## Lolatility

When one writes binary numbers, it’s pretty tempting to view them as computer chat in which someone is laughing chronically. Here’s 85 (decimal) in binary:

lololol

Okay, yes. I put lowercase L’s instead of 1s. But I probably made my point. Let’s define the lolatility of a number, L(x), to be proportional to how lol-y a number is, made well-defined by assigning it to the number of groups of 0s that are between 1s. So, for example L(85)=3 (1010101 has three groups of 0s), because there’s three groups of 0s with 1s on both sides. L(200)=1 (11001000 has only one group of 0s between the second 1 and third 1 from the left; the 0s after the third 1 don’t count since they don’t have a 1 after them).

Of course, after defining a function, we get our hands dirty and plug in some numbers and see what we get. L(0)=0, L(1)=0, L(2)=0, L(3)=0, and L(4)=0. L(5)=1, the smallest positive lolatile integer (binary: 101). (A lolatile number is hereby defined as a number with positive lolatility.) The next lolatile integer is 9 (binary: 1001), and then 10 (binary: 1010) and 11 (binary: 1011). All of these numbers so far that are lolatile have a lolatility of 1. The first positive integer with a lolatility greater than 1 is 21 (binary: 10101): L(21)=2.

Notice that I have started using “integer.” We could easily extend the lolatility function to the set of the real numbers. For example, L(5/8)=1, since 5/8 in binary is 0.101; 8/5, on the other hand, is infinitely lolatile, since it has a 0 in its repeating block in binary; L(8/5)=∞. L(10.375)=2, since 10.375 in binary is 1010.011.

Here are some properties of the lolatility function, L(x):

• The function maps the real numbers to the nonnegative integers, with the exception of the infinitely lolatiles. The function is surjective but is neither injective nor bijective.
• L(x) is an even function; in other words, L(x)=L(-x). Appending a hyphen does not change the lolatile appearance of a string of digits.
• L(x)=L(2x) for all x, since in binary multiplying by 2 is only the moving of the radix point one place to the right.

Here are some properties of the lolatility function restricted to the positive integers:

• The smallest positive integer with lolatility n is $\sum\limits_{k=0}^n 2^{2k}$.
• Call LC(x) the lolatile count function, which outputs the number of lolatile positive integers x digits long in binary. Its domain is the positive integers. As a recursive function, LC(1)=0, LC(2)=0, and LC(n) where n>2 is $2^{n-2}+\text{LC}(n-1)-1$.
• The sum of the reciprocals of the lolatile positive integers diverges.
• The sum of the reciprocals of the non-lolatile positive integers converges, but is difficult to compute. We can see that every non-lolatile number is either all 1s or a group of some number of 1s followed by a group of some number of 0s. Notice that all numbers in the latter case are a power of two multiplied by a number in the first case. Thus, all non-lolatile nonnegative integers can be written in the form $2^a(2^b-1)$ for some nonnegative integers a and b. Thus, the sum is $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ plus $\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\frac{1}{24}+...$ plus $\frac{1}{7}+\frac{1}{14}+\frac{1}{28}+\frac{1}{56}+...$ etc. Applying the infinite series formula $S=\frac{a_1}{1-r}$, we get that the sum is $\frac{1}{1-\frac{1}{2}}+\frac{\frac{1}{3}}{1-\frac{1}{2}}+\frac{\frac{1}{7}}{1-\frac{1}{2}}+\frac{\frac{1}{15}}{1-\frac{1}{2}}+...=2(1+\frac{1}{3}+\frac{1}{7}+\frac{1}{15}+...)=2\sum\limits_{n=1}^\infty \frac{1}{2^n-1}$. This sum is very tough to compute, but is approximately 3.2133903048… in decimal (and is thus probably an infinitely lolatile number; it approximates as 11.0011011010… in binary).

Also, here is a badly drawn picture of a loling floor tile. It’s probably loling at you because your multivariable calculus textbook made you find its area using the Jacobian when you could have just rotated the coordinate grid.