Several people (read:Dad and Mom) have been asking me if it's Math and Other Adventures, where's the math? I finally gave in and decided to start a mathematics column.
Thought #1: In mod n, the left diagonal of its Cayley multiplication table is a palindrome.
Mod (modular) is a system of arithmetic that is repeated. In other words, it goes round and round.
Ex: (mod 6)
012345
012345
012345
012345
012345
...
A number in mod is restricted between 0 and n. If it got larger, it would get reset back to the beginning.
Ex:
012345
012345
012345
...
Bold number 5 = 17
Now, here is a more succinct definition of modulus:
k(mod n)(that means k in mod n) = remainder of k/n
Here's why:
Imagine two sticks, a and b'.
Imagine b' is bigger than a.
Now say we cut off part of b' so there's a part b equal to a:
By definition, c is the remainder of b'/a.
But wait! If b is equal to a, b is equal to 0:(in mod a)01234...a-1(because then total number of numbers is a)
0
(Bold 0 = b)
But that means that if you add c to 0, you get c! Therefore, c is what b' is in mod a. (This also works when b is a multiple of a, as multiples of a are 0 as well.)
Now, what is a Cayley table? A Cayley table looks like the below. This is a Cayley multiplication table,which is basically a multiplication table for modulus.
Now take a look at the left diagonal of this table:(next picture). As you can see, it's a palindrome (1410141).Here's why:
The left diagonal is made out of squares(1,4,9,16,25,...) with the last square being (n-1)^2. Now, n-1is the same as -1 in mod n. n-2 is also the same as -2 , and so on. Around some point k is when x gets close to -x.
Now, we know that (-x)^2=x^2.
That means than (n-1)^2=(-1)^2=1^2,(n-2)^2=(-2)^2=2^2, and so on. This means that the values in the left diagonal's left side is mirrored by the values on the left diagonal's right side. But this means the left diagonal is a palindrome. So we have finally proven that the left diagonal is a palindrome.
Thank you for reading!
Dash
Note: Similar logic can be applied to find that the left diagonal and pandiagonals are palindromes.
