What are some examples of mathematical errors in computer algebra systems (CAS)?

Question

Computer algebra software is easily and freely available for students nowadays but they still have mathematical errors. For instance, ask them to solve the simple equation $a x - b = 0$ for $x$. The answer given is typically $b/a$, but that is wrong if $a = 0$, for instance. I am collecting other mathematical errors in CAS systems. Any examples or references are welcome, the simpler, the better.

Answer 1

I find it very helpful in my classes to include examples of calculators giving obviously wrong answers to answer the constant question "why do I need to learn this if my calculator can do it?"

You can find some examples in Patrick Honner's talk "When Technology Fails": https://www.youtube.com/watch?v=tRT3QIt7w20

Some answers from math.stackexchange: https://math.stackexchange.com/questions/1360953/when-would-a-graphing-calculator-make-an-error-while-plotting-a-function

Some examples I've collected in Desmos: https://www.desmos.com/calculator/yjw4awkdej https://www.desmos.com/calculator/h1ctokasxm

Answer 2

Here is an example using Wolfram|Alpha. Note that $1/\lfloor x \rfloor < 2/x$ for all $x > 1$. However, Wolfram|Alpha seems to think otherwise.

Input:

Solve 1/floor(x) < 2/x for x > 1

Output:

Solution over the reals

$\displaystyle x = \frac{99}{5} \approx 19.8000$

A different solution appears if the terms are rearranged.

Input:

Solve x < 2 floor(x) for x > 1

Output:

Solution over the reals

$\displaystyle x = \frac{759}{10} = 75.9000$

One can try changing the range of $x$ (such as $x > 20$, or $x > 76$, to avoid the previous solutions) to get some other values as the answer.

I don't know how Wolfram|Alpha internally defines the floor function, but it clearly has trouble handling some inequalities that use it.

Answer 3

"solve the simple equation ax−b=0 for x. The answer given is typically b/a, but that is wrong if a=0"

But is it truly wrong?

One gets the same result manually.

Taken as a limit where a ⇒ 0 it's correct.

Granted it's undefined at that exact value and that has to be taught, but the question implicitly brings Numerical Methods into play as it involves computers. When dealing with floating point numbers, i.e. non-integers, the very concept of equality is a mistake.

In the computer language APL, it defines 0/0 = 1. Ironically APL was created specifically for math. In the mathematical abstract it's undefined, but in numerical methods it often makes more sense than what most languages do, which is to crash, because more often than not the computation is approaching zero in the denominator when it ran out of bits.