting system for all of NYC, Long Island, Upstate New York, and Connecticut probably not existing Monday morning...

Additionally, since we don't usually get hurricanes through here, buildings are not necessarily designed to handle them. For example, my apartment has 9' high, by 20' wide windows across its exterior walls. The entire building is similar to this setup. Thus, flying debris may be an issue.

Anyway, I'm battening down for my first hurricane experience. Hopefully it will be a good show outside, with none of the drama coming in.

So:

Come on Irene,

Toora Loora Toora Loo-Rye-Aye!

~Alex

A Response to Lane's Letter

Lane-

I must respectfully decline your challenge. As intriguing as it is, I will not take on your challenge as I know I will be unable to complete it. This inability is not from a lack of the necessary tools to complete it (though I will admit that creative writing has never been a particular strength or interest of mine), but from a lack of resources. I have neither the time nor the energy to undertake such a large project as a novel. My lackluster contributions to this blog ought be evidence enough of that. While it is true that I have plenty of time on the train, I have found it quite pleasant to read when I wish to and sleep other times. I am not yet ready to devote that time to writing.

However, while I will be unable to join you in the challenge, were you to undertake it yourself, I would be happy to assist you. So, I offer my assistance to read drafts, consider ideas, and whatever other help you might need. I would also be honored to read the final product.

With that, I hope that you choose to undertake this regardless of my inability to join you, but know that I will assist you if I can.

Sincerely,

~Alex

A Challenge/Open Letter to Alex

Dear Alex:

This is a challenge.  What are the terms?  I challenge you to write a work of fiction of at least 50,000 words.  It doesn't have to be good, just relatively long.

Seems pretty simple, right?

Wrong.

In addition, it must start with the sentence "He could not grow a beard".  You must not start before 12:01am September 1, 2011 (EST).  You must finish before 11:59pm September 30 (EST).  You may plan out/outline your story before then as well as choose a title.

If you agree, I will complete the challenge with you and we can trade novels afterwards.  I also agree to work within the rules written above.  There are a thousand excuses, but only one reason to accept: it will be awesome.  You should have plenty of time while you're commuting, like you mentioned here.

I am eagerly awaiting your reply,
Yours Sincerely,

Lane

Training, Week of 8/21

This week was an interesting one in terms of training.  I started the week off well with a couple medium-length runs until Friday, when I left Connecticut for New Hampshire and went on a hike with my cousins, uncle, and dad.  It was really fun (and you can read about it here) but I didn't get a run in.  I slept over night in the woods and then hiked back and did manage a 4-mile run in the afternoon.  Sunday rolled around and I was going to go for an easy 10 to get 45-ish miles for the week but got lost in the heat and ended up with a 16.2 miler for a total of 50.8 miles in 6:12:49 (7:21/mile average).

I'm glad I got the distance in for the week.  Hopefully I will be able to build on that momentum going forwards.  I've already started the next week out really well with a 10+ miler at the Reservoir Trails in West Hartford (more on that in about a week!).  Spoiler alert: it might be my second favorite running spot yet!

-Lane

A Long Walk, Interrupted

This weekend I went for a hike with my dad, uncle, and two cousins (one of whom is only 6).  It was a relatively short hike on the Monadnock-Sunapee Greenway.  The plan was to leave on Friday afternoon, walk about 3.5 miles, make camp and stay the night, and then hike the remaining 3.2 miles the next day.  My dad and uncle finished the entire Long Trail earlier in the summer and had a blast, so I was very excited to be able to share in this trip.

We got going about 3:00pm.  We parked at the very first Seventh Day Adventist church.  Here we are, getting ready to leave:

We decided to bring smaller backpacks since this was a short trip.  We had plenty of water, a tent, and some rather unusual ingredients for dinner in camp.  Off we went!

It was a tough walk the first day with a lot of up-hill climbing, though I am informed it couldn't hold a candle to the Long Trail.  When we got to the campsite, the only tent platform had been taken by a single-person tent and all the other good locations had been taken by various groups, so we pitched our tent at the fork in the trail.  One trail led to the water and the other led to the latrine.  There was lots of thunder the first night and we were expecting rain but counted ourselves lucky that we only got a couple sprinkles.  As for the unusual ingredients?  They were required for the delicious braised-beef stew in a tomato paste, beef broth, balsamic vinegar, and red wine base with vegetables and potatoes.  We also "baked" an apple pie with some charcoal briquettes.  Simply amazing.

Did I sleep well?  Considering the circumstances, yes, but I did miss my bed.  I didn't want to carry my sleeping bag, so I brought long nylon pants and a fleece sweatshirt and a sleeping pad.  The plan was perfect except that I neglected to put socks on before bed and somehow I spent all night with my feet sticking out the tent door.  I woke up several times during the night with ridiculously cold feet but in barely-waking state I couldn't figure out why.

We were up and moving by 6:00am and on the trail by 7:00am.  The weather was greatly improved and we made very good time as the trail was mostly down-hill.  Then we hopped in a car that was waiting for us and went home.

As far as additional photography, there were a couple cool views along the way and I have always been fascinated by piecing together multiple photographs to form a panorama, though I have not yet been very successful.  I did take two series of photos that I was hoping to put together.  This is the first series, taken on the first day:

 In real life it's about 14000x2000 pixels and looks pretty cool though obviously it's not seamless.  I have some theories on why this is a problem and how to fix it.  Perhaps this is the next thing I'll be experimenting with!  All in all, a very good weekend and the great thing about taking Friday off is that I have another day to relax and recharge before going back to work!

-Lane

P.S. Click the panorama for a somewhat larger version!

Mini-Golf Power Couple

My fiancé has taken up golfing.  I tried to play a year or two ago and found that I really just did not enjoy it.  The learning curve was a bit too steep and I had neither the time, the energy, or the patience to pursue it.  I also don't like being bad doing things for a long time.  So I gave up on golf.

Bri, on the other hand, is very enthusiastic about it and I'm glad that she has found something that she enjoys doing!  One tangentially related activity that I also enjoy is mini-golf and we have vowed to become a mini-golf power couple.  We have played twice in the past two weeks (she won once, I won once) and we're looking to hone our skills and become an unstoppable force... if only I could get the ball past the windmill!

-Lane

Racing? I Guess Not...

A few weeks ago I wrote about how I had found a cool weekly race I found in the next town over. I wrote that I was excited to start going to it and meeting people in the running community around where I live. Sadly, I must report that I have not made it back since. In fact, I suspect that I will not make it back at all. Now, this doesn't have anything to do with my motivation, though that has been a bit of an issue as I mentioned here, it is just a schedule thing.

Since I started actual work I have been working solidly over 9 hours a day (typically getting in at about 8 and leaving between 6 and 6:30, with a 15-30 minute lunch break in the middle). On top of that I have been going to a building that is farther away from my house than my normal office which has left me with a commute that is about 90 minutes each way. When you total that all up I'm spending 12-13 hours a day involved in something related to work. Add in 8 hours for sleep and 1 hour to get ready in the morning, I'm left with 3 hours in the evening to run, eat, and relax. I usually get home between 7 and 8, and I aim to be in bed by 10. Since the races start at 7, obviously I can't get to them. As sad as that is though, the bigger problem is that I am having trouble finding time to run. Today I got home at 7 and went right out for a 3 mile run. Now it is 9:30, I am still eating dinner, and need to be in bed in 30 minutes... So, I guess this continues my quest to find time to run.

I know at this point that I will not prevail tomorrow. My Dad will be in NYC for business and we are going to meet up for dinner. There goes running, out the window. Oh well, it will be good to see my Dad, so I don't mind

~Alex

Training, Week of 8/14

This past week was a good one in terms of training, though I took one day off and only ended up with 45.6 miles in 5:36:36 (7:24/mile average).  I started the week off well with a nice, slow long run on Monday (~14 miles) and I ran well on Thursday in the Corporate 5k Challenge in Hartford (4th place in 16:40) and helped my team to a 3rd place finish.  I also bought a new pair of running shoes (for my thoughts, see: New Balance Minimus Trail) and got a pretty quick ~8 miler in on Saturday.  A little more quality and a little less quantity than I would like, but I'll take it.

I pretty much used up what I would have said about how I'm feeling about running right now in an earlier post on motivation, so I recommend checking that out.  I guess my next challenge is to actually start getting up in the morning and running to put a few more miles in and then there's the next race sponsored by the company I work for in September!

-Lane

Training Update

As I've said before, I don't give regular running updates (and probably don't plan to in the near future), but Lane's post on Motivation got me thinking about my running as I have started work.

I too have been finding it difficult to motivate myself to get out and run after work. Though, one thing to note that differentiates me from Lane is that I don't have the same history of consistent training and racing. While I have been running for 7 years now, I have been plagued with injuries and have had to take significant portions of time off. Even now, I am trying to come back from what is nearing a 2 year break due to injury. The result of this is that I have found myself firmly outside of the habit of running daily. Thus, it is very easy for me come home after work, feel tired or hungry, and just decide to skip running. The thing is, I do love running and racing, and I almost always feel better after a run. But, it is still difficult to find the motivation to get out at 7:00PM when I get back from work.

So, how do I break this block? I'm not sure yet. Maybe once I get settled into my job and my routine a bit more I will find the motivation to get back to running. Maybe I just need to toughen up and force myself out everyday. The option that I want to avoid the most is just taking a break and saying that I'll come back to it eventually. I have already taken a long enough break for injury, and I know how difficult it is to get back in the running habit. So, I know that this route is a slippery slope to ending my running career. One piece of advice that my college coach gave me was to keep running after my scholastic racing career was over. He said that he felt burnt out and done with running when he finished school and decided to take a break from running. Since then, he has never really gotten back into running. He told me that it was a good idea to tone the intensity of training down, just keep casually running and forget about racing. But, he said to keep running regardless. So, I know that I need to keep going, I just need to figure out how to get myself out the door.

~Alex

New Balance Minimus

I was in the market for new running shoes today.  My flats are getting older (~450 miles) and my other pair is nearing what I estimate to be half of their life (~150 miles).  So I headed to the store.

I ended up bring home the New Balance Minimus Trail.  Now to give some background information, I have been a little bit wary of the latest barefoot-running fad.  I haven't seen very many people who are faster than I am really committing to it and I haven't seen anyone who was slower than I am commit to it and pass me so I haven't  been buying into it.  To be fair, I am pretty much already enjoying the alleged benefits of barefoot running (quick cadence, low injury rate, mid-foot strike) so I can't really speak to the benefits to people who are heel-strikers and/or prone to injury.  In any case, I opened the box when I got home and found a warning to run only 10% of total mileage in the shoes.  To heck with that!  My first run in them was an 8.51 run over a hilly course on the roads.

All I can say is "wow".  They felt very light and despite the lack of cushioning, I didn't feel the pounding on my legs. They're like running in ballet shoes with a thick rubber sole!  My usually quick cadence (176 strides/min) upped a bit to 192 strides/min though that may have been due to the excitement of new shoes.  We'll have to see how I feel tomorrow.  In conclusion, I am not a convert to the cult of barefoot running as a cure-all.  I am already poised to take advantage of really light shoes and have been training in flats for a long time.  On the other hand, I do love these shoes and although I probably won't wear them every day, I will enjoy running them as much as possible.

What Motivates Me

What motivates me?  I find that this is an interesting question to explore because if I really take a hard look at what is really motivating me, not what I thought was motivating me or what I told myself motivates me, I can learn a lot about myself and hopefully that will lead to some self-improvement.

What motivated me to work hard when I was in school?  Impressing my teachers or making my parents proud wasn't a true motivator (though I am pleased that I seem to have done both).  Even the objective measures of success, grades, didn't really seem to motivate me too much.  The longer I was in school, the less I was upset by lower-than-usual grades even though I tended to improve throughout my entire scholastic career.  Continuing to dig deeper, I think that doing a good job, in and of itself, is what started me off.  Later, success became a "habit" and I was able to continue working hard on a particular task, even if it no longer interested me.

What motivates me to work hard at work?  I have had a number of jobs that didn't test me or fully utilize my skills, some of which I liked, some of which I didn't like.  Regardless, I wanted to do a good job and what I finally found worked as motivation was to find the co-workers that I cared about and work "for them".  Whenever I had to do an unpleasant, menial, or boring task, I would remind myself that the harder I worked, the easier and more pleasant my friends' work would be.  Hopefully they would reciprocate and work hard "for me" but I was okay if they didn't.  Now that I have a challenging job, I enjoy working hard to stretch my abilities and really utilize my gifts as much as possible.

What motivates me to work hard at running?  I used to train to run faster times, to place higher in important races, and to help my teammates beat other teams.  Those were easy reasons to find to get motivated to run when I didn't feel like it, when it was hot or cold or snowy.  Now that I'm out of college, I don't really have those reasons pushing me out the door and I really can't put my finger on what it is that keeps me running.  Racing with my co-workers doesn't really cut it for motivation and I'm not really finding it in myself to push the way I would need to to lower my times further.  Perhaps success at running has become another "habit" because I'm not sure that I could stop if I wanted to.  Perhaps there's something else there, but the thing I don't understand is that the discipline and hunger that I used to have is missing.  I have resolved a number of times to start running in the morning and have failed pretty much every morning.

What precipitated this introspection?  Strangely enough, it was a streak of recognition for success.  I was given an appreciation award at work for some work that I did and I also finished 4th in a local 5k in 16:40 which I believe is my third-fastest time ever (and fastest on the roads).  Now that I have shown what I am capable of, I suppose I'll have to keep it up.  One more piece of motivation!

-Lane

Work Starts!

This post is only half about one of the topics that I said I was going to post about last time. That's because there has been plenty going on this week to write about, so I'm going to let most of the stuff go.

The one topic that I do want to mention though is Professionalism. I was going to write a whole post about the professionalism (or lack thereof) of my peers during our orientation, but I'll just gloss over it quickly. In short, I was much less than impressed with the profession conduct of many of my peers. The most noticeable problem was punctuality. Training was scheduled to start at 8:00AM sharp everyday, yet people (who were staying in the same hotel as the training) were utterly unable to arrive on time. Additionally, throughout the day we would have breaks to go to the bathroom, get coffee, etc.. These breaks were usually 10-15 minutes, but invariable turned into 20-30 minute breaks when people simply did not return to the training. Not only were many people terrible at arriving on time, they also enjoyed going out drinking on some nights. This resulted in more than one morning where there was a sizable portion of the group that was hung over for training the next day. All in all, I was disappointed in my peers and a bit worried how my workplace would actually work when I arrived there. Fortunately, when I arrived at work on Monday I was greeted with a very professional and respectful group. Everyone in the team has been welcoming and helpful as I try to rapidly learn all the systems, acronyms, and processes that are already in place.

Which brings me to the other half of this post, my thoughts on my new job. So far it has been a lot of fun, but really crazy. My team is stretched a bit thin right now as we just had a very good employee (coincidentally the guy who hired me) take a different job within the firm. We are also in the process of developing a new system while still trying to support the existing systems. So, I've been learning a lot, but much of it has been sort of trial by fire. I'm simply given small tasks and projects and told to do them. The good thing is, unlike engineering where I would have to learn a proprietary programming language or figure out the proper equation to use, much of my job is looking at a system and determining what needs to be changed or updated to make it easier to use. This is good because I am generally good at looking at a project and seeing what needs to be changed, but I tend not to be quite as good at actually implementing the change. In short, the job seems to be exactly what I thought it would, and what I hoped it would. The strange thing though is how little I understand about the system as a whole, or the firm for that matter. My degree doesn't give me any background in software, finance, or business analysis. The main applicable skill is that I know how to learn and think critically. That part I am making very good use of. To make a long story short, my job has been great so far and I'm looking forward to learning lots more about the industry.

Just don't get me started on having a consistent place to sit...over the past 4 days I've sat at 3 different desks in two different buildings. Another story for another post though.

~Alex

Math, The Way it Should Be (Unit 2.6: Summary and Philosophy)

As you (probably) well know, the final lesson in each unit includes a summary of previous lessons and then some personal philosophy from me.  But before we get to that, I would like to point anyone new to this series to the first post in the series.  I usually also link to the first lesson in the unit, but I'll be getting to that soon enough.

So this unit was pretty much a refresher on geometry and a little bit of what is usually covered in trigonometry classes.  We started with the very of geometry, angles.  The most important thing to take away from that lesson is how radians work.  They're not the most intuitive but once you get used to them, they're well worth it, as you will see later.  After angles came triangles.  The most important thing to remember about those three-sided shapes is that they're the key to doing all kinds of things with vectors.  In fact, you can make an infinite number of triangles with a given vector... it's choosing the right one that is difficult (and the right triangle is usually right!).  Then we covered functions that are basically little machines that turn one thing into another thing.  Trigonometry rounded out July.  The take-away message there regarded the trigonometric functions (most importantly: sine, cosine, and tangent).  The penultimate lesson dealt with areas and volumes.  I wasn't very happy with that lesson but I consider that most of geometry education deals with memorization of formulas that becomes largely obsolete with an understanding of calculus, so I kind of glossed over it, (hopefully) leaving you with exactly enough to understand some of the things that come next.

So here's the philosophy: I don't really like geometry or trigonometry.  My real interests lie elsewhere in mathematics.  Unfortunately, like all of mathematics, those more interesting areas are built on the foundation provided by less complex mathematics, specifically algebra and geometry.  So even though I don't like them, I  have realized that they are necessary to fundamentally understand what is going on in more complex operations.  To me, things like geometry and the multiplication tables are like the alphabet: they're not that interesting in and of themselves, but combine them together and you can form the words that describe the whole universe in a way that very few people understand.

-Lane

Training, Week of 8/7

This past week, there was some bumpiness at the beginning of the week and the long run got bumped until today (Monday) and so I only managed 40.3 miles in 4:51:56 (7:15/mile average).  The Thursday Night Track Workout went well despite the fact that I didn't feel that great.  We started an 800m repeat every 5:00 (6 in total) which creates the paradoxical effect of the faster people getting more rest than the slow people (I averaged 2:25-2:27).  I didn't feel that great, but I didn't feel worse as the workout went on.

During the workout, I did experience one of the hardest running decisions yet.  The guy who runs the workouts offered to time me and the other "fast" guy for a seventh interval.  I didn't want to because I was tired and my legs hurt and I was nauseous, but I felt obligated.  Eventually we decided not to continue, which I was very thankful for.

Now I'm looking ahead to the Corporate 5k to Benefit Special Olympics, CT.  I want to help Pratt win another team trophy and I think that there's probably going to be a trophy in it for me which is always nice.  Not to mention I get to hang out with the other runners at Pratt and Whitney, which is super cool.

Now on to a bigger and better week!

-Lane

Training Ends

Whew...haven't been here for longer than I would have liked. The last week of training for my new job was absolutely crazy. It took place at three different locations around Manhattan, and involved a couple new topics that were both brand new to me and very technical in nature. The long and short of it was that I did not have time for much more than just a quick run, dinner, and bed. Thus, no posts from Alex...

Anyway, the last three weeks of training have generally been good. I gained a solid foundation in basic financial instruments, and introduction to the programming language Python, and an overview of the main initiative that I will be a part of as I start actual work. I learned a lot through the training and met a bunch of people who will help to build my network as I start. However, I am glad that it is done and that I can now move on to learning my role in my actual job, which starts tomorrow morning.

I will likely post a couple more posts about the training to relay some of my observations about both the training and all the people I spent the last 3 weeks with. Topics might include: professionalism, networking, learning, etc. Not exactly sure what, but there is certainly more to come.

~Alex

PETA

If you haven't heard, the internet has just discovered that PETA (People for the Ethical Treatment of Animals) actually end up "saving" but ultimately killing lots and lots of pets.  PETA absolutely LOVES controversy, but I haven't heard a whole lot from them about this.  Essentially, they rescue abused pets and then determine if they are "adoptable".  The pets deemed un-adoptable are then euthanized, to the tune of 85%.

The reason that PETA is so quiet on the issue is that they believe that they have a moral position that is superior to any that could possibly be held by anyone else.  From this perspective, their decisions are beyond question because they care more about animals than anyone else.  This is a dangerous position to take (for anyone).  Even if there were tons of demonstrations and protests against PETA but I am sure they would not change their behavior.  Once a person/group of people think that they're above being questioned they start to go a little crazy and I think that's one reason that PETA is falling more and more out of the mainstream.

Anyways, I agree (to an extent) with their message that we shouldn't be unnecessarily cruel to animals but it's not what is said, it's how it's said.

Math, The Way it Should Be (Unit 2.5: Areas and Volumes)

As usual for this series, if you haven't seen them before, you should check out the first post in the series and the first post in the unit.  This unit has kind of stretched on a lot longer than I intended but it is an important one because although I'm not a big fan of geometry and trigonometry, it does set up one of my favorite topics, calculus.  In light of that, we'll have this lesson and a summary and then I want to tie up some loose ends and then we'll get to the fun stuff, though perhaps not in the way you might think.

The first thing to be covered right now is the area.  Consider the old, familiar Cartesian plane:

Now imagine drawing a shape on this plane.  It would have two dimensions.  When we talked about vectors, we talked about their length, that is, the "space" that they take up in one dimension.  When we want to talk about the "space" taken up in two dimensions, we call it the area.  Many regular shapes have special formulas to help figure out their areas and in the interest of brevity, I will allow the reader to tackle that on his or her own.  What shapes really interest me right now are rectangles and circles.  The reason is because you can make any shape with small enough rectangles or circles.  This may not be obvious but trust me, I will explain later.

Circles (more on them here) have an area proportional to the square of their radius:

\(A = \pi r^2\)

I will prove this to you later, I promise, for now, just please believe me.

Rectangles (4-sided, closed shapes with all right angles) have an area equal to the product of the length of their sides:

\(A = ab\)

Where \(a\) and \(b\) are the lengths of the two adjacent sides.  This makes a lot of sense if you go back to the explanation of multiplication earlier in the series.

Volume is the amount of space an object takes up in three dimensions.  Again, regular shapes have their own special formulas, which you can find in a number of places on the internet.  For volumes, I basically only care about the three-dimensional analog of the rectangle, which is the prism.  It is has all right angles and six faces, all of which are rectangles.  In essence, the prism is a box and the volume of the box is as follows:

\(V = abc\)

Where \(a\), \(b\), and \(c\) are the lengths of the three sides that are perpendicular to each other.  

So those are the basics of areas and volumes.  I didn't spend a whole lot of time on them because I really think that memorizing a bunch of formulas for all sorts of shapes isn't very useful because you can just reconstruct them if you know the formulas for rectangles, circles, and prisms and just one other thing (just wait for unit 4!)

Homework: No homework for this lesson because I'm a) feeling benevolent and b) want to go to bed.

-Lane

Cowboys & Aliens: A Review

Note: Originally I meant this to be a review of the movie but it turned into a movie rant.  If that doesn't bother you, keep reading.

*WARNING: SPOILERS FOLLOW (NOT LIMITED TO JUST THE MOVIE AT HAND)*

Over the weekend, I went to see Cowboys & Aliens.  My fiancé said that she wanted to see it after seeing the preview that played before the last Harry Potter movie.  While I expected that it would be a fun movie to watch from an action point of view, I was pretty sure that the plot would be terrible.  After all, with a name like Cowboys & Aliens, what chance did it really have?

Well, I was wrong (and you won't hear me say that very often).  The plot was very good and even allowed me to suspend my disbelief in the fact that band of cowboys and Native Americans could possibly defeat the advance scouts of a space-faring race of aliens.  I won't give a plot re-cap here because that can so easily be found elsewhere.  What I do want to say is that I appreciated the ending of this movie more than the usual Hollywood ending.

Usually a movie ends with one character selflessly giving up his/her life to eliminate the threat to everyone else and everyone else pretty much survives and goes on to live happily ever after.  The main character goes largely untouched and everything is just hunky-dory.  Take the most recent Harry Potter movie: despite being at the center of the near-destruction of the wizarding world, witnessing the deaths of countless friends, and going toe to toe with the nastiest wizard ever, Harry Potter still has this wonderful perfect life afterwards (don't even get me started on the death and back-to-life thing.  Worst ending to an epic series ever).  On the other end of the spectrum would be Oedipus Rex.  At the end of the story, everything goes completely wrong for the hero: he learns that he killed his dad and married his mom and puts his own eyes out.  I can't always hope for an end like that, but I can hope for something a little more like Lord of the Rings, where Frodo returns, broken, to the Shire to find that he can't adjust back to life and eventually fades into the distance with the elves.

This is a little like what I found in Cowboys & Aliens.  While there was the typical sacrifice of a character near and dear to the protagonist to seal the antagonists' doom, the main character just wasn't quite able to put his life back together.  All he can do is just ride out of town without a plan or a friend.

It may be strange to not want a happy ending at the movies, but here's why: I can't believe in the perfect hero who saves the day and emerges unscathed.  I can't believe in a person like Superman or Harry Potter.  I can believe in a hero who tries his/her hardest (and may, like Frodo, ultimately fail) and emerges scarred.  The fact that a hero can exist, even if he is not invincible, is what gives me hope.

-Lane