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September 10

Enormous number digits puzzle

How many digits does $10^{111}+(-2)^{111}$ have? What are the first ten digits? What is the last one? Widener (talk) 01:27, 10 September 2011 (UTC)[reply]

Quick answer: WolframAlpha. To get the answers to your first two questions, note that 2 raised to a power is much less than 10 raised to the same power. To get the last digit, use modular arithmetic with a modulus of 10. Exponentiation by squaring will be useful too. —Bkell (talk) 01:33, 10 September 2011 (UTC)[reply]

(small special cases of) Fermat's little theorem and the Chinese remainder theorem would also work. Algebraist 12:46, 10 September 2011 (UTC)[reply]

Will Bkell get a share of the prize if you win? Looie496 (talk) 16:07, 10 September 2011 (UTC)[reply]

What do you mean? Widener (talk) 01:15, 11 September 2011 (UTC)[reply]

Also note that log₁₀ 2¹¹¹ = 111 log₁₀2 ≐ 33.41, so 2¹¹¹ has 34 digits (when represented in base-10) and your sum will start with 111 - 34 = 77 nines. Even without a calculator, everyone knows that 3 < log₂ 10 < 4 because 8 = 2³ < 10 < 2⁴ = 16, so 1/4 < log₁₀ 2 < 1/3, so 2¹¹¹ has between a quarter and a third as many digits as 10¹¹¹. See common logarithm and logarithm. -- 110.49.65.26 (talk) 13:05, 11 September 2011 (UTC)[reply]

Limit question

$\lim _{x\to 0^{-}}x^{x}$ : http://www.wolframalpha.com/input/?i=limit+of+x^x+as+x+approaches+0+from+below

Since x^x is not a function when x is small and negative how does WolframAlpha calculate the limit from below as 1? How do you choose which branch to consider when calculating the limit? Not all branches have real parts approaching +1. --41.135.50.237 (talk) 07:12, 10 September 2011 (UTC)[reply]

Where did you get that it would not be 1 on some branch? See Indeterminate form. x is analytic and non-zero in that neighbourhood and therefore the limit is 1. Dmcq (talk) 11:32, 10 September 2011 (UTC)[reply]

x^x is calculated as e^{x log x}. It doesn't really matter which branch you start from for the log here provided you keep to it. Dmcq (talk) 11:40, 10 September 2011 (UTC)[reply]

The article you linked says that for the limit to be 1, f (the denominator) "must not be negative in the domain of the limit". Also, if n is odd and negative then the real root of

(1/n)^{1/n}

approaches -1 as n increases to infinity. --41.135.50.237 (talk) 17:34, 10 September 2011 (UTC)[reply]

Sorry you're quite right, the value of the limit is undefined for negative reals. I don't consider power as being defined for negative reals, only nth roots and so was considering that the principal value of the complex power was being used. Also if you choose a cut which winds around 0, or else just assume z is continuous winding round zero, you can get z^z to go to any point in the unit circle so it is undefined too even with complex powers if you don't use a straightforward branch cut. Dmcq (talk) 16:09, 11 September 2011 (UTC)[reply]