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:


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.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s