In your example of the volume of the sphere you are confusing a fractional
number with three constants in an algebraic expression. Write it
differently and the fraction isn't so obvious. Like 4 * π * (r^3) /3.
The 4, the 3 and π are constants and r is the variable. You can
manipulate the solution any way and then round the result. If r is 3, the
r^3 is 27. 27 divided by 3 is 9. 9 times 4 is 36. 36 times π is 113.097
335 5 (As far as the calculator displays) .
Sure, but in doing such manipulation you are drawing on your knowledge and
understanding of fractions and what the expression 4/3 actually means.
What is a fraction (arithmetically) if it isn't one number divided by
another?
You can round this result to whatever significant digits are necessary.
Would you leave the result as 36π because you don't want to see an
irrational result?
I don't dispute the fact that ultimately the result can always, for
practical purpose, yield a result in decimal form. I just think that in
getting to that stage you have to motivate a great deal of conceptual
knowledge that understands x/y equally as well as 0.ddd ...
When performing a calculation, you leave all of your constants in their
basic form, perform the calculation to the decimal result, apply the rule
of significant figures and that is your answer. You don't leave a number
in a fractional form, as in truth, a fraction is a step of division not
completed. Why don't I write the number 4 in a result as 12/3? So why
would I want to leave the number 0.5 as 1/2?
That all depends on how you are going to use the resultant information. 0.5
literally means 5/10, take your pick.
Even whole numbers have significant figures. Is 1/2 always 0.5? Or can
it also be 0.500000 or 0.500000000000? Where do the number of zeros
really end? In reality a number like 1/2 is just as irrational as 2/3,
but we consider the zeros to be of no value so we drop them. But, are
they?
Again it depends on context but a number written as 0.500 say implies that
it (could) represent an actual measured (or measurable) quantity in the
range 0.4995 and 0.5005, or if you will, 0.500 +/- 0.0005
In the world of number theory the zeros may not be of value, but in the
world of manufacturing, they may be.
May be we're getting closer now to the real heart of this debate. Is you're
real contention that we shouldn't teach anything that might be construed as
pure mathematics and that the world of ideal numbers are not relevent? I get
the feeling that what we're really talking about is the difference (if there
is any) between pure and applied mathematics.
If you'll bear with me I'd like to don my pure mathematician's hat for a
moment.
I could describe all those horrible and useless looking numbers, such as
17/37 etc, as elements of the set of rational numbers (usually symbolised by
the letter Q) which is the set of all numbers of the form x/y (x, y
integers, y > 0) such that they are all in their lowest terms (e.g. 6/21 is
not distinct from 2/7) The integers are in the set Q
I would then point out that the set of all rational numbers in decimal form
are a proper subset of Q. I would describe the decimals (meaning numbers
with at least one non-zero digit to the right of the decimal point) as
numbers of the form x/10^k (x, k integers, k > 0) I'll call this set D
I would show (I won't go into actual proof) that there are rational numbers
of the form u/v say, such that if v has at least one prime factor other than
2 or 5, then u/v is not in the set D You can write a sequence of numbers in
D such that they converge toward u/v getting as close as you like but never
actually equal to it. These are the recurring decimals, e.g 0.3, 0.33,
0.333, ... converging to 1/3 or put the other way 3/10, 33/100, 333/1000,
...
Now my point is that it isn't the numbers being referred to as fractions
that are incomplete.
It is the reverse that is true since D < Q
Ok all this is way above school mathematics and not really the subject of
normal college courses either. However I find it hard to concieve of a
satisfactory program of study that only teaches how to deal with the set D
and puts the set Q - D virtually off limits except for approximations, no
matter how able the individual student may be.
Phil Hall
