https://padlet.com/andrewmis/stuffs & things en-us 2020-07-08 21:22:02 UTC 2025-10-23 23:58:58 UTC hello@padlet.com andrewmis https://padlet.com/andrewmis/stuffs/wish/652303422 the smallest things in our lives often regarded with a perhaps at most a proportional sense of significance are actually some of the most important.  i disagree with the macro vs micro dichotomy. whenever i hear "i just want to understand the big picture" my reply is "but you cannot understand the bigger picture without first understanding each smaller part."  before we calculate the sum of the series we must understand how each element relates to its sequence. oh yes, infinity seems like such a long time and yes, some things seem so infinitesimally small.  but try it, take the time and look closely, and just maybe you will see there are things within you never even knew were there that are so beautiful you come out seeing the whole in a greater way than you could have previously ever imagined.]]> 2020-07-11 14:45:16 UTC https://padlet.com/andrewmis/stuffs/wish/652303422 andrewmis https://padlet.com/andrewmis/stuffs/wish/658975091 We can write any integer number as a sum of powers of 10 times each of its digits:

A = (a₀ x 10⁰) + … + (a_s x 10^s) + … + (a_t x 10^t) + … + (a_n x 10ⁿ)

A’ = (a₀ x 10⁰) + … + (a_t x 10^s) + … + (a_s x 10^t) + … + (a_n x 10ⁿ)

Important to note t - s = 1 since transpose means the digits getting their orders switched are right next to each other.

So to show A - A’ is divisible by 9.. Notice all the terms we didn’t mix up cancel out when subtracting leaving only the two from the transpose left:

A - A’  = (a_s x 10^s) + (a_t x 10^t) - (a_t x 10^s) - (a_s x 10^t)
= (a_s - a_t) x 10^s + (a_t - a_s) x 10^{s + 1}
= 10^s x (a_t - a_s) x (10¹ - 10⁰)
= 9 x [(a_t - a_s) x 10^s]

Since a_t - a_s is an integer we can conclude A - A’ is divisible by 9! ]]> 2020-07-21 04:49:38 UTC https://padlet.com/andrewmis/stuffs/wish/658975091