e

I spent some time last night reviewing some math fundamentals in an attempt to get my brain moved from el mundo de Español and back into the world of functions, algorithms, and the basics of calculus. You should know, too, that I have spent the summer collecting urls for math sites that might be of some use to me as I travel deeper in the abstraction jungle. One site last night was particularly helpful…..on the subject of e, the mathematical constant. If you are like me, you might ask, what is e? And, what is it good for? (Not absolutely nothing…..The song. Sorry….)

Believe me when I tell you that my calculus textbook is worthless in terms of an explanation. For a simpler (?) explanation, let’s go to Wikipedia:

The mathematical constant e is the unique real number such that the function e^x has the same value as the slope of the tangent line, for all values of x.^[1] More generally, the only functions equal to their own derivatives are of the form Ce^x, where C is a constant.^[2] The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain seriesrepresentations of e, below). (see

The number e is one of the most important numbers in mathematics,^[3] alongside the additive and multiplicative identities 0 and 1, the constant π, and the imaginary unit i.

The number e is sometimes called Euler’s number after the Swiss mathematician Leonhard Euler. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called simply Euler’s constant.)

Since e is transcendental, and therefore irrational, its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of e truncated to 20 decimal places is:

2.71828 18284 59045 23536…

e is the unique number a, such that the value of the derivative (the slope of the tangent line) of the exponential function f (x) = ax (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).

Got that?

Good, neither did I.

But this guy does, and does a helluva job explaining it.

In a nutshell:

e is the base amount of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.

Just like every number can be considered a “scaled” version of 1 (the base unit), every circle can be considered a “scaled” version of the unit circle (radius 1), and every rate of growth can be considered a “scaled” version of e (the “unit” rate of growth).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.

Now we’re getting somewhere. I understand the principle of compound interest. Who knew it came from calculus?

To continue:

Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket?

Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:

n          (1 + 1/n)^n
------------------
1          2
2          2.25
3          2.37
5          2.488
10         2.5937
100        2.7048
1,000      2.7169
10,000     2.71814
100,000    2.718268
1,000,000  2.7182804
...

The numbers get bigger and converge around 2.718. Hey… wait a minute… that looks like e!

Yowza. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:

This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2.718.

But what does it all mean?

The number e (2.718…) represents the compound rate of growth from a process that grows at 100% for one time period. Sure, you start out expecting to grow from 1 to 2. But with each tiny step forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2.

Now, of course, I understand why the bankers and financial advisers get so stoked about compound growth rates. This also gives me a peek behind the curtain as to why calculus offers so much to the rest of the natural sciences. Every discipline seeks answers about rates of change, and by golly, it sure does look like e helps them all measure those rates. To wrap things up…..

The big secret: e merges rate and time.

This is wild! e^x can mean two things:

• x is the number of times we multiply a growth rate: 100% growth for 3 years is e^3
• x is the growth rate itself: 300% growth for one year is e^3.

Won’t this overlap confuse things? Will our formulas break and the world come to an end?

It all works out. When we write:

the variable x is a combination of rate and time.

Let me explain. When dealing with compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).

• 10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year.
• 1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop.

The same “30 changes of 1%” happen in each case. The faster your rate (30%) the less time you need to grow for the same effect (1 year). The slower your rate (3%) the longer you need to grow (10 years).

But in both cases, the growth is e^.30 = 1.35 in the end. We’re impatient and prefer large, fast growth to slow, long growth but e shows they have the same net effect.

So, our general formula becomes:

If we have a return of r for t time periods, our net compound growth is e^rt. This even works for negative and fractional returns, by the way.

Example Time!

Examples make everything more fun. A quick note: We’re so used to formulas like 2^x and regular, compound interest that it’s easy to get confused (myself included). Read more about simple, compound and continuous growth.

These examples focus on smooth, continuous growth, not the “jumpy” growth that happens at yearly intervals. There are ways to convert between them, but we’ll save that for another article.

Example 1: Growing crystals

Suppose I have 300kg of magic crystals. They’re magic because they grow throughout the day: I watch a single crystal, and in the course of 24 hours it creates its own weight in crystals. (Those baby crystals start growing immediately as well, but I can’t track that). How much will I have after 10 days?

Well, since the crystals start growing immediately, we want continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: 300 * e^(1 * 10) = 6.6 million kg of our magic gem.

Example 2: Maximum interest rates

Suppose I have $120 in a count with 5% interest. My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years?

Our rate is 5%, and we’re lucky enough to compound continuously. After 10 years, we get $120 * e^(.05 * 10) = $197.85. Of course, most banks aren’t nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don’t like you.

Example 3: Radioactive decay

I have 10kg of a radioactive material, which appears to continuously decay at a rate of 100% per year. How much will I have after 3 years?

Zip? Zero? Nothing? Think again.

Decaying continuously at 100% per year is the trajectory we start off with. Yes, we do begin with 10kg and expect to “lost it all” by the end of the year, since we’re decaying at 10 kg/year.

We go a few months and get to 5kg. Half a year left? Nope! Now we’re losing at a rate of 5kg/year, so we have another full year from this moment!

We wait a few more months, and get to 2kg. And of course, now we’re decaying at a rate of 2kg/year, so we have a full year (from this moment). We get 1 kg, have a full year, get to .5 kg, have a full year — see the pattern?

As time goes on, we lose material, but our rate of decay slows down. This constantly changing growth is the essence of continuous growth & decay.

After 3 years, we’ll have 10 * e^(-1 * 3) = .498 kg. We use a negative exponent for decay — we want a fraction (1/e^rt) vs a growth multiplier (e^(rt)). [Decay is commonly given in terms of “half life”, or non-continuous growth. We’ll talk about converting these rates in a future article.]

More Examples

If you want fancier examples, try the Black-Scholes option formula (notice e used for exponential decay in value) or radioactive decay. The goal is to see e^rt in a formula and understand why it’s there: it’s modeling a type of growth or decay.

And now you know why it’s “e”, and not pi or some other number: e raised to “r*t” gives you the growth impact of rate r and time t.

I think I understand e a little better now. How about you?

The Future?

Sometimes, in the dark of night, when the nay-sayers are gathered together on my shoulder and whispering negative thoughts in my ear, I get anxious about my future. Who wants to hire an old guy, who’s probably stubborn, who likely won’t take kindly to some 30 year old criticizing his work product, who might be a touch slower than the rest of the crew, and who most assuredly thinks his ideas are great most of the time?

Yeah, the nay-sayers can make some pretty good arguments in the throat of the night, and their continued existence is like the inevitable drip of water from the leaky faucet – a sign that there is a larger problem.

If I let them overtake me, then this might be my future…………..

Hey, wait a minute…..that don’t look so bad!

Pic via Maggies Farm.

Spanish Technology

The dog days of August signal the end of Summer School, and a short pause before the madness of another ‘real’ semester.

But every learning exercise has its own reality, with moments of tension and drama, and Spanish 102 is no different. Today was the day for my ‘oral presentation’; my three minutes in front of the class, alone but for my powerpoint presentation, an opportunity to declaim on a subject of our own choosing.

Yes, not only did I have to memorize about 381 words, but I had also to present audio-visual support! No big deal for you corporate warriors, or you students who have been doing slides since you were old enough to mash keys. For those of us in a certain age range and career skill sets, PowerPoint has always been something to be afraid of.

In the end, of course, it wasn’t that bad. Office 2007 makes PP fairly straightforward, and, well, I have been a fairly big boy in the corporate sense, so putting together a presentation isn’t a totally new concept. I was slightly pleased to see that, for all of its horrors, my business life did help me put together something with a little more, ahem, polish that some of my classmates.

There was, naturally, one student who did not bring a thumb drive with his slides; no, he went to his website and linked to a presentation that include complex graphical manipulations. But, you know, it was about food in Peru. The only thing missing was the music……

On a side note, I went to the bookstore to pick up the last book needed for my fall classes. While there, I couldn’t help but look at a business statistics textbook for a class that I will likely take in a semester or two. Not like any statistics textbook I have thus far encountered; no calculus, no integrals or derivatives, just a CD with Excel and many, many sets of statistical problems that will have to solved using Excel and, yes, PowerPoint.

XKCD

Because this says what happens to me in so many of my math classes:

How To Shoot Yourself in The Foot

With apologies to those who will not get the joke, and with appreciation for those of you who will, herewith a purloined bit of prose created by a frustrated programmer.

How to Shoot Yourself in the Foot:

Java

You locate the Gun class, but discover that the Bullet class is abstract, so you extend it and write the missing part of the implementation. Then you implement the ShootAble interface for your foot, and recompile the Foot class. The interface lets the bullet call the doDamage method on the Foot, so the Foot can damage itself in the most effective way. Now you run the program, and call the doShoot method on the instance of the Gun class. First the Gun creates an instance of Bullet, which calls the doFire method on the Gun. The Gun calls the hit(Bullet) method on the Foot, and the instance of Bullet is passed to the Foot. But this causes an IllegalHitByBullet exception to be thrown, and you die.

MS-SQL Server
MS-SQL Server’s gun comes pre-loaded with an unlimited supply of Teflon coated bullets, and it only has two discernible features: the muzzle and the trigger. If that wasn’t enough, MS-SQL Server also puts the gun in your hand, applies local anesthetic to the skin of your forefinger and stitches it to the gun’s trigger. Meanwhile, another process has set up a spinal block to numb your lower body. It will then proceeded to surgically remove your foot, cryogenically freeze it for preservation, and attach it to the muzzle of the gun so that no matter where you aim, you will shoot your foot. In order to avoid shooting yourself in the foot, you need to unstitch your trigger finger, remove your foot from the muzzle of the gun, and have it surgically reattached. Then you probably want to get some crutches and go out to buy a book on SQL Server Performance Tuning.

¿Era diferent la vida? (Was Life Different?)

The title of this post refers to the chapter currently under study in our Spanish class. While learning to speak in the imperfect past tense (I used to drink a lot of beer), and learning to use the vocabulary and grammar that makes comparisons (I don’t drink as much beer as I used to, and I am fatter today than when I was 25), I have been forcibly marched down memory lane.

It is a source of some amusement to the class and the professor that people (like their parents) actually voluntarily wore bell bottoms. Several in-class recitations have drawn on the allegedly heroic amount of drugs consumed by the ‘older generation’ during the ’60s and ’70s as a source of humor and ridicule.

I write this with a semi-forced grin on my face. On the one hand, the behavior of my generation does, in retrospect, seem a little ridiculous, but I don’t recall any of us pushing back against the eternal tide of group behavior. And, in truth, I see the same forces at work on the generation that sits in my classes and flows around me on the campus sidewalks.

Just yesterday some chica flashed her underwear to several of us as she turned around in her chair. Her skirt was impossibly short and I can’t imagine how uncomfortable it must be for her to sit, get up, walk, etc. in such gear. Do you think her mother, who must have bought the skirt for her, was momentarily transported to the halcyon days on yon when she, too, slipped into her first mini-skirt? Are girls really all that different from their mothers in these days of “We’re best friends” or “what are you guys doing?” when speaking of the parent-child interaction?

When I heard one of my male classmates say to another PYT (pretty young thing in my generation-speak): “Like, I had to take an adderall last night to study for this test”, did I have a momentarily flash of revulsion for the scientifically approved medication of this generation, or did I flashback on the days, not that long ago it seems, when everyone knew the guy who had access to “black beauties” during exam time?

The more things change the more they remain the same.

Francis Fukuyama wrote about the end of history in 1992. I am not so sure that he is onto something. This generation, like all generations and those who write about generations, wants to believe that cultural evolution is a fact of the generational passage of time. But we may be living in a time that would be very familiar to our antecedents who lived before the industrial revolution, before the accelerated rate of change in the human existence; when generations had much more in common than not.

I’m not sure this generation is much different than mine…….and that’s pretty scary.

The Lowering of the Larynx

I love my small school. Here’s another reason: My professor for the second semester of Spanish also teaches linguistics. If I took this class during the regular school year I would not have a snowball’s chance in hell of having her as my teacher. She only teaches Spanish majors taking high level courses, who have an interest in linguistics.

So today, while explaining the past imperfect indicative, or simply, the imperfect tense, and walking us through the conjugation of the various forms, one of the students had some trouble with pronunciation. Or to be clear, your scribe stumbled badly while trying to say trabajábamos (we used to work…..no kidding!).

This led to a brief explanation about the role of the larynx in speech; who knew that the human larynx drops as we age, and that the dropping creates our ability to speak. And who knew that until the larynx drops, at a young age, the baby cannot choke…..the raised larynx acts to block food and water from the windpipe.

The details of this fascinating bit of infovoration* can be found here. A teaser:

The larynx works like a valve, opening and closing to let air pass. When it is shut, food can pass into the esophagus at no risk to the lungs. The best place for such a seal is right at the top of the trachea so that no food or drink accidentally goes even a little ways down it, but humans have a second use for the valve. We work it like a musical instrument shaping the sounds made by passing air as we speak. The musical valve works best if we pull it a bit down into the trachea so that the air wave shaped by the larynx can resonate before leaving the mouth.

At birth the human larynx is in the normal, animal location, enabling babies to nurse without risk of choking. The larynx typically begins to move lower at about three months of age and reaches its final position by age four. People familiar with children’s speech will notice that the start of the relocation is also when infants start to coo. The end is about the time the children finally become clearly intelligible to well-meaning strangers. The lowered larynx lets humans produce a much wider variety of sounds, particularly vowel sounds, than apes can generate.

I’m not sure I would have picked up that bit tasty morsel during the regular semester when my professor would be trying to teach 4 sections of unruly, disinterested freshman the rudiments of Espanol.

*Infovoration – Product which is consumed by an infovore

The End of Summer (Part I)

The first summer session is over. I have some slight command of the Spanish language, knowing about 200 words, have some rudimentary understanding of grammar, and possess a slight ability to discern words in a conversation if spoken slowly enough. I assume that I passed the first course, based on the e-mail from my professor who wrote that I “did a good exam”, and also based on the fact that, so far, I am still enrolled in the second course that begins Tuesday (that is, I have not received a communication from the school telling me that I can’t take the second course).

The mathematical sabbatical has been a very good idea indeed. The ‘A’ that I expect from the first session will certainly help the GPA regain some of its lost value, and there is a reasonable expectation of a similar result in part 2. Plus, the realization that I can still memorize material relatively quickly is an enormous confidence booster for the expected rigors of Biology that await in the Fall term. The brain still works, if not in an abstract manner.

Staying in the Spanish milieu for this post, here is a representation of how I felt at the end of the Spring semester:

I was being gored, tossed about like a rag doll, and receiving absolutely no respect from any of my courses……..

Today, with an all but certain victory in a class, and another likely to follow, my state of mind can best be expressed with this image:

¡patear el culo y algunos nombres teniendo!

A little confidence is a great thing……….

The Scientific Method Pushes Back…..

In my last post, here, I linked to a very interesting article by Chris Anderson, of Wired Magazine. Anderson posited that Google is fundamentally changing science and the scientific method.

Well, it didn’t take long for the scientific community to weigh in on the issue:

From Ars Technica, the other side of the argument:

Every so often, someone (generally not a practicing scientist) suggests that it’s time to replace science with something better. The desire often seems to be a product of either an exaggerated sense of the potential of new approaches, or a lack of understanding of what’s actually going on in the world of science. This week’s version, which comes courtesy of Chris Anderson, the Editor-in-Chief of Wired, manages to combine both of these features in suggesting that the advent of a cloud of scientific data may free us from the need to use the standard scientific method.

…Overall, the foundation of the argument for a replacement for science is correct: the data cloud is changing science, and leaving us in many cases with a Google-level understanding of the connections between things. Where Anderson stumbles is in his conclusions about what this means for science. The fact is that we couldn’t have even reached this Google-level understanding without the models and mechanisms that he suggests are doomed to irrelevance. But, more importantly, nobody, including Anderson himself if he had thought about it, should be happy with stopping at this level of understanding of the natural world.

Obviously, there is a lot more, so follow the link for the full post.

I’m not a scientist, I’m a student. Nevertheless, it is fascinating to see the dynamics of conflicting viewpoints that arise from the inevitable conflicts between orthodoxy and revolution. I suspect that the way forward in this discussion will bring us to a harmonic convergence of new research methods and a revision to the hallowed Scientific Method.

Correlative Analytics♠

Once again, Kevin Kelly explains the intersection of computer science, mathematics, large datasets, and science in a way that few can. The link will take you to the entire post, but these juicy tidbits are here to tease:

There’s a dawning sense that extremely large databases of information, starting in the petabyte level, could change how we learn things. The traditional way of doing science entails constructing a hypothesis to match observed data or to solicit new data. Here’s a bunch of observations; what theory explains the data sufficiently so that we can predict the next observation?…

In a cover article in Wired this month Chris Anderson explores the idea that perhaps you could do science without having theories.

This is a world where massive amounts of data and applied mathematics replace every other tool that might be brought to bear. Out with every theory of human behavior, from linguistics to sociology. Forget taxonomy, ontology, and psychology. Who knows why people do what they do? The point is they do it, and we can track and measure it with unprecedented fidelity. With enough data, the numbers speak for themselves.

Petabytes allow us to say: “Correlation is enough.” We can stop looking for models. We can analyze the data without hypotheses about what it might show. We can throw the numbers into the biggest computing clusters the world has ever seen and let statistical algorithms find patterns where science cannot.

There may be something to this observation. Many sciences such as astronomy, physics, genomics, linguistics, and geology are generating extremely huge datasets and constant streams of data in the petabyte level today. They’ll be in the exabyte level in a decade. Using old fashioned “machine learning,” computers can extract patterns in this ocean of data that no human could ever possibly detect. These patterns are correlations. They may or may not be causative, but we can learn new things. Therefore they accomplish what science does, although not in the traditional manner…

My guess is that this emerging method will be one additional tool in the evolution of the scientific method. It will not replace any current methods (sorry, no end of science!) but will compliment established theory-driven science. Let’s call this data intensive approach to problem solving Correlative Analytics…

Perhaps understanding and answers are overrated. “The problem with computers,” Pablo Picasso is rumored to have said, “is that they only give you answers.”  These huge data-driven correlative systems will give us lots of answers — good answers — but that is all they will give us. That’s what the OneComputer does –  gives us good answers. In the coming world of cloud computing perfectly good answers will become a commodity. The real value of the rest of science then becomes asking good questions…

This is the clearest expression yet of what I think the Discovery Informatics degree at my school can offer to those interested in these emerging fields. And remember, where science leads, business opportunities follow closely behind. There is much to be done…………….