410: Math Paper
Title text: That's nothing. I once lost my genetics, rocketry, and stripping licenses in a single incident.
The math paper Cueball is in the process of describing in this comic turns out to be nothing but an elaborate setup for a joke about imaginary friends by taking the concept of "friendly numbers" into the complex (imaginary) plane, which comprises complex numbers that have both a real and an imaginary part (see details below).
Cueball is challenged on this setup by his superiors, specifically the Cueball-like guy sitting at the end of the table, who look straight through his first line-up for the joke, and ask him directly if this is just a build-up for this joke. Cueball tries at first to look like he has no idea what he's talking about, then lowers his head, in shame, and finally tries to state that it might not be such a setup. But it is too late now.
Such a pun is both so obvious and so terrible that Cueball's superiors deem that he should no longer have a license to math, and they thus revoke Cueball's "math license." Of course you do not need a math license, but that is part of the comic's concept along the lines mentioned here below and further elaborated in the title text.
The title text takes the joke a step further, with the added hilarity of making the audience question exactly how Cueball/Randall was able to work a striptease into a presentation about genetic engineering and astrophysical rocket study (or possibly genetics and rockets into a striptease), and then even manage to lose all three licenses in one go. This is what TV Tropes calls a "noodle incident."
The whole comic is basically Randall making the joke that Cueball never got around to, but packing it up so we think it is about something else. Randall has often made such feeble jokes, but by putting them into a context where someone listening may comment on how bad that joke is or have to explain the joke, it somehow becomes alright, and he can get away with these jokes anyway. (See for instance 18: Snapple).
An imaginary number is a number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = -1 (an impossibility for regular, "real numbers," for which all squares are positive). The name "imaginary number" was coined in the 17th century as a derogatory term, since such numbers were regarded by some as fictitious or useless, but over time, many applications in science and engineering have been found.
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi (the formula shown at the bottom of Cueball's slide ), where a and b are called, respectively, the real part and the imaginary part of the complex number. If a and b are both integers, the complex number is called a Gaussian integer (as Cueball mentions). The complex plane is an X-Y plot with a on the X axis and b on the Y axis. (Such a plane is shown at the bottom of Cueball's slide).
Joel Bradbury (once) had the below cited and wonderful explanation of friendly numbers on his site:
- What are Friendly Numbers?
- We need first to define a divisor function over the integers, written σ(n) if you're so inclined. To get it first we get all the integers that divide into n. So for 3, it's 1 and 3. For 4, it's 1, 2, and 4, and for 5 it's only 1 and 5.
- Now sum them to get σ(n). So σ(3) = 1 + 3 = 4, or σ(4) = 1 + 2 + 4 = 7, and so on.
- For each of these n, there is something called a characteristic ratio. Now that's just the divisors function over the integer itself: σ(n)/n. (This is the formula shown at the top of Cueball's slide). So the characteristic ratio where n = 6 is σ(6)/6 = 12/6 = 2.
- Once you have the characteristic ratio for any integer n, any other integers that share the same characteristic are called friendly with each other. (This is what is written in the frame in Cueball's slide, spelling friendly numbers as friendly #s). So to put it simply, a friendly number is any integer that shares its characteristic ratio with at least one other integer. The converse of that is called a solitary number, where it doesn't share its characteristic with anyone else.
- 1, 2, 3, 4, and 5 are solitary. 6 is friendly with 28; σ(6)/6 = (1+2+3+6)/6 = 12/6 = 2 = 56/28 = (1+2+4+7+14+28)/28 = σ(28)/28; this is due to both of them being perfect numbers.
- [Cueball, holding a pointing stick, is using it to point at an equation on a panel. He is looking to the right. There are several parts of the panel that can be read. At the top, there is a formula. Below is a frame with text. Below again to the left is an X-Y plot with small dots all over all four quadrants, probably indicating the complex numbers with b on the Y and a on the X axis. Finally, right of this is yet another formula.]
- Cueball: In my paper, I use an extension of the divisor function over the Gaussian integers to generalize the so-called "friendly numbers" into the complex plane.
- σ(n)/n = d(n)
- Friendly #s share d(n)
- For a + bi...
- [The audience to the right of Cueball consist of two Cueball-like guys (one in front and one in the back), and between them are Hairbun, with glasses, and Megan. They sit around a table; only Hairbun is on the near side. The Cueball-like guy sitting to the right is at the end of the table, the other two are on the far side. The Cueball at the end of the table is talking, the other three have turned to look at him:]
- Guy at the end of the table: Hold on. Is this paper simply a giant build-up to an "imaginary friends" pun?
- [Back to Cueball, who stands speechless.]
- [One more beat panel with Cueball, who now looks down.]
- [Zoom out to Cueball and the front end of the table with the Cueball-like guy who has not spoken yet and Hairbun who now looks at Cueball. Cueball looks up again and speaks. The guy at the end of the table speaks off panel.]
- Cueball: It might not be.
- Guy at the end of the table (off panel): I'm sorry, we're revoking your math license.
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