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Study of accuracy of calculators

 

Introduction: (Initial Observation)

Since the first electronic calculators were made until today, there has been a lot of changes and improvements on the speed and accuracy of calculators. Specially when it comes to programmable and engineering calculators, accuracy is very important. 

When working with large numbers and repeated calculations, inaccuracies will add up and may get so high to make the results unusable.

In this project you will learn different methods of testing calculators for their accuracy. You will then perform experiments on old and new calculators that you may have access to, and measure and compare their accuracy.

Information Gathering:
Gather information about your project. If you are a basic or advanced member of ScienceProject.com, your project advisor may prepare the initial information that you need and enter them in this section. In any case it is necessary for you to read additional books, magazines or ask professionals who might know in order to learn more about the subject of your research. Keep track of where you got your information from.

This project does not have a project advisor, that is why we have moved it to this section. Try it at your own risk. It is a good and educational project related to mathematics and physics, but unfortunately no support is available on that.

Methods proposed in the experiment section are the only thing that you get and the rest is up to you.

Question/ Purpose:
What do you want to find out? Write a statement that describes what you want to do. Use your observations and questions to write the statement.
Identify Variables:
When you think you know what variables may be involved, think about ways to change one at a time. If you change more than one at a time, you will not know what variable is causing your observation. Sometimes variables are linked and work together to cause something. At first, try to choose variables that you think act independently of each other.
Hypothesis:
Based on your gathered information, make an educated guess about the answer to your question or the result of your experiment. 
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

 

Materials and Equipment:
Extract the list of material from the experiment section and write them here.
Results of Experiment (Observation):
Experiments are often done in series. A series of experiments can be done by changing one variable a different amount each time. A series of experiments is made up of separate experimental "runs." During each run you make a measurement of how much the variable affected the system under study. For each run, a different amount of change in the variable is used. This produces a different amount of response in the system. You measure this response, or record data, in a table for this purpose. This is considered "raw data" since it has not been processed or interpreted yet. When raw data gets processed mathematically, for example, it becomes results.
Calculations:
If you do any calculation for your project, write your calculations in this section.

Summary of Results:
Summarize what happened. This can be in the form of a table of processed numerical data, or graphs. It could also be a written statement of what occurred during experiments.

It is from calculations using recorded data that tables and graphs are made. Studying tables and graphs, we can see trends that tell us how different variables cause our observations. Based on these trends, we can draw conclusions about the system under study. These conclusions help us confirm or deny our original hypothesis. Often, mathematical equations can be made from graphs. These equations allow us to predict how a change will affect the system without the need to do additional experiments. Advanced levels of experimental science rely heavily on graphical and mathematical analysis of data. At this level, science becomes even more interesting and powerful.

Conclusion:
Using the trends in your experimental data and your experimental observations, try to answer your original questions. Is your hypothesis correct? Now is the time to pull together what happened, and assess the experiments you did.
Related Questions & Answers:
What you have learned may allow you to answer other questions. Many questions are related. Several new questions may have occurred to you while doing experiments. You may now be able to understand or verify things that you discovered when gathering information for the project. Questions lead to more questions, which lead to additional hypothesis that need to be tested.

Possible Errors:
If you did not observe anything different than what happened with your control, the variable you changed may not affect the system you are investigating. If you did not observe a consistent, reproducible trend in your series of experimental runs there may be experimental errors affecting your results. The first thing to check is how you are making your measurements. Is the measurement method questionable or unreliable? Maybe you are reading a scale incorrectly, or maybe the measuring instrument is working erratically.

If you determine that experimental errors are influencing your results, carefully rethink the design of your experiments. Review each step of the procedure to find sources of potential errors. If possible, have a scientist review the procedure with you. Sometimes the designer of an experiment can miss the obvious.

 
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