Re: devreden
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: 8 Nov 2006 13:57:16 -0800
Bob Kolker wrote (in part):
Is their something in the air or water. When I was a kid
everyone (including me) knew what a convergent series was.
This was my experience also, although only for convergent
geometric series. Apparently, high school texts don't
cover this like they used to, because the a/(1-r) formula
for the sum of a + ar + ar^2 + ar^3 + ... was quite basic
when I went to school -- basic as in before quadratic
equations ever came up. I've ranted about this issue
at least a couple of times in sci.math, about 3.5 years
ago (the first post below) and 4 months ago (the second
post below).
*********************************
|-|erc wrote (in part):
now prove to me 1/3 does not equal .33 recurring,
then I'll believe you are a mathematician.
Huh? I don't know about you, but about all that would
cause me to believe is that the person is taking (or has
taken) a beginning algebra course, the kind that children
age 12 to 15 take in school. There was a whole section
in my high school algebra 1 text about converting repeating
decimals to fractions. It appeared in the chapter on solving
linear equations (e.g. equations such as 4x = 12, 3x+2 = 20,
5x-1 = 8, etc.), which was near the beginning of our text.
We knew from grade school long division that some fractions
give rise to repeating decimals (and if this is all that you're
asking, then I'll revise my previous age estimate of 12 to 15
down to age 8 to 10), but it wasn't until the second or third
month of our beginning algebra course that we were shown how to
reverse the process (i.e. obtain the fraction equivalent to a
repeating decimal). Later in our text another approach was shown
using geometric series, but since this was a small rural school,
we didn't get to that until second year algebra. However, in my
school there wasn't a rule that said you couldn't read ahead if
you wanted to (or, heaven forbid, even check out math books from
the school library if you wanted to), so three or four of us
already knew the geometric series method before the other
students saw it in second year algebra.
http://www.mathwizz.com/algebra/help/help16.htm
*********************************
Related to this, I've been wondering for quite a while
now (several years, at least) why there seems to be so
much confusion over infinite decimals. I wonder if it
has something to do with the use of calculators, where
all decimals are 8 or 10 digits? Infinite decimals
never seemed to be a problem when I was in school,
but this was before calculators. Everyone back then
seemed to be very aware that, except for a few "special
fractions" (those whose denominators have no prime
factors besides 2 and 5, although not many of us
actually knew this result), the division you do to
change a fraction to a decimal goes on forever.
We always put a bar over the repeating part to
show this, also.
Something that seems a little funny now is that we
used to consider problems (and I'm mostly thinking
of word problems) with fractions easier than those
with decimals. This is because the fractions rarely
involved numerators and denominators with more than
two digits, so the numbers weren't that hard to work
with. No matter what you had to do with the two fractions,
it reduced to just multiplying and/or adding integers:
a/b +/- c/d = (ad +/- bc)/(bd), (a/b) x (c/d) = (ac)/(bc),
(a/b) / (c/d) = (ad)/(bc). None of these involved very
much computation when a, b ,c, and d were one or two
digit numbers. Such fraction computations were MUCH
easier than, say, dividing 40.62 by 3298.
*********************************
Dave L. Renfro
.
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