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) . 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?
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?
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?
In the world of number theory the zeros may not be of value, but in the
world of manufacturing, they may be.
Dan
----- Original Message -----
From: "Philip S Hall" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>; "U.S. Metric Association" <[email protected]>
Sent: Sunday, 2005-10-16 10:59
Subject: Re: [USMA:34902] Re: Approximations (was fractions)
You choose the number of useful digits by knowledge of the situation. A
person who is properly taught how to apply the rules of significant
digits knows how many digits apply. A number left in fractional form is
not an answer. If I have a number like 2/3, what does that mean if I'm
trying to build something with it?
2/3 is no less an anwer than 0.67
Admittedly the problem presented is purely numerical with no context.
Even if you have a number such as 2/3, you still have to assign a level
of accuracy to it. There is no way you can make something exactly 2/3 of
something. You are always going to have to state a plus/minus something
else.
Alright, but it may not be an end result. It could be an intermediate step
involving a fractional coeifficient. Take as an example the formulae for
the volume of a sphere - 4/3 * pi * r^3
In any case, if the figure of 2/3 was an approximation for something, with
a known error bound, then by substituting a decimal approximation you
introduce a further error.
Phil Hall
--
No virus found in this incoming message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.12.1/136 - Release Date: 2005-10-15
