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Floor Functions
Posted:
Jan 17, 2009 12:17 PM


I am/was/still doing a research [ not the only 1] on these types of functions [ check wikipedia if you don't know what they are ]. I was a university student and I liked math alot. However things in real life has made me loose interest lately [ lately is not quite right, 4 years maybe?], and I had to withdraw [not cause of laziness].
I really wish to share some stuff with you :). Tell me if you like it or not :). Most of my work is nothing special, but math has been my favorite hobby since I was young and sharing stuff with people makes me happy :). I have never been on online forums about math before so not sure if I am posting this in the right place.
I discovered floor functions before I even learned them. I was proud that I managed to find stuff before I even learned it, and it was right. Of course I managed to also find some stuff thats interesting :).
check this formula
I think you may like this :) .
I will use the symbol [ .. ] for floor functions rather then the normal symbol because its easier to type.
A) a floor function to convert numbers from base 10 to other bases.
let assume you want to convert base 10 to base X.
then { the n in the dn should be a subscript }
dn = [ U/ b^(n1)]  b[ U/b^n]
where, U = value of number in base 10. , b = the base you want to convert the number n = the digit position of the number in BASE X.[ n=1 is left to radix point, n=0 is right to radix point ] dn = the value of the digit at digit position n in BASE X.
where , U 10 = ....{ d3 }{ d2 }{ d1 }.{ d0 }{ d1 }.... x ( by those brackets I mean a digit )
[10 and x should be small letters]
Example: 123 base 10 to base 4.
using formula.
its a known fact that the number in base 4 will not have any decimals or fraction, so its useless trying n= 0 or less.
let n = 1,
d1 = [ 123/4^(11)]  4[ 123/4^1] d1 = [123]  4[30.75] = 123  4(30) = 123 120 = 3
let n = 2,
d2 = [123/4^(21)]  4[ 123/4^2] = [ 30.75]  4[7.6875] = (30) 4 (7) = 2
let n = 3,
d3 = [ 123/ 4^(31)]  4[ 123/4^3] = [ 7.6875]  4[1.921875] = (7)  (4) = 3
let n = 4,
d4 = [ 123/ 4^(41)]  4[ 123/4^4] = [1.921875] 4[0] = 1
putting n= 5 or more, always shows 0., putting n = 0 or less, always shows 0, because there is no decimal.
this means the number in base 4 is, 1323
It works with all numbers , as long as b is a whole number. It also converts fractions. It still faster to use algorithym, but mathematicians may prefer formulas :).
Nothing important but its still interesting :). { if you interested in proof reply bk }
[ I am doing a research about floor function integrals. If you are interested I will tell you about it :).]
I will keep posting similar curiouse stuff from time to time if you want :).



