# 410: Math Paper

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:What are Friendly Numbers? | :What are Friendly Numbers? | ||

− | :We need first to get define a divisor function over the integers, written σ(n) if | + | :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. | :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 | + | :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 | + | :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. | + | :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. |

==Transcript== | ==Transcript== |

## Revision as of 11:15, 22 May 2014

Math Paper |

Title text: That's nothing. I once lost my genetics, rocketry, and stripping licenses in a single incident. |

## Explanation

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 ask just how the hell Cueball was able to work a striptease into a presentation about genetic engineering and astrophysical rocket study. This is what TV Tropes calls a "noodle incident".

### Math

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 *i ^{2} = -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.

Joel Bradbury has a wonderful explanation of friendly numbers on his site:

- 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.

## Transcript

- [Lecturer points to equations on the board.]
- 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.

- Cueball: Hold on. Is this paper simply a giant build-up to an "imaginary friends" pun?

- [Lecturer stands speechless for two panels.]

- Lecturer: It MIGHT not be.
- Cueball: I'm sorry, we're revoking your math license.

**add a comment!**⋅

**add a topic (use sparingly)!**⋅

**refresh comments!**

# Discussion

Shouldn't it say something about the whole math licence, and that you don't actually need a licence to do math? 108.162.231.228 21:01, 31 October 2013 (UTC)Synthetica

Despite what this comic implies, the divisor function is defined over the Gaussian integers. There still is a problem, though. If a divides b, then so does -a, along with ai and -ai. The divisors will inevitably sum to zero. You could get around this by ignoring all the numbers that aren't in a given quadrant. I personally like the idea of using ones where the real part is greater than the imaginary part (although that still does become a problem with multiples of 1+i). This way, a friend of a natural number will also be a natural number (though it's only the same as what you'd get normally if all the factors are three mod four). 199.27.128.167 (talk) *(please sign your comments with ~~~~)*

- I do not agree. Here is how it should work. You should define that a divides b if and only if there is a natural number n such that an = b. This way, natural numbers don't get new divisors when you move to the Gaussian plane. Consequently the extended sigma function gives the same value as the classical one when applied to a natural number. So, natural numbers will be friends according to the new definition if and only if they are friends according to the old definition and we are indeed allowed to say that the new definition extends the old one (Burghard von Karger). 162.158.93.28 (talk)
*(please sign your comments with ~~~~)*

Does someone care to hazard a guess about the alt text? Are there any possible papers that could have been written about those three fields? 173.245.53.186 15:04, 5 January 2014 (UTC)

I think no one's noticed the "Straight man's deadpan stare" in panels 3 and 4. After being rudely interrupted in panel two, the lecturer stares silently at the person who stopped him in panel 3. Note the shape of the lecturer's head: his forehead and chin are facing towards the right as revealed by the angled features at the top and bottom of his head. In panel 4 the lecturer turns and stares at the reader and stands silent. Note the completely round shape of the head. This is the classic deadpan stare. Also, am I the only one who assumed that Cueball himself is the lecturer and the person interrupting him is likely the proffessor of the class?ExternalMonolog (talk) 20:14, 21 January 2014 (UTC)ExternalMonolog

- I don't know. In panel three DOES look straight at the commenter in the audience in a deadpan way, like he'd been caught unawares by the person's observation, but in panel four seems to look down, head tilted towards the floor as if wondering "aww shit, now what do I do?" Or like he felt slightly ashamed of being caught. 108.162.216.76 22:56, 27 February 2014 (UTC)
- I thought it was pretty plainly evident that Cueball is the presenter - going along with the XKCD theme of banning/revoking Cueball's privilege due to "inappropriately" wasting people's time at an ostensibly serious event. Also, in relative agreement with 108.162.216.76, I think that Panel 3 is his feeble attempt to continue to look serious (or straight/deadpan) while thinking of a way to respond, followed in Panel 4 with a slightly defeated shoulder/head slump (giving up with trying to perpetuate his premise). Panel 5 could be viewed as a plain admission, but I prefer to see it as a final, hail-Mary attempt to maintain the graces of the audience without
*actually lying*to them. -- Brettpeirce (talk) 16:09, 5 March 2014 (UTC)- They are probably beat panels more than anything else. -Pennpenn 108.162.250.155 00:13, 27 March 2015 (UTC)

- I thought it was pretty plainly evident that Cueball is the presenter - going along with the XKCD theme of banning/revoking Cueball's privilege due to "inappropriately" wasting people's time at an ostensibly serious event. Also, in relative agreement with 108.162.216.76, I think that Panel 3 is his feeble attempt to continue to look serious (or straight/deadpan) while thinking of a way to respond, followed in Panel 4 with a slightly defeated shoulder/head slump (giving up with trying to perpetuate his premise). Panel 5 could be viewed as a plain admission, but I prefer to see it as a final, hail-Mary attempt to maintain the graces of the audience without