| Point is, no matter how well you know the precision of one factor (sqrt(3)),
| if the other factor is only known to two or three significant digits, any
| product of the two can really only be known to two or three significant
That is true. Just be careful not to add to the error of the lack of
precision by prematurely reducing the precision of the well defined
constant. Otherwise the product ends up with more error than either
of the multiplicands.
120 * 1.7320508075688772935274463415058723669428052538103806280558 = 208
Of course, this example is extreme precision. This is not a boundary case.
If it were a boundary case, a precise value of the square root of three
enough to avoid the boundary issue, would be wise to use. I typically use
more precision than just 1.732. If I have to type in this value by hand,
I use 1.732050807568877 because I have that much memorized, unless I feel
lazy in which case I just go with 1.7320508 :-)
|> Back when I was in junior high school, without the aid of any calculator
|> or computer, I pondered the meaning of the frequency 3.58 MHz as it
|> to the TV broadcast standards (which at the time I "knew" to be 15,750 Hz
|> horizontal and 60 Hz vertical. But I found a book in the school library
|> that gave the value as 3.579545 MHz. Just that much information allowed
|> me to "reverse engineer" this number to determine it came from 5 MHz times
|> 63 divided by 88, and really had "454545" repeated (3579545.45[45..] Hz),
|> and that the horizontal frequency was really 15734.265734[265734..] Hz,
|> and that the vertical frequency was really 59.940059[940059..] Hz.
| But with all your 'refinements', you're still starting from '5 MHz'. And
| just how accurate is the 5 MHz crystal considering the ambient temperature
| of the crystal is pretty much uncontrolled? At the broadcast studio, I'm
| sure their's are more precise. But the one in the TV set? If I'm not
| mistaken, that's why they use a PLL circuit.
| You've assumed the '5 MHz' is exact, and therefore the 3.58 MHz is wrong.
| Why couldn't it have been.....
| 5,000,634.92063 Hz * 63/88 = 3,580,000 Hz
Or 5,047,213.114754 Hz * 61/86 = 3,580,000 Hz
The thing is, none of these had any particular meaning. But once I had
that value of 3,579,545 Hz, the resultant 4,999,999.365 became a lot more
relevant. Then it became clear that the value most likely came from the
exact 5 MHz reference. For a while, though, I called it 315 MHz / 88.
| (see, you're not the only one with an arbitrary precision calculator :-)
There are plenty around. Knowing how to use them correctly can elude some.
| You're not trying to blow smoke and claim that the color burst frequency in
| an old TV is derived from multiplying 5,000,000.000 Hz times exactly 63/88
| ??? Like to see the analog circuit that produces such exact multiplication.
| Sure wasn't in my old RCA set that I tore into a couple of times :-)
No. The _definition_
is. It might be more practical, if you wanted to
produce it from a WWV locked oscillator, to use 15 MHz * 21 / 88.
A cheap receiver only needs to be close enough to get a lock and stay
stable enough to keep the color reasonably consistent. A broadcaster
didn't even need to, as getting a crystal in an oven to stay within 10 Hz
of 14318181.818 Hz would good enough. It could be tested or calibrated
by doing a phase comparison at 315 MHz between the the 21st harmonic of
15 MHz and the 22nd harmonic of 14.318181818 MHz.
|> Do you do any computer programming? If so, do you just add up a long
|> list of floating point values in the order given, or do you sort them
|> so you accumulate the sum by adding the lowest values first?
| Yes I do quite a bit of programming thank you very much. Since most
| floating point numbers are already an inexact representation of 'real'
| numbers, they can be inherently flawed (hint, use 'doubles', there are more
| significant digits). Yes, you point out correctly (as anyone who has
| studied "Numerical Methods" can tell you) that when adding floating point
| values of widely different decades, the exact order of operations can have
| an effect on the exact outcome. In general, the more floating operations
| you go through, the fewer and fewer significant digits you can rely on.
| That's why there are BCD and 'arbitrary precision' techniques.
| By the way, if you use something other than base 10 for your fractional
| representation (or base 2 as used in IEEE-754 floating point format), you
| can represent some numbers more precisely.
Indeed. I have used base 16 for that, as well as scaling numbers to get
all the digits I want to have as integers.
And I use the GMP package in my Mandebrot fractal program to take my zooms
to extreme levels.
|> How many digits do you want for the square root of three expressed as a
|> ratio of two integers with a precision in digits equal to or greater than
|> the SUM of the digits in the numerating AND denominator?
|> Anyone can just say:
|> divided by:
|> but that is only 77 digits of precision for 154 digits expressed.
|> But if I give you:
|> divided by
|> then you can be sure you have 154 digits of precision. Try it.
|> Remember 355/113 for the value of PI? I have way better fractions. You
|> won't _need_ them, of course. But I have them.
| But again, you can take as much precision of a mathematically defined value
| as you want. When you multiply / divide it into something that is measured
| to only two significant digits and try to claim the result is 207.8461 and
| is 'more accurate' than 208, you've wasted a lot of everyone's time.
I use 208 or 207.846 or 207.84609690826527522329356 for different purposes.
| To claim....
| 120 * sqrt(3) = 207.8461
| Is pure nonsense. You cannot possibly improve upon the accuracy of the
| original measurement. Yet that is what this sort of statement is claiming.
| "I measured the voltage to the nearest ten volts (120), and thus I know the
| accuracy of the line-line wye voltage to the nearest ten-thousandth of a
| volt (207.8461)."
I disagree ... depending on the context of use. If the multiplication is
done to an actually measured value, I'll keep the product at the same level
of precision as the measured value (as scaled by the multiplication).
123 * sqrt(3) = 213
123.1 * sqrt(3) = 213.2
| Now, if you had said....
| 120 * sqrt(3) = 207.8461 plus or minus 17.3205
| I would agree with that. And everyone would see that the 'answer' is not
| that well known (could range from about 190 to 220). But look at what all
| your 'accuracy' has accomplished.
Plus or minus 10 in the original measurement, if that is a measurement
by a voltmeter, is not that accurate at all. That voltmeter sucks.
OTOH, that could well be the allowable range of the voltage provided by
a utility or derived from a generator. So:
120 +/- 10 * sqrt(3) = 207.8460969 +/- 17.32050808
But at that point, given in definition accuracy, precision in the final
expression really is pointless. So:
120 +/- 10 * sqrt(3) = 208 +/- 17
Still, in cases where I would have to calculate the values, I would use
a lot of digits of sqrt(3), well more than the measured L-N voltage, then
round the L-L voltage.
| I see this a lot when doing unit conversions as well. If you go to a
| definitive source for the conversion from one set of units to another,
| you'll find that some conversion factors are given as *exact*
, while others
| are approximations to some number of significant digits. For example, an
| inch is defined by NIST as *exactly*
25.4 mm. So a foot is *exactly*
| mm or 0.3048 m. But 1 meter is only *approximately 3.28083989501 feet.
The numeric base system we use and/or the units of measure we use do have
an impact. It is basically a quantization effect. Given the same level
of meter accuracy, but instead, doing the work in base 2 instead of base
10, changes things "a round a bit".
| Phil Howard KA9WGN (ka9wgn.ham.org) / Do not send to the address below |
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