math class is getting to me head
check out this sequence of numbers:
A0 = .1
A1 = .10
A2 = .101
A3 = .10110
A4 = .10110101
A5 = .1011010110110
A6 = .101101011011010110101
A7 = .1011010110110101101011011010110110
A8 = .1011010110110101101011011010110110101101011011010110101
A9 = .10110101101101011010110110101101101011010110110101101011011010110110101101011011010110110
&c
basically the way you generate the next number is by attaching the second-to-last one to the end of the last one.
i'm remembering this sequence from a book about the golden ratio, where it was presented not as a schematic for a repeating decimal but just as an idea for a string of 1's and 0's, boundless both to the left and the right. The original idea was to start with a 1 and apply the following pattern: for every 0 substitute a 1, and for every 1 substitute a 1 and a 0. Guess what? The total number of digits is a fibonacci, as are the respective counts of 1's and 0's. The book also demonstrated the possibility for all manner of ingrained patterns in such a string - don't want to get into it now, but suffice it to say they looked pretty cool.
I started sketching it in the middle of math. our topic for a while has been real numbers and limits so I quickly realized that if made a decimal expansion, this would converge quickly on some sort of limit. So let's call it A: define A as lim A(n).
What can we determine about A? Is it possibly, even possibly, an algebraic number, i.e. the solution of an integer polynomial equation? Also, the example implies I'm talking about base 10: what if it's in binary? base 3?
This is getting too complicated for vagueries. The problem deserves a better definition:
define {A} as the sequence (1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, ...) (note: an algorithm by which to define {A} would probably help. It would probably use a mix of recursion and the same process I used to define the sequence (1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,...), i.e. deriving the value of an entry via its place-value's place in another sequence, here the fibonaccis and in the quoted case the triangle numbers 1,3,6,10,...)
define B(x,n) as the sum from i=1 --> n of A_i/n^i, in other words as some kind of limited sum related to A. Given an x>1 and a sequence of integers n, will the corresponding "B sequence" converge?
Since x>1, the difference between subsequent items in that sequence decreases exponentially. From there it wouldn't be very hard to show that it's a Cauchy sequence, and hence that it does converge. As such we can define a limit:
define f_x as the limit of B(x,n). (the infinite decimal I gave at the beginning of this post would correspond to f_10, but I'm also pretty interested in f_2. note that f_1 is excluded from discussion since x>1, and besides a value of 1 would clearly cause this shit to diverge.)
What can we determine about f? it's continuously descending, and that's about all i can think of.
Can we determine anything about individual entries in f?
If I knew anything about matlab I might be able to graph some of this shit. Christ I am a nerd.
A0 = .1
A1 = .10
A2 = .101
A3 = .10110
A4 = .10110101
A5 = .1011010110110
A6 = .101101011011010110101
A7 = .1011010110110101101011011010110110
A8 = .1011010110110101101011011010110110101101011011010110101
A9 = .10110101101101011010110110101101101011010110110101101011011010110110101101011011010110110
&c
basically the way you generate the next number is by attaching the second-to-last one to the end of the last one.
i'm remembering this sequence from a book about the golden ratio, where it was presented not as a schematic for a repeating decimal but just as an idea for a string of 1's and 0's, boundless both to the left and the right. The original idea was to start with a 1 and apply the following pattern: for every 0 substitute a 1, and for every 1 substitute a 1 and a 0. Guess what? The total number of digits is a fibonacci, as are the respective counts of 1's and 0's. The book also demonstrated the possibility for all manner of ingrained patterns in such a string - don't want to get into it now, but suffice it to say they looked pretty cool.
I started sketching it in the middle of math. our topic for a while has been real numbers and limits so I quickly realized that if made a decimal expansion, this would converge quickly on some sort of limit. So let's call it A: define A as lim A(n).
What can we determine about A? Is it possibly, even possibly, an algebraic number, i.e. the solution of an integer polynomial equation? Also, the example implies I'm talking about base 10: what if it's in binary? base 3?
This is getting too complicated for vagueries. The problem deserves a better definition:
define {A} as the sequence (1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, ...) (note: an algorithm by which to define {A} would probably help. It would probably use a mix of recursion and the same process I used to define the sequence (1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,...), i.e. deriving the value of an entry via its place-value's place in another sequence, here the fibonaccis and in the quoted case the triangle numbers 1,3,6,10,...)
define B(x,n) as the sum from i=1 --> n of A_i/n^i, in other words as some kind of limited sum related to A. Given an x>1 and a sequence of integers n, will the corresponding "B sequence" converge?
Since x>1, the difference between subsequent items in that sequence decreases exponentially. From there it wouldn't be very hard to show that it's a Cauchy sequence, and hence that it does converge. As such we can define a limit:
define f_x as the limit of B(x,n). (the infinite decimal I gave at the beginning of this post would correspond to f_10, but I'm also pretty interested in f_2. note that f_1 is excluded from discussion since x>1, and besides a value of 1 would clearly cause this shit to diverge.)
What can we determine about f? it's continuously descending, and that's about all i can think of.
Can we determine anything about individual entries in f?
If I knew anything about matlab I might be able to graph some of this shit. Christ I am a nerd.