The square roots of recurring '1's

Identity · 2009-04-13 03:13:16

Try taking square roots of numbers of the form 111...1111 with different numbers of '1's. Notice anything interesting?

What happens when the number of '1's tends to infinity?

Can you predict what the square root of 111...111 (1000000 '1's) will be with great accuracy?

Can anyone explain why this is?

I thought it was interesting

mathsyperson · 2009-04-13 07:19:17

111...111 with n 1's is very close to 10^n/9, whose square root is 10^(n/2)/3.

So when n is even, there's a very close rational approximation to the number you want.
Your example would be close to 333....333 1/3, with 500000 3's in the integer part.

When n is odd, probably the best way to do it would be to use n-1 as above, and then multiply the answer by √10.

Identity · 2009-04-13 15:03:08

Hmm that is pretty cool!

Are there any other numbers whose square root has interesting properties?

Math Is Fun Forum

#1 2009-04-13 03:13:16

The square roots of recurring '1's

#2 2009-04-13 07:19:17

Re: The square roots of recurring '1's

#3 2009-04-13 15:03:08

Re: The square roots of recurring '1's

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