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From: Guy Steele
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Date: Tue, 22 Sep 92 11:50:22 EDT
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To: mb8d+@andrew.cmu.edu
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In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu>
Subject: cubes are great
Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
From: Matthew John Bushey
Does anyone out there know what is the cubed root of 81?
Just wondering....
Well, the "root of 81" is 9 (recall that when you don't say
what kind of root you want, the default is "square"), and
9 cubed is 729.
... Eh? Oh, you meant the "cube root", not the "cubed root"?
Well, that's another kettle of fish entirely. The n'th root
of x is equal to x raised to the power 1/n. I fed this to
my friendly Common Lisp system:
> (expt 81 1/3)
4.3267487109222245
If I were you, I wouldn't trust the last few digits of this
approximation, but fifteen decimal places ought to hold you
for now.
Here's how you could estimate it in your head.
Note that 81 = 3 to the fourth power, so
1/3 4 1/3 4/3 1/3
81 = ( 3 ) = 3 = 3 ( 3 )
Now, the cube root of 3 is surely between 1 and 2,
because 1 cubed is 1 and 2 cubed is 8. So the cube
root of 3 is 1 plus some smaller fractional amount x.
3 2 3
So 3 = (1 + x) = 1 + 3 x + 3 x + x (binomial expansion).
3
Let's ignore the x term, which is probably small because
x is sort of small. Then
2 2
1 + 3 x + 3 x = 3 so x + x = 2/3 .
2
Hm... if x = 1/2, then x + x = 3/4, which is a bit
2
too big. So figure x is about 0.4; then x + x = .4 + .16 = .56
which is too small. So probably x is about 0,45 or so.
So the cube root of 3 is about 1.45, and the cube root of
81 is 3 times that, or about 4.35 -- not a bad approximation.
--Guy STeele