410: Math Paper
Title text: That's nothing. I once lost my genetics, rocketry, and stripping licenses in a single incident.
This comic is a set up to use the joke about imaginary friends by taking the concept of "friendly numbers" into the complex plane, which comprises numbers that have both a real and an imaginary part. Such a pun is both so obvious and so terrible that Cueball's superiors deem that he has lost the right to carry a "math license".
This is a recurring theme in earlier xkcd comics, being banned from holding presentations at conferences because said presentations are just elaborate puns. The title text takes the joke a step further, with the added hilarity of making the audience question exactly how Cueball was able to work a striptease into a presentation about genetic engineering and astrophysical rocket study. This is what All The Tropes calls a "noodle incident".
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, 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.
- 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 . 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. 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.
- [Cueball points to equations on the board.]
- 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.
- Professor: Hold on. Is this paper simply a giant build-up to an "imaginary friends" pun?
- [Cueball stands speechless for two panels.]
- Cueball: It MIGHT not be.
- Professor: I'm sorry, we're revoking your math license.