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.
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).
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.
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.
- What are Friendly Numbers?
- We need first to get 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 it’s 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.
- Lecturer: 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.
- [Lecturer points to equations on the board.]
- Guy in room: Hold on. Is this paper simply a giant build-up to an "imaginary friends" pun?
- [Lecturer stands speechless.]
- Lecturer: It MIGHT not be.
- Guy in room: I'm sorry, we're revoking your math license.