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The e manifesto: e > pi > tau

Flying slightly below the radar of most normal people is a raging mathematical controversy with serious implications (for raging mathematicians, anyway). So it’s time to jump in to enlighten the general public and contribute to the discussion. And along the way, we’ll introduce the solution to this whole mess in a bold ploy for personal fame and glory.

The controversy involves some people’s favorite number, pi. That’s the 3.14 etc etc number that’s also written as π. This number is featured in one of the possibly three most famous equations known to the masses (note that it’s good to have something “squared” if you want to invent a famous equation):

1. E = mc2 (Einstein’s big winner about energy, mass, and the speed of light)
2. a2 + b2 = c2 (Pythagorean theorem for right triangles)
3. A = πr2 (Area of a circle in terms of its radius) Some upstart folks have starting lobbying that it’s better to talk about a number “tau” (written as τ) instead of our lifelong friend π. All this for a number that is simply equal to 2*pi. That’s it. So tau is about 6.28 while pi is about 3.14. These upstarts want to replace π in equations with τ (well, with τ/2 to keep things accurate).

This might not be a captivating matter to all readers. And that’s OK because this post is actually about a number better than either π or τ. So hang on a moment.

First of all, the tau advocates are passionate that circle size (circumference) should be thought about in terms of its radius (C=τr) instead of its diameter (C=πd). And they claim that this also makes some equations a little simpler. Their full case can be read at this LINK. Take some tranquilizers before reading that so you won’t get too worked up about suggestions that we relegate π to second class citizen status. This is almost as unthinkable as deciding that Pluto is no longer to be considered a planet. As with Pluto, it will never happen. In any case, a rebuttal to the τ manifesto can be found HERE.

Ending the pain: e-volve

Enough of that. It’s time to end all that numerical saber rattling by moving those two numbers aside for the number that actually deserves top honors. It’s known as ‘e’, and like π its value is written by an endless string of decimal place numbers that doesn’t repeat: 2.718281828459 … and so on forever.

This number is at the heart of an exponential equation that has the following curious characteristic: at any point, the value of the equation is equal to the rate at which the equation is changing. Yes I know, that’s profound and at least 75% life-changing. The rather simple equation is: y = ex, as shown in the picture. So here are approximately 3*e (8) reasons to go with ‘e’ over its competitors.

1. It’s used in equations relating to exponential growth or decay, such as population growth or compound interest. People do not use pi when analyzing how bacteria can multiply to take over the world, or how many years it will be after they’ve died before they can actually afford to retire. It can even be used solving ‘derangements‘ which is obviously a cool word even though we don’t know what it means.

2. We are living in exponential times, the theme of this video, and e is all about exponential. We are not living in pi or tau times.

3. Its name has a nice history, involving other letters.

While the concept of e has existed in the universe from way back (since the times when people could count to almost 3), it only started to be used and written down in the 1600’s. An early reference comes from work of one John Napier. He forgot to name it, and later in the century Jacob Bernoulli ‘discovered’ the constant explicitly and it was named ‘b’ for a while.

But in 1727 or 1728, the 21-year old Leonhard Euler came along and figured out a number of important properties about e. But mainly he started calling it ‘e’ in an act of extreme humbleness. There are reports that it was also briefly called ‘c.’

4. As a number, e is both irrational and transcendental. Like most of our bosses. Admittedly, π and τ also have these properties, but who’s counting. And transcendental is another nice sounding word that we don’t understand but will use freely.

5. e is involved in two of the easiest possible math questions that can make you look smart even if you aren’t. Here’s a tidbit from calculus: It turns out that the derivative of ex is ex, and also the integral of ex is ex (ignoring the inevitable constant that comes along with integration). So if someone asks what the derivative or integral of ex is, you don’t have to remember the details, just confidently mimic back the answer: ex (pronounced “e to the x”).

Let’s see π or τ be such an easy answer to a complicated sounding question.

6. Because of the previous point, e is involved in an excellent gimmick that a math teacher can play on students.

Teacher: What’s the derivative of ex?

Student: ex?

Teacher: Yes, that’s what I’m asking, ex. What’s the derivative of ex?

Student: ex

Teacher: Are you deaf?? Yes, I want the derivative of ex. Now tell me the answer!

Student (emphatically): ex!

Teacher: All right, if you’re not going to answer, let’s try a different one: What’s the integral of ex? Ignore the constant in the answer.

Student: ex

Teacher: Yes I’d like you to tell me the integral of ex.

Student: ex

Teacher: [etc. etc.]

After a while, the teacher expresses exasperation, rolls eyes, and says with resigned drama: OK, let’s switch gears. What’s the 4th derivative of sine of x?

Student: sine of x

Teacher: Yes, tell me its 4th derivative

Student: sine of x

Etc.

7. Google’s IPO in 2004 sought to earn e-billion dollars, or \$2,718,281,828 (and some change). If Google likes e, it must be good. I verified that with Bing.

8. Smaller is bigger

While 2.718… is smaller than 3.14… which in turn is smaller than 6.28…, this discussion on ranking allows us to accurately but paradoxically write,

e > π > τ

This is illustrated vertically in the figure above. Click on that figure to see how e feels about that.

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Teaser for future post: an easy way to quickly remember and recite more digits of e than even 99.9% of mathematicians can recite. Can’t wait.