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.

## 364.4 ± an εar

In October of 1958 Oliver Smoot became a measurement's namesake and part of MIT lore. As part of his fraternity pledge task his body was to be used to measure the Harvard Bridge - a span connecting Cambridge and MIT with Boston proper. At five foot seven Smoot was the shortest of the pledges - a feature that would create the most labor. An added advantage that smoot sounded like a proper unit of measure.

The plan was to have Oliver lie down a few times - perhaps ten - and mark the distance with chalk. This would calibrate a string which would be used to finish the measurement. An onlooker stumbled onto this and, given an audience, the plan was abandoned for something more insane .. Smoot the ruler would lay down for each sixty seven inches of the way. This went on for about half the bridge's span with a line marking every ten smoots. Oliver was now exhausted, so his fraternity brothers carried and dragged him to each new point along the way. In the end he had to move or be moved three hundred and sixty four times with about another forty percent of his height remaining. Being proper engineers they added an error estimate of an ear written as εar - epsilon is often used to denote an error measurement. Being beginning engineers they were wrong.. but it became famous and is maintained to the day. The smoot is a non-standard unit of measure set exactly to 67 inches or about 170.28 centimeters. If you query Google for length you can request the answer in smoots. Siri doesn't measure up (yet)...

Oliver Smoot went on to a career technical standards and measurements and became the chair of the American National Standards Institute before becoming President of the International Organization for Standardization. Another Smoot, George, is known for one of the most beautiful measurements in all of science - the anisotropy of the cosmic background radiation. It is generally regarded as the point where cosmology - a field with roots thousands of years long across many cultures - became a precision science. George's other cool factor is that he was one of two contestants to win the million dollar prize on the game show

Are You Smarter than a 5th Grader?Oh ... he also won the Nobel Prize in physics.Oliver's smoot is undoubtedly wrong although the unofficial definition is fine. They should have sorted out the error in using Smoot as a ruler. What about his posture? What about his height laying down during the measurements? What about his angle with respect to a straight shot across the bridge? You're probably one to two centimeters taller if your height is measured in the hour hour or so before gravity compresses your spine after you wake up. Lots of what ifs to consider. To do this properly an error has to be estimated for each component of the measurement and a grand totaling know as the error propagation that considers them as a whole. Part of the coursework of anyone early on the a natural sciences course.

How the errors are presented and judgements made can be problematic. People talk about a crises in science where measurements, usually in the social and medical sciences, are often not repeatable. I don't think there's a real crisis, but perhaps it is fodder for another post down the road.

^{1 }It applies in many places -- I have worries that some of what has become to be known as data science is done without regard to understanding errors... It's presentation is often beautiful and can lure people towards misjudging the result. One I did some post mortem work caused over a hundred million dollars of hurt.A hallmarks of trade has been the standardization of measurements. If you find yourself in a good reference library track down the history and evolution of a measurement. A few years ago I hunted the foot in the stacks of the New York City Library with the help of a great reference librarian who specialized in technology and math. In recent history - about the time the smoot was defined - the foot became standardized and tied to the meter - 0.3048 meters exactly. It has been part of many systems for obvious reasons. What surprised me is it varied from city to city from about 250 to 330 millimeters! Merchants had to carry around books full of conversion factors (some beautiful examples exist in the library). In one case a despot wanted to take tall young men from a recently suppressed area in Poland for his army. His region defined a larger foot than the area with soldiers who appeared to be much taller on paper than they really were. If you are six modern feet your height could be anything from nearly seven foot three to a bit over five foot six. And that is in the areas that divided a foot into twelve inches .. some used sixteen digits..

The enlightenment gave us the metric system - official adopted everywhere except the United States, Myanmar and Liberia.. We may as well use smoots.

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^{1}It has much to do with how science deals with and learns from failure - also that medicine and social sciences are extremely messy given human subjects that are often impossible to completely control for. Many, myself included, feel they are real sciences (yet). That doesn't mean they aren't worth doing! Math isn't a science and neither is engineering.__________

Recipe Corner

I love brussels sprouts, but in a pan or the oven.. never a microwave. Until now. If you're pressed for time try this (sent by Chris). It isn't too bad...

I would clean the sprouts, cut part of the bottom stem off, then slice them in half. Then, I would get a small glass rectangular dish that was about 1 1/2 inches deep. I would place the sprouts cut side up in the bottom of the dish along with a tablespoon of water. Now, I would melt some butter in the microwave oven and spoon a little over each of the sprouts. A very slight pinch of sea salt and white pepper over the butter. Now, the most important part - the wrap you cover the dish with. I do not use a lid for reasons I will explain. I use a good clingy wrap (Stretch - Tite, available at Costco), and place it tightly over the dish and down the sides. Then I place them in the microwave (one that has a rotating platter), set it on a medium - high heat (65 to 70 percent), and cook them for about 2 to 2 1/2 minutes, depending on the power of your microwave. I let them sit for about three minutes, then set the microwave at the same heat for about a minute and a half. You should see the stretch wrap rising like a bubble above the dish. Watch them to make sure they do not brown. When cooking is finished, remove them from the microwave and the stretch wrap will suck itself down around the sprouts, almost like they were vacuum sealed. Leave them like this for a few minutes - it is at this point that the butter, salt, and pepper will be forced into the sprouts by the apparent vacuum. Cooking them this way retains most of the color of the sprouts and retains their wonderful flavor without the smell during cooking. If one has to watch their sodium intake, leave the salt off, as the butter has a small amount

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