410: Math Paper
explain xkcd: It's 'cause you're dumb.
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{{comic | {{comic | ||
| number = 410 | | number = 410 | ||
| − | | date = | + | | date = April 14, 2008 |
| title = Math paper | | title = Math paper | ||
| image = math_paper.png | | image = math_paper.png | ||
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==Explanation== | ==Explanation== | ||
| − | + | This comic is a set up to use the joke about Imaginary Friends by taking "friendly numbers" into the complex (imaginary) plane. | |
| − | + | Imaginary numbers on the complex plane are of the form '''a''' + '''b'''''i'' where '''a''' and '''b''' are constants and ''i'' is the square root of negative 1 (an impossibility in the plane of "regular" numbers). | |
| − | + | Joel Bradbury has a wonderful explanation of Friendly Number on his site http://joelbradbury.net/notes/friendly_numbers. The following explanation of Friendly Numbers is taken from his site: | |
:What are Friendly Numbers? | :What are Friendly Numbers? | ||
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: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. | :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 | + | :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== | ||
| − | :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. [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. [Points to equations on the board.] |
| − | :Guy in room: Hold on. | + | :Guy in room: Hold on. Is this paper simply a build-up to an "imaginary friends" pun? |
| − | :[Lecturer stands speechless] | + | :[Lecturer stands speechless.] |
:Lecturer: It MIGHT not be. | :Lecturer: It MIGHT not be. | ||
| − | :Guy in room: I | + | :Guy in room: I'm sorry, we're revoking your math license. |
{{comic discussion}} | {{comic discussion}} | ||
| − | + | [[Category:Math]] | |
| + | [[Category:Cueball]] | ||
Revision as of 03:05, 12 March 2013
| 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 "friendly numbers" into the complex (imaginary) plane.
Imaginary numbers on the complex plane are of the form a + bi where a and b are constants and i is the square root of negative 1 (an impossibility in the plane of "regular" numbers).
Joel Bradbury has a wonderful explanation of Friendly Number on his site http://joelbradbury.net/notes/friendly_numbers. The following explanation of Friendly Numbers is taken from 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 = 6, 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: 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. [Points to equations on the board.]
- Guy in room: Hold on. Is this paper simply a 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.
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