the natural language of growth

A ninth grader asked me what e was all about.  She had seen it looking through her brother's calculus book.  I took out my iPhone and opened a calculator (rpn of course;-) and rambled on more or less like this...

 

Imagine you're going to deposit a dollar in an extremely generous bank that gives 100% interest a year.  At the end of the year you have your original dollar plus a dollar interest.  Now imagine a slightly different rate ...  50% interest, but now every six months..  At the end of six months you have $1.50 and at the end of the year that increases another 50% giving you $2.25 ... not bad.  Now you're interested so you inquire about the monthly rate.  You see they offer 8-1/3% interest - a twelfth interest - compounded monthly.  So at the end of the first month you have $(1 + 1/12) or   $1.083..  One the second month it is (1 + 1/12)*(1+1/12) = (1 + 1/12)² .. a bit over $1.17 ..   You see the pattern.  At year's end your total is $(1 +1/12)¹² .  $2.613 .. things are getting better.  

 

They don't offer daily interest, but you figure it out anyway and add it to the little table you've been making

 

(1 + 1) = 2  

(1 + 1/2)² = 2.25

(1 + 1/12)¹² = 2.613

(1 + 1/365) ³⁶⁵  = 2.7146

 

You're doing better, but the rate of increase is slowing down.  What about every hour - ( 1 + 1/8760)8760 ?  Your calculator shows $2.7181.  A slight improvement.  You could try every second, every microsecond, every femtosecond...  but what if the bank compounded every instant?  

What is (1 + 1/n)ⁿ if n is infinity?

Euler worked it out.  It is the irrational constant e.  He didn't name it after himself, but it's a convenient way to remember who did it .  There are a variety ways to calculate it.. he showed 1 + 1/2 + 1/6 + 1/24 + ... + 1/n!    He managed to work it out eighteen digits 2.71828182845904523536028...  

 

Some of the important numbers are buried in geometry ..  π is the ratio of the circumference of a circle to it's diameter, √2 is the hypotenuse of right triangle with side lengths of one.  The Greeks loved geometry and found rich pickings.    e, on the other hand, is not geometrical.  Rather it is fundamentally associated with growth.  

 

Try plotting y = e^x ..  Notice y =1 at x =0 and e  at x =e .  If you look a little deeper the gradient of the curve at e turns out to be e and the area under the curve up to e is, you guessed it, e.  In fact for e^x the gradient is e^x and the area under the curve is e^x.   Remarkable - this is the only function with the value, gradient and area under the curve is the same at every point.  This property makes it the natural language of calculus - the math of rate of change, areas, etc.  So if you study anything involving area, or growth .. finance, physics, biology...  e has a deep and fundamental importance.  It makes the math much easier.  If you force yourself to avoid it the math gets complicated in the most ugly way imaginable.

 

You can use it for geekspeak...  A friend would indicate something is huge by saying "e to the something"    brrr ... it's e to the cold outside.  Or "Trump is e to the narcissistic"...  

 

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Recipe Corner

 

How to roast potatoes.  Some tricks.  Chop them into golf ball or smaller sized chunks and parboil them in salted water for about ten minutes. Drain in a colander and shake them up to rough up the edges a little.  Toss in  some olive oil (I used about three tablespoons per kilogram of potatoes) and roast at 425°F or so for about 40 minutes turning a few times as you go.  The real trick is the parboil and roughing step.

 

Here's a seasonal recipe with another trick

 

Ingredients

 

° about four pounds potatoes  ( I like red potatoes . use what you like to roast)

° two oranges

° bunch of sage

° 6 tbl olive oil

° 8 cloves of garlic

 

Technique

 

º oven to 425, chop potatoes to golf ball size chunks

° parboil for 10 min in salted water, drain and rough up a bit

° peel long strips of orange peel  (use a spread peeler!) and pick the sage leaves

° pour the oil on a large roasting tray over a couple of range elements or burners on low heat and add the sage and garlic and orange peel.  Let them fry for about a half minute .. now add the potatoes and toss everything with tongs.  Pop in the oven and broil for 40 to 50 minutes get them crip and golden. 

 

 

Comments

[Jheri]

I love how clear your explanations are!