/me picks up gauntlet...
Ok, here we go. 10033 is the Holy Grail. Yep.
First, lets say it's not 10033, but 100.33. Now, bear with me here for a sec.
100.33
100 +33/100
100 +1/(100/33)
100 +1/(3+1/33)
[100,3,33]
What I've done here is reformated the decimal number into a
continued fraction, which are useful for representing real numbers with infinite decimal digits after the decimal point, including some numbers with non-periodic decimal digits (irrational). A number expressed in continued fraction is of the form a+1/(b+1/(c+1/(d+1/...))) etc., where the list form is obtained by replacing the '+1/(' pattern that repeats all the time with a comma, and putting the remaining list of numbers inside brackets. Thus, [a,b,c,d,...]. An easier way to visualize it is:
Code: Select all
1
a + ------------------- = [a,b,c,d,...]
1
b + -------------
1
c + -------
d + ... etc.
But that takes a lot of space. The first number 'a' can be any integer, but all the others must be positive integers.
Notice how in that intertpretation of 100.33 all the numbers in the continued fraction list form use the same digits, 0, 1, and 3. This is not always the case for a continued fraction, of course, regardless of which digits are used in the decimal representation. Even with the same 10033, take an alternative interpretation:
1.0033
1+33/10000
1+1/(10000/33)
1+1/(303+1/33)
[1,303,33]
Uh... ok that one also worked with every digit being 0, 1, or 3. I must have been looking for another one. Lets try this:
10.033
10+33/1000
10+1/(1000/33)
10+1/(30+10/33)
10+1/(30+1/(33/10))
10+1/(30+1/(3+3/10))
10+1/(30+1/(3+1/(10/3)))
10+1/(30+1/(3+1/(3+1/3)))
[10,30,3,3,3]
Huh... well that took a bit more work but, I'll be damned, it still uses the same digits... hang on I got to show you this is not always the case:
1003.3
1003+3/10
1003+1/(10/3)
1003+1/(3+1/3)
[1003,3,3]
Oh dear... only one more to try or I'll have to eat my words with this...
.10033
0+10033/100000
0+1/(100000/10033)
0+1/(9+9703/10033) ahh there we go, we got a 9!...
0+1/(9+1/(10033/9703))
0+1/(9+1/(1+330/9703))
0+1/(9+1/(1+1/(9703/330)))
0+1/(9+1/(1+1/(29+133/330)))
0+1/(9+1/(1+1/(29+1/(330/133))))
0+1/(9+1/(1+1/(29+1/(2+64/133)))) holy crap this is a long one!
0+1/(9+1/(1+1/(29+1/(2+1/(133/64)))))
0+1/(9+1/(1+1/(29+1/(2+1/(2+5/64)))))
0+1/(9+1/(1+1/(29+1/(2+1/(2+1/(64/5))))))
0+1/(9+1/(1+1/(29+1/(2+1/(2+1/(12+4/5))))))
0+1/(9+1/(1+1/(29+1/(2+1/(2+1/(12+1/(5/4)))))))
0+1/(9+1/(1+1/(29+1/(2+1/(2+1/(12+1/(1+1/4))))))) phew!
[0,9,1,29,2,2,1,1,12,1,4]
Aha, so there you have it. It could end up being any numbers. But more appropriately, if 10033 was meant to be represented with a continued fraction, it probably means that the actual number has infinite digits after the decimal point. One possibility would be 100.333333... with indefenitely many 3's:
100.333333333...
100+1/3
[100,3]
Another one might be:
1.0033003300330033...
1+1/303
[1,303]
If you're up to it you can figure out other similar interpretations with infinite decimal digits, but I'm tired of all these fractions. Oh, uh, alright. Just one more I just thought of. Last one, I promise:
100/33
3+1/33
[3,33]
Ha, that's it! Remove the comma and you get 333, the triple trinity, or a half-beast...
There, I'm done and fried.
](./images/smilies/eusa_wall.gif "Brick wall"){kind=small}