Photo Credit: www.cut-the-knot.org… but I helped, too.
Hi blog people,
Cryptography is hard to do by hand. I’m just throwing that out there. The reason I’m realizing this now may or may not be the fact that I have had to encipher the Marquis de Lafayouhavenochanceofencipheringmyquote’s quote, a total of 150 characters, with four different keys. I started building a program to encipher the code, then realized two of my TI-84′s shortcomings (well, three really):
1. The TI-84 has no modulo function. After Googling “modulo function on the TI-84″ I found a three-line substitute. FOR A SINGLE #!&%*!$ COMMAND. Which, by the way, I had to write out 52 times(26 times for addition for every letter, 26 for multiplication).
2. The TI-84 can’t convert numbers into variables… at least not when every single variable has a value. There are 27 total possible variables on the TI-84, and I had already used all 27. Converting it back would be impossible… think trying to make x=x+2.
3. Overall, BASIC is old, difficult, and hard to work with. Kind of like that old man who sits on his rocking chair and yells, “Get off my lawn!”
*has evil thought*
*builds program that says “Get off my lawn” every time you try to put something into the calculator*
Anyways, I was looking at the modulo tables for addition, and I noticed something: without fail, the numbers in the top left corner (1 times 1) and the bottom right corner (the number of the table-1 times the number of the table -1) are always 1.
So, modular multiplication tables. They’re the same as regular multiplication tables, except the numbers being multiplied have to be less than a given number (we’ll call it n). Also, every number in the table must be “modulo n”, meaning that what really goes in the box is the remainder of what used to go in the box when it’s divided by n.
It sounds complicated, and, well, it is. Lucky you have me, then.
For example, let’s sat we’re in the modular multiplication table for 5. In the box where the four and the three coincide, the 4 times 3 box, there should be a 12. But since we’re in a modulo table, and our special number is 5, we have to take 12 mod 5, or the remainder when 12 is divided by 5. 12/5 gives a remainder of 2, so a two goes in the 4 times 3 box.
By the way, modular tables inspire individuality, while regular tables don’t. They improve the self-esteem of numbers, because every number gets its own table, whereas regular multiplication tables lump all of the numbers together. I had a whole spiel prepared, but, seeing as you’re not paying attention to anything I write COOKIES <—(except that because it’s bold) and just skimming this without really reading, I’ll get to the point.
What was my point again?
Oh, yeah, the right-corner-one thing.
So what this implies is that for any number n, (n-1)(n-1) mod n = 1. Like in the table, . Complicated? With Sam on your side, you won’t be <insert word here that rhymes with side>!
Simplifying the original equation, we get . So, using the definition of mod, what we need to do now is divide by n. n can divide and -2n, but it can’t divide 1, so we are left with a remainder of 1. Here, let me repeat that, for those of you sleeping in the back. WE HAVE A REMAINDER OF 1. Which proves that, no matter how many times you click the little up arrow next to the online modulo table, the number in the bottom right corner will always, infuriatingly, be 1.
I just proved something that wasn’t in my textbook! Yay!
Even though technically, this whole unit isn’t in the textbook. But still.
In case you live under a rock (CECILIA!), a touchdown celebration dance is when a player scores, and they show off to the crowd that they scored by doing a little dance. As if the crowd didn’t know already. So I decided that I wasn’t getting enough joy out of completing a proof (the little black box at the end just didn’t cut it), and decided to compile a list of possible “Proof Celebration Things”. PCTs for short.
1. Shove your proof (figuratively) into other peoples’ faces.
2. Shove your proof (literally) into other peoples’ faces
3. Read the proof in your most dramatic voice possible. See if anyone stops what they are doing.
4. Laugh and taunt whatever forced you to do the proof (teacher, textbook, etc.)
More PCTs will come soon.
A PCT of the week section, perhaps?
5. Eat your next meal with a spork,
I actually did numbers 1 and 2 after solving #6 on the homework.
You should, too.
I assume no liability for any negative outcomes. Especially for #4.