Heard while watching Almost Famous….
“You’re a slave to the groove…”
Yep……slave to the groove…….
A Student Returns to College After Thirty Years
Before I resumed my education and began this journey in Discovery Informatics, I did as much research as possible. Among those efforts was a meeting with the Assistant Chairman of the Mathematics Department. I disclosed my dream, my background, and then got to the point: Could I, at my age and with my lack of background in math, possibly get through the DI program? His response was brief, brutal, and very honest. If you struggle with pre-calculus and algebra, you probably shouldn’t be in the program.
Fair enough. The A in algebra boosted my spirits, but the C+ in pre-calculus scared me. Then it was on to Calculus I…….a mightly battle from which I emerged scarred, and, to a certain extent, wiser.
Today, I walked into my Calc II class. Yes, there stood my old friend, the Assistant Chairman. He began the class with a brief slide presentation; the last dozen or so semesters of Calc I students whol earned either A, A-, or B+ in the class. Know from my Statistics classes that they represent a sample of sufficient size so that we can assume a normal distribution, aka, the bell curve, in the grade distribution. Note, too, that he did not include in his sample population those students who earned a grade less than B+ (like me). He then showed a grade distribution of those students in Calc II.
The median was a C+. There were plenty of grades worse than that (I know, and you should too, the median is the 50th percentile). Some freshman whippersnapper, fresh off his AP SAT score, and thus placed in this class, and heretofore considered by his high school classmates as a genius, stated to the professor that he would, without doubt, get an A. The prof begged to differ, stating that half of us will drop or fail, and of the rest, only 2 or 3 will get an A. Added the prof, You might get an A, and I hope you do, but numbers don’t lie.
Whatever sangfroid I might have felt disappeared completely during this exchange of data, to be replaced with that old familiar sensation….gut wrenching fear. Pulse racing, blood pressure elevated, the room suddenly became too warm and I struggled to breathe. I thought that I had trained myself to suppress these periods of anxiety (that primarily arrived just before any tests), but NO!
So the battle resumes. Visits to the math lab, visits to the professor’s office, Sundays spent studying, and anxiety like you don’t know in the days before each test (4 and a Final that is cumulative); these will be my routines this semester.
Wish me luck, I’m gonna need a lot of it……..
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 ex has the same value as the slope of the tangent line, for all values of x. More generally, the only functions equal to their own derivatives are of the form Cex, where C is a constant. The function ex 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 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.)
- 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).
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?
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.
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/ert) 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.]
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 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.