12 replies to this topic

[phil1882]

let heads be 1 and tails be 0. can you flip a coin enough times to form an irrational number?

[bushindo]

let heads be 1 and tails be 0. can you flip a coin enough times to form an irrational number?

This puzzle statement may be undercontrained,

Spoiler for it seems to be

Let f_i be a binary variable representing the i-th coin flip. There is an extra parameter that is required to form a number from the sequence of f_i. Consider the following equation for using f_i to generate the natural numbers, A^_N, from N coin flips


or this equation to use f_i to generate a random number between 0 and 1


Both of these methods return rational numbers for all N, but both of the above implicitly uses a factor of 2 in constructing the number A_N. There is nothing that really requires us to use 2, though. We can easily define a number generating scheme that returns an irrational number as follows


Under the scheme above, we are guaranteed to get an irrational number with any finite number of coin flips.

[superprismatic]

Spoiler for ....OR.....

Just put your generated sequence inside the square root function. You'll get an irrational number soon enough!

[phil1882]

okay maybe some clarification is in order.
clearly no finite number of flips will be enough, assuming you're not allowed to apply a function to it.
but can you mathematically flip an infinite number of coins to represent an irrational number?
i guess what I'm asking is, is the digits of a single irrational number countably infinite?

[jim]

Any real number can be expressed by a sign, a finite number of binary digits before a binary point, and then a possibly iinfinite and definitely countable number of binary digits after the binary point. 0.1010010001... would be an example of an irrational number in binary where every run of 0"s countains one more 0 than the previous run and is followed by a single 1.

[TheChad08]

Spoiler for

I suspect the answer is in binary.
I'd say 16 flips total to get "Pi"

[phil1882]

I'm very interested in knowing how you get 16 flips for the value pi, since pi is transcendental.

[superprismatic]

I'm very interested in knowing how you get 16 flips for the value pi, since pi is transcendental.

Spoiler for I think TheChad means

You can make the two ASCII characters "Pi" with 16 bits.

[bonanova]

TheChad:

[EventHorizon]

okay maybe some clarification is in order.
clearly no finite number of flips will be enough, assuming you're not allowed to apply a function to it.
but can you mathematically flip an infinite number of coins to represent an irrational number?
i guess what I'm asking is, is the digits of a single irrational number countably infinite?

Spoiler for My thoughts on this...

As you know, the integers are countably infinite. You can simply map each nth integer to the nth digit after the decimal place to see that the digits of an irrational number are countably infinite.

You ask if you can flip an infinite number of coins to represent an irrational number, and the answer is yes.

If you represent the number in binary and have a coin per decimal place, you'll **always** end up with an irrational number. To be rational, the digits would eventually need to continue to repeat through infinity. By this construction, each digit is random and therefore no repetition can be continued through infinity. In other words, the probability you end up with a rational number is 0%, and the probability you end up with an irrational number after flipping an infinite number of coins representing its digits is 100%.

Can you end up with a rational number using a fair coin and an infinite number of flips? In theory, yes (it could, for example, come up heads every time). In practice, no... it is impossible.

Can you end up with a rational number using a fair coin and an infinite number of flips if you allow reordering of digits/coins? Yes. In fact, you can end up with **any** rational number whose repeated pattern contains at least one one and at least one zero... and you'll still use every coin... eventually... if you want to.