# This will Break your Brain

Courtesy of Brishen:

‘Imagine a race of talking ants. The ants can compress the infinite digits of pi in an interesting way. For example, let us imagine that the ants can speak by manipulating their crude jaws. The first ant in the long parade of ants screams out the first digit, “3”. The next yells the number on its back, a “1”. The next yells a “4”, and so on. Further imagine that each ant speaks its digit in only half the time of the preceding ant. Each ant has a turn to speak. Only the most recent digit is spoken at any instant.

‘If the first digit of pi requires 30 seconds to speak (due to the ant’s cumbersome jaws and little brain), the entire ant colony will speak all the digits of pi in a minute! (Again, this is because the infinite sum 1/2 minute + 1/4 minute + 1/8 minute + … is equal to 1 minute.) Astoundingly, at the end of the minute, there will be a quick-talking ant that will actually say the “last” digit of pi! The geometer God, upon hearing this last digit, may cry, “That’s impossible, because pi has no last digit!”‘

## 4 Replies to “This will Break your Brain”

1. brishen says:

Of course that last ant also has 1/infinity time to say that digit, but as long as the math works.. 🙂

2. brishen says:

Of course that last ant also has 1/infinity time to say that digit, but as long as the math works.. 🙂

3. brishen says:

And actually, here is an easy proof of the above statement:

Lets say that the infinite sum does equal some number, we’ll call that S:

S = 1/2 + 1/4 + 1/8 + …
now divide S by 2:
S/2 = 1/4 + 1/8 + 1/6 + …
take the first equation and minus the second:
(S-S/2) = 1/2 (1/4 – 1/4) + (1/8 – 1/8) + …
=> S/2 = 1/2 => S=1

Simple really and quite cool. But yes, how does it approach reality? That’s where Zeno comes in and the answer that we really haven’t been able to explain in a good way for the past 2000+years.

4. brishen says:

And actually, here is an easy proof of the above statement:

Lets say that the infinite sum does equal some number, we’ll call that S:

S = 1/2 + 1/4 + 1/8 + …
now divide S by 2:
S/2 = 1/4 + 1/8 + 1/6 + …
take the first equation and minus the second:
(S-S/2) = 1/2 (1/4 – 1/4) + (1/8 – 1/8) + …
=> S/2 = 1/2 => S=1

Simple really and quite cool. But yes, how does it approach reality? That’s where Zeno comes in and the answer that we really haven’t been able to explain in a good way for the past 2000+years.

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