In the W19 term, a number line was added to the north-northeast stairwell in MC (the one near the DC and M3 bridges), putting CC’s 2018 review of vertical transportation mechanisms in MC out of date. Well, things haven’t changed that much. If anything, the number line has only further solidified the stairwell’s position as the nicest one in MC. Perhaps now it is also the most educational stairwell in UW.

But is it really educational, or just an expensive spiral of black paint? What can we learn from this piece of art (and the wealth of mathematical knowledge on the Internet)? I invite you out for a walk: MC north-northeast stairwell, basement level. Bring this article with you.

Floor $0$

“This place exists?” Of course there’s a Floor $0$. What kind of heretics would start their number line at $1$? You look around, and there’s not much down here other than a really loud pipe and a door you’re not allowed to open. It’s as if this landing exists solely to host the start of the number line.

To the left of the big pink $0$, the black number line gives you a taste of the negative numbers, until it runs into the wall around $-0.2$. (You remark that they could have fit $\flac{-1}{12}$ in there, and wonder why they didn’t.)

Slightly to the right of the big pink $0$ is a silvery plaque:

$\epsilon$

arbitrarily small positive number

After taking calculus, the response is automatic: there exists a positive $\delta$ which is sufficiently small to guarantee that something else is less than $\epsilon$. In that sense, $\delta$ is also an arbitrarily small positive number, but disappointingly, they seem to have left it out.

Or maybe it’s just a bit further along. You turn right and begin to follow the line. The tick marks count up by the hundredths, and you count $11$ before you get to the next number. Unfortunately, it’s not $\delta$.

$L$

Liouville’s constant
$0.11000100000\dots$

(For more digits, see OEIS A012245.) A transcendental number is a number that is not the root of any polynomial with integer coefficients. Transcendental numbers are some of the most interesting numbers, and include superstars like $\pi$, $e$, and $e^\pi$. Meanwhile, Joseph Liouville was a 19th century French mathematician, known for proving the existence of transcendental numbers. The proof goes like this:

1. Lemma: Liouville numbers exist.
2. Lemma: all Liouville numbers are transcendental.
3. Profit.

But what are these magical Liouville numbers? To discuss them, we need a concept of approximation for real numbers. Take Liouville’s constant ($L$), for example. One way to approximate $L$ is with a sequence of rational numbers that converges to it. For a first attempt, let the $q$th term be the closest rational number that has the denominator $q$.

$$\tfrac{0}{1}, \tfrac{0}{2}, \tfrac{0}{3}, \tfrac{0}{4}, \tfrac{1}{5}, \tfrac{1}{6}, \dots, \tfrac{1}{12}, \tfrac{1}{13}, \tfrac{2}{14}, \dots, \tfrac{2}{22}, \tfrac{3}{23}, \dots, \tfrac{3}{31}, \tfrac{4}{32}, \dots$$

At most, we’re off by $\flac{1}{2q}$, so the approximations get closer to $L$ as the sequence goes on… really slowly. What if we only kept “good approximations”: the fractions with denominator $q$ that are within $\flac{1}{q^2}$ of $L$?

$$\tfrac{0}{1}, \tfrac{0}{2}, \tfrac{0}{3}, \tfrac{1}{8}, \tfrac{1}{9}, \tfrac{2}{18}, \tfrac{3}{27}, \tfrac{10}{91}, \tfrac{11}{100}, \dots$$

Liouville numbers are those impossibly rare numbers. They are defined as all numbers which have infinitely many rational approximations within a margin of $\flac{1}{q^n}$ — for any $n$. Even if I take $n = 5040$, I can find a fraction $\flac{p}{q}$, which is within $\flac{1}{q^{5040}}$ of Liouville’s constant. And then I can find another, and another, and another… infinitely many of them.

Liouville proved the existence of his numbers by showing that for any integer $a > 1$, the following is a Liouville number:

$$\sum_{k=1}^{\infty} a^{-k!}$$

If you plug in $a = 10$, you get $L$, Liouville’s constant. (If you would rather work in binary, plug in $a = 2$, and check out OEIS A092874.)

Exercise: find a rational number $\flac{p}{q}$ which is within $\flac{1}{q^{42}}$ of $L$.

Floor $0.5$

The sound of fluids rushing through pipes is a bit quieter now. You are a mere ten steps up from the great pink $0$, which you can still see. Next to you is a much more imposing pink $0.5$. A scientist would grumble that this style is inconsistent, because it gives $1.5$ more significant figures than $1$. An artist would grumble that this style makes $0.5$ looks more important than $0$. A mathematician would grumble that symbolically, $0.5$ is a construct of our arbitrary base $10$ system. At least it looks cool.

To the left and right of the questionable pink number, are two plaques:

$P$

Prime constant
$0.4146825098\dots$

(For more digits, see OEIS A051006.) Depending on how many digits of the square root of two you know, you do a double take. Is that $\sqrt{2} - 1$? It isn’t, but you wouldn’t be wrong about the remarkable two-ness of this number. If you convert the prime constant to base $2$, you get $0.01101010001010001010\dots$ (for more digits, see OEIS A010051). The prime constant is known to be irrational, but Wikipedia says (without citation) that nobody knows whether it is transcendental. (Perhaps you can find out?) Exercise: why is it called the prime constant?

$\gamma$

Euler-Mascheroni constant
$0.5772156649\dots$

But what is $\gamma$? Everyone who hasn’t forgotten first-year calculus should be familiar with the pain of Riemann sums:

$$\ln(n) = \int_1^n \tfrac{1}{x} dx \approx \tfrac{1}{1} + \tfrac{1}{2} + \tfrac{1}{3} + \dots + \tfrac{1}{n-1} = \sum_{i=1}^{n-1} \tfrac{1}{i}$$

But this is pretty nifty: if we don’t have a calculator, we can add up a bunch of simple reciprocals by hand to approximate $\ln(n)$. The only problem is, we overshoot by a bit, because this is an upper Riemann sum. At $n = 7$, the error is more than $0.5$; at $n = 70$, the error is more than $0.57$. Keep increasing $n$ and eventually the error exceeds $0.577$, then $0.5772$, then $0.57721\dots$ and so you get the digits of the Euler-Mascheroni constant. (This is not a good way to compute $\gamma$.)

$$\gamma = \lim_{n\to\infty} \left(\sum_{i=1}^{n-1} \tfrac{1}{i} - \ln(n)\right)$$

In a nutshell, the Euler-Mascheroni constant is the error term of an ambitious crossover event between the discrete and continuous worlds of mathematics. Because of this, it pops up in all sorts of weird places. Perhaps the coolest example is Robin’s Theorem. If you find a number $n > 5040$ whose positive divisors (including $1$ and itself) sum to more than $e^\gamma n \ln(\ln(n))$, you will have refuted the Riemann hypothesis. If you prove that no such number exists, then you prove the Riemann hypothesis. Exercise: prove or disprove the Riemann hypothesis.

Floor $1$

Climbing another ten stairs, you return to civilization from the depths of the basement. A big pink $1$ greets you with congratulations. This is where we leave off, but before you can exit, another plaque introduces itself.

$G$

Gauss’s constant
$0.8346268416\dots$

(For more digits, see OEIS A014549.) Like Euler, Gauss needs no introduction. However, his constant likely does. Rooted in the depths of very hard integrals, Gauss’s constant is involved in a great number of seemingly unrelated problems under topics like:

• the gamma function
• the arc length of an $\infty$-shaped curve called the lemniscate of (Jakob) Bernoulli
• definite integrals of $\sqrt{\sin x}$ and $\sqrt{\cos x}$

In fact, the constant is named after Gauss because he proved that $G$ is the value of yet another very hard definite integral. Thus, with regard to undergraduate math, Gauss’s constant is probably the most esoteric one so far. Nonetheless, its definition is well within our grasp. $G$ is the reciprocal of the arithmetic-geometric mean of two simple numbers: $1$ and $\sqrt{2}$.

$$G = \tfrac{1}{\text{agm}(1, \sqrt{2})}$$

The magic behind $G$ is the "$\text{agm}$" part of its definition. The arithmetic-geometric mean (AGM) of two positive numbers $x$ and $y$ is the limit of a pair of sequences $(a_n)$ and $(g_n)$ defined like this:

1. Let $a_0 = x$ and $g_0 = y$.
2. $a_{i+1}$ is the arithmetic mean of $a_i$ and $g_i$: $(a_i + g_i)/2$.
3. $g_{i+1}$ is the geometric mean of $a_i$ and $g_i$: $\sqrt{a_i g_i}$.

Exercise: prove that $(a_n)$ and $(g_n)$ have the same limit. The sequences converge very quickly, so within a few operations on a normal calculator, you can compute $G$ to high precision. Just remember that back when Gauss was your age, he had to do it by hand.

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So there we go, the first floor of a multi-floor series dedicated to the constants that decorate the north-northeast stairwell in MC. Next time, we pick back up at Floor $1$ where we left off.

Exercise: don’t take the elevator.

Footnotes