17 Oct
2011
17 Oct
'11
1:46 p.m.
During the span of my teaching, I watched the transition from VERY few
people remembering how to compute a square root ("well, it looks a like
I read the method a few times, but found I prefered Newton's method :
Guess a sqaure root of the givien number Y and call it X(0). Any positive
guess will do, you cna use 1 in all cases, it just takes a little longer
to converge
Now keep on calculating X(N+1) = (X(N) + (Y/X(N)))/2 until X(N+1) and
X(N) are 'close'. Than that's the square root.
For example, in RPL, define 'ITER' as
\<< OVER OVER / + 2 /\>>
(Takes Y as second on stack, X(N) on top of stack, returns Y (unchanged)
in second on stack, X(N+1) on top of stack).
Now put 2 (value to find square root of) and 1 (initial guess) on the
stack and repeadedly execute ITER.
The X(N) values are : 1, 1.5, 1.4166666666, 1.41421568628, 1.41421356238,
and that remains unchanged (on my HP48). It converges very fast...
But of course
such amchines are not mathematically perfect, there are
rounding erorrs and the like [1]. I feel that if you'r not going to
teach how to do soemthing by hand becuase everybody uses a machine to do
it, you should instead teach why the machine doesnt' always get the
mathematically correct resualt, and how to handle the common pitfalls.
"Get a better calculator/computer/floating point co-processor"
"Just use 'double precision'"
Just making them divide 1 by 3, write down the result, and then multiply
what they had written by 3 actually helped!
Am I allowed to work in any number system I like? If so, I'll pick base 3
and write 1/3 = 0.1 :-)
that PI was EXACTLY 22/7 ! (half a century ago, in
elementary school, I
got into "big trouble" for telling a teacher that PI was NOT 22/7 !)
I
have never understood the windespread use of that approximation. My parents knew better, but would not believe my
protestations that
disagreeing, and refusing to retract, was all that I had done wrong.
In the eyes of the school system, any kid who knows more than 3 digits of
PI in 5th grade needs to be brought back into compliance with the norm.
(no, that did not remain my only manifestation of maladjustment)
You may have guessed that one of my main occupations at school was driving
so-called teachers insane...
> 'How I need a drink, alcholic of course after the heavy lectures involving
> quatnm mechanics'
Argh!. I checked and double checked that and still made a typo...
BUT, you have 3.141592653589 6 9
instead of 3.141592653589 8 9
^
7, surely?
in spite of presumably knowing a bit about quantum
mechanics.
Do I? I may be a particle physicist, but I can assue you I did a lot more
with transputers and ECL than with quarks...
-tony