Science Forums 2

[Turtle]

___Here we are again then. I want to clarify that I said two principles in the K function, but mentioned three. Just so, three priciples:addition, repetition, & comparison, ie conditional testing. Because we soon will start compounding these principles, it is not incorrect to say the course is fractal inasmuch as it is self similar.
___Now strictly speaking, a proof is required for my claim K(n) is congruent n mod 9. Instead, I prepared a recreational demonstration. Down below, you see I arranged some little boxes, 3 rows of 28. The bottom row is open empty boxes, the middle row is closed empty boxes with blank tops, & on top, closed boxes of rocks with the first twenty eight integers expressed in Base ten. A pile of rocks on the ground in front which I gathered up from the cave floor. Finally, in the top row, the number of rocks in each box is the same as the number on its lid, & the ellipsis says of course add as many boxes as you please.
___This wouldn't be any kind of expedition (as if following a turtle into a cave isn't daring enough), if you didn't get your hands a little dirty, & I know everyone wants to try their shiny new K functions. So, take each written integer,n, from the top row lids & use the K funtion on it & write the result on the lid of the box directly below your integer n. Do this for the whole row.
___Now out of each box in the top row, take out all the rocks of one box, & seperate them into piles of nine. Any rocks left over, drop into the empty box in the third row directly beneath the box you took them from. Diascard your piles of nine on the pile on the ground.
___When you compare how many stones in a bottom row box(which of course is the remainder, or residue, from casting out nines), with the number written on the middle row lid above it, you see they agree, i.e. congruent. Now, just one last point before I give you a rest. The range of the K function does not return zero as does the modulo function, but rather 9. With the katabatak function, you get rocks in your box. Adieu until next time, when we start pointing our K functions at squares.