6 days to Tau Day

Even now, a select group of mathematicians and mathophiles are counting down to what should (in my opinion) be one of the principal holidays on the mathematician’s calendar.  You’ve all heard of Pi Day, some of you even celebrate it in one way or another (the number of my facebook friends signed up for “The Only Pi Day of Our Lives” surprised me), and it grieves me to say so, but it’s overrated.  At best it should be a minor math holiday.  If you celebrate only one circle-constant-related holiday in a year, it should be Tau Day.  And if you use only one Greek letter as shorthand for a particular constant which is related to circles and mathematically ubiquitous, better that it be $\tau$, the circumference of a circle with unit radius (approximately 6.28318531, which is twice that other number).

Tau Day is 6/28 (by analogy with Pi Day Half Tau Day, 3/14), so our countdown stands at 6 days ($\lfloor \tau\rfloor$ days, if you will).

I won’t get into the details here, but the key contention is that τ, not π, is the circle constant we should focus on.  Instead of $C = 2 \pi r$ (or $C=\pi d$), it would be better to write $C = \tau r$.  For a quick and cute introduction, check out the Pi is Still Wrong video from everybody’s favorite mathemusician, Vi Hart.  For the whole manifesto, check out Michael Hartl’s wonderful site (updated a few months ago for Pi Day Half Tau Day).

Especially for the non-mathematicians here, “should” and “better” might seem strange word choices.  Today I want to focus on what exactly they might mean in this context.

(Remark: If you’re more science-minded, this is a little like the oft-heard-in-some-circles assertion that the convention for positive and negative charges is “backwards”, the opposite of what it “ought to be”.  Apropos xkcd comic!)

Rather than circle constants, let’s look at the example of exponential functions and logarithms.

The exponential functions, such as $f(x)=2^x$, $g(x)=10^x$, and $h(x)=19^x$, are a family of continuous functions characterized by their continuity and the property $f(x+y) = f(x)f(y)$ (and let’s add non-constancy, to rule out the trivial $f(x)\equiv 1$).  They have many interesting properties in common, and in many ways if you understand one you understand them all.  If $f(x)$ is an exponential function and $g(x)$ is another, then there is a nonzero constant so that $g(x)=f(kx) \forall x$.  (That is, the graphs of all exponential functions are the same, apart from horizontal rescaling and reflection across the y-axis.

The logarithm functions, such as $f(x)=\log_2 x$, $g(x)=\log_{10} x$, and $h(x)=\log_{10} x$, are a family of continuous functions from the positive numbers to the real numbers characterized by their continuity and the property $g(xy) = g(x)+g(y)$.  They are the inverse functions of the exponential functions, and vice versa.  Again, their properties are very similar to one another, and in many ways if you understand one you understand them all.  If $f(x)$ is a logarithm function and $g(x)$ is another, then there is a nonzero constant so that $g(x)=kf(x) \forall x$.  (That is, the graphs of all logarithm functions are the same, apart from vertical rescaling and reflection across the x-axis.

So there are lots of functions that have earned the right to be called an exponential function (or a logarithm function).  Indeed, for each positive number $b\neq 1$, there is an exponential function $b^x$ and a logarithm function $\log_b x$; different $b$‘s give different functions.  So when mathematicians refer to the exponential function $\exp(x)$ or the logarithm function $\log x$, which functions are those?  Does that even make sense?

Indeed it does, and mathematicians have with great consensus identified the exponential function, which is to say a canonical choice of $b$, which is somehow the best or more natural or most beautiful choice.

This is a somewhat unusual and provocative assertion, not least because it occupies a strange place between the objective and the subjective.  Why is one choice best?  It’s certainly not a fact, like $2+2=4$ or $\frac{19}{95}=\frac{1}{5}$.  But it’s also not just a “mere” opinion, like “Grape jelly isn’t as good as strawberry.” or “Lilacs are the best-smelling flower.”

There are lots of reasons to single out a particular function from each family, and they just about all lead to the same choice.  I’ll just give a couple here.

With basic calculus, it’s not hard to show that, if $f(x)$ is an exponential function, then the derivative has the form $f'(x)=kf(x)$ for some $k$ (which would be different for different functions).  That is, if a population is growing exponentially, it’s rate of growth at any moment in time is proportional to its size at that time.  Well, what’s the constant?  As I said, it depends on the function.  If $f(x)=2^x$, then the constant is about 0.693147.  If $f(x)=10^x$, then the constant is more like 2.302585.  Whatever the constant is, it starts showing up all over the place once you start doing serious analysis or problem-solving with the function.  You know what would be a good constant?  1.  Then the rate of change of the function is precisely the function itself, there is no correction factor, and we have the simplest formulas.  The exponential function $\exp(x)=e^x$ is characterized by the property that there is no correction factor.

On the logarithm side, if $f(x)$ is a logarithmic function, then we can show that $f'(x)=\frac{k}{x}$, i.e. the rate of change is proportional to $\frac{1}{x}$.  Of all the functions proportional to $\frac{1}{x}$, which is the most natural?  Yup, it’s $\frac{1}{x}$.  Choosing the base so that the corrective scaling constant disappears again gives the base $e$The logarithm function is $\log x = \log_e x$, often called the natural logarithm (and confusingly written as $\latex \ln x$ in grade-school textbooks).

So it’s not an arbitrary choice.  You might say it’s the “most convenient” choice.  I like to think that I have more poetry in my soul than that, and say it’s the “most beautiful” choice.  Somehow that base just fits best.  The formulas come out cleaner.  Everything just works with a little less noise.  That’s (part of the reason) why the number $e$ (approximately 2.7182818284590452) has a reserved letter and a name (Euler’s constant; I’ll let you decide whether e stands for Euler, or exponential, or something else).

Subjective-objective-wise, it’s something like the situation in music theory, where it is said that certain combinations of notes or progressions of chords “sound good”.  This looks superficially like a personal value judgment, but it isn’t really.  It’s a shorthand for a statement about the ways sound waves combine which is hard or impossible to actually express in words, but you know it when you hear it.  It’s not that hard to imagine a person of highly idiosyncratic musical taste who just doesn’t like, just can’t stand the sound of the “standard” harmonies.  But such a person would probably still know what “that chord progression sounds good” means.

Likewise, you could imagine a person who loves the number 7 — how it looks, how it smells, how much fun it is to write — so much that she would prefer to use $7^x$, and the aesthetic joy of all those 7s.  Such a person, whatever her country or planet or universe of origin, would still see what I see about the base e, would still know why it is called the exponential function.

It’s just right.