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Re: computing times and output display

• To: mathgroup at smc.vnet.net
• Subject: [mg66962] Re: computing times and output display
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Mon, 5 Jun 2006 03:48:21 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <e5tqd6$cv2$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Steven Shippee wrote:
> [...]
> Anyone know why 8^8^8 (or 9^9^9) takes several minutes to "compute" and 

Here, the computation -- the surrounding double quotes are not needed -- 
   is done in _exact_ arithmetic and is entirely managed by software 
[1]. What one gets is infinite precision, but at a slow pace.

> display results, but in a new fresh kernel/environment 9^9^9. (decimal after 
> the last number 9) one gets an almost instantaneous response?  Is this 

Here, the computation is managed by the hardware [2]. Limited precision, 
hardware dependent, but fast.

Indeed, the whole section 3.1, "Numbers", of the Mathematica Book is 
worth reading [3].

> merely the difference in preparing for the output of the computation to the 
> screen ...

Say you want to display the result of 10^10^10 in exact arithmetic. 
10^10 is equal to 10 billions. Now, ask yourself how many digits are in 
10 raised to 10 billions? To gain some perspective, compare the number 
of digits of this number and the number of atoms in the Universe [4].

Note that a similar question was posted in the Mathematica student 
support forum a couple of months ago [5].

Regards,
Jean-Marc

[1]: http://documents.wolfram.com/mathematica/book/section-3.1.1

[2]: http://documents.wolfram.com/mathematica/book/section-3.1.6

[3]: http://documents.wolfram.com/mathematica/book/section-3.1

[4]: http://www.cs.umass.edu/~immerman/stanford/universe.html#q70

[5]: http://forums.wolfram.com/student-support/topics/10541