Calculators: The Ugly

I've posted  about "Calculators: The Good" as well "Calculators: The Bad", but I think Calculators the Ugly deserves it's own dedicated page. It will be a place to accumulate/document specific problems with calculators and mathematical technology in general.

1. Calculators/technology can give the wrong answers on purpose: $latex 0^0$. See this post where  $latex 0^0$ is discussed as being set equal to the wrong answer, 1, even though it's actually defined. The reason?: other software gives the wrong answer, too. It makes you wonder if there are effects related to a limit approaching $0^0$ which then gets evaluated incorrectly as 1. Or are there other programming decisions made to give you the wrong answer that you/we aren't aware of. The Google calculator (which is in the search bar: just type 5+7 instead of searching with words) makes this $latex 0^0$ error as well (though I don't know if it's on purpose).

2. Calculators/technology can give wrong/misleading answers: $latex x^{1/3}$. Many calculators graph $latex x^{1/3}$ for nonnegative values only. The Coolmath calculator, Sage, and other technology make this mistake. Sage does it on purpose, so unless you've used it a lot you probably aren't aware that you need to enter something different to get the answer you want. In general, the newer calculators have corrected this mistake. However, almost all calculators will give the wrong output for: $latex x^x$ and, from my experiments playing around, even more problematic is $latex \sin(x^x)$. Likewise, $latex \sqrt{1111111111111111111}=1054092553$ on some calculators even though $latex \sqrt{1111111111111111111}$ isn't an integer asTingyao Zheng notes on page 6-7. A similar problem with $latex 5^{14}/4$ is mentioned here. Many students forget that the calculator's answer for $latex 70!$ only an approximate answer while graphing $latex \frac{\sin(x)}{x}$ has the y-axis covering over the gap at (0,1). Graphing $latex \sin(40x)$ over [-360,360] for TI-73, 81, 82,and 83 calculators is mentioned here. Similar problems are on other calculators. I once gave a worksheet on calculators where I asked students to graph $latex \sin(1/x)$ on the graphing calculator and determine how many times the graph crossed the x-axis from [-1,1]. Typical answers ranged from 7-14. That was one of my problems to motivate the discussion that calculators can't replace your thinking.

A consequence of 1 and 2 is that: Different calculators give different answers to the same problem.

3. Every calculator uses a finite number of numbers to represent the infinite set of real numbers. Therefore, all calculators will be subject (with varying degrees) to problems with overflow, underflow, and precision. So 1000000000 + .1 = 1000000000 on many calculators.  From Lies My Calculator Told Me, try $latex (10^{15}+7.2-10^{15})*100$

4. Calculators graph by plotting points and figuring out how to connect them properly, so some calculators will show the asymptotes on some graphs: $latex \tan(x)$ or $latex \frac{1}{(x-3)}$ on some graphing calculators, depending on the plotting parameters.

5. User errors/entry problems: people don't know the orders of operation correctly, plot something they didn't want, and then don't recognize the answer is wrong: x^1/3. If you know the order of operations then you'll know this is $latex \frac{x}{3}$ and not the cube root of x. Similar problems where the user hit the wrong key and don't have the math sense to correct the error.

6. Disagreement with the orders of operation; for example, on $latex 2^{3^2}$. Some calculate this as $latex 8^2=64$ and others as $latex 2^9=64$. Consider this, where the same brand of calculator gives different answers to a^b^c. See the section on "Exceptions to the standard", too. It's even worse with  3^2! because factorial isn't mentioned in orders of operation. So is it 9! or $latex 3^2$ which is 9? There's a huge difference.

7. Never forget, graphing calculators are running software and will have bugs just as all complicated software does. See, for example, here. Just search on your favorite calculator along with "buglist".

8. Calculators don't recognize important numbers when you use them $latex \lim_{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^n$

9. Zooming in too much results in a graph which goes crazy (precision errors). From Lies My Calculator Told Me, the  graph of  $latex y=\left(1+x\right)^{1/x}$

10. Some graphs can't be rendered: f(x) = 1 if x is irrational and 0 if x is irrational.

11. Problems with matrix calculations if the determinant of the matrix is close to zero.

12. Problems with negative bases in can cause all sorts of problems from calculator to calculator. Try $latex (-1)^0$, $latex (-8)^{-\frac{1}{3}}$, $latex (-1)^{-1}$, and $latex (-1/3)^{-1/3}$ and see if anything breaks. Of course, it varies from calculator to calculator.


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