Experiment
Design:
Design an experiment
to test each hypothesis. Make a step-by-step list of what you
will do to answer each question. This list is called an experimental
procedure. For an experiment to give answers you can trust, it
must have a "control." A control is an additional experimental
trial or run. It is a separate experiment, done exactly like
the others. The only difference is that no experimental variables
are changed. A control is a neutral "reference point"
for comparison that allows you to see what changing a variable
does by comparing it to not changing anything. Dependable controls
are sometimes very hard to develop. They can be the hardest part
of a project. Without a control you cannot be sure that changing
the variable causes your observations. A series of experiments
that includes a control is called a "controlled experiment."
Scientific (programmable) calculators
should be accurate! Here are a few ways to determine their accuracy and
some results. Naturally, if you test a calculator not mentioned here, I
am eager to learn the results you get: Please e-mail.
One way to test the accuracy of a
scientific calculator is to perform the following formula and see when
the answer turns out to be ONE (1).
Y =lim ( X / sin X)
where X descends to zero.
If Y is 1 upon X = 0.01 then the
accuracy will be classified ´low´.
Simular with X = 0.001, accuracy will
be ´below normal´.
With X = 0.0001, accuracy ´normal´. X
= 0.00001, accuracy ´good´. X = 0.000001, accuracy ´very good´. X =
0.0000001 or smaller, accuracy ´extremely good´.
A few examples of calculators I have
that are still in working order: The TI-57 and TI-25 score NORMAL. The
TI-58, HP-33E, FX-502P and FX-6000G score GOOD. The HP-48SX scores VERY
GOOD and the TI-92 scores EXTREMELY GOOD.
Another way to test the accuracy is to
see how many digits will be correct (or give a correct rounded result)
for the following calculations:
Cos 0.184 (rad) =
0.983119705665017
Sin 3.8° (deg) =
0.066273900400000
Atn 0.445 (rad) =
0.418688151438
The TI-57 scores 7, 7, 7 digits.
The FX-6000G scores 10, 10, 10 digits.
The HP-33E scores 10, 11, 10 digits.
The FX-502P scores 11, 11, 11 digits.
The TI-58 scores 12, 10, 12 digits.
The HP-48SX scores 12, 11, 12 digits.
The TI-92 scores 14, 14, 12 digits.
Of course the number of
accurate digits never exceeds the number of digits used by the
calculator internally. Digits that are not shown by the display can be
made visible by subtracting the visible result. For example: The TI-92
gives the following result for Cos 0.184 (rad): 0.983119705665. If we
subtract 0.983119705665 from this answer we will see the remaining
digits: 2 E -14. Put together the result is 0.98311970566502, which is a
correct (be it rounded) result for all 14 digits.
(Of course, on the TI-92
there is an easier way to see the full 14 digits of the result, but that
is beside the point here.)
A Simple Accuracy Test for Calculators
Back in the early days of handheld
calculators the 'Commodore' test was a simple demonstration of
trigonometric accuracy. It was invented by the Commodore Business
Machines (CBM) marketing department to highlight the accuracy of their
machines. Althought it only tests a few functions, over the years it has
proven a useful way to get a 'feel' for the overall accuracy of
calculators.
At the time the test appeared in CBM
adverts good machines would return values a few thousands of a
percentage out while there were some machines on the market which would
be a few percent in error. (Yes, I owned a Sinclair Oxford
300!) Nowadays it is unusual to find a machine that is more than a
billionth of percent in error.
To test a calculator set it to work in
degrees and then enter the following calculation:
asin(acos(atan(tan(cos(sin(29))))))
which should return a value of
'29'. To get the percentage error carry out the following calculation:
(asin(acos(atan(tan(cos(sin(29))))))-29)/29*100
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