Calculator contradicting true statement

Question

There are things called Fermat's near misses. These, according to a regular calculator, are "solutions" to Fermat's Last Theorem. The reason why they're not solutions is simply because there is a limit on how many digits the calculator can display.

I was also pleased to find out that calculators "think" that addition is not associative. In particular, here's a "counterexample":

$(10^{30}+(-10^{30}))+1 \neq 10^{30}+((-10^{30})+1)$

The LHS is $1$ (as one would expect), but the RHS, according to the calculator, is $0$.

Are there any other interesting examples of something like this?

Answer 1

This is from The Simpsons cartoon series:

$3987^{12}+4365^{12}=4472^{12}$

If you try this on an 8- or 10-digit calculator it seems to be correct.

The full value of the left-hand side is $$63976656349698612616236230953154487896987106$$ and that of the right-hand side is $$63976656348486725806862358322168575784124416$$ so they agreee on the first ten digits.

Answer 2

The YouTuber Stand-up Maths has a nice video concerning what appears to be (according to his calculator) confirmation that $\pi$ is in fact rational, along with some other examples and a broader discussion of why calculators sometimes get things wrong like this.

So far as I am aware, these are all just rounding errors, nothing particularly deep. In your example of associativity 'failing', the calculator rounds $(-10^{30})+1$ to $-10^{30}$.

Answer 3

Consider the sequence given by $a_1=a_2=\pi$, $a_n=20a_{n-1}-19a_{n-2}$ for $n=3,4,5,\dots$. Obviously, this sequence is just $\pi,\pi,\pi,\dots$. But on a calculator, the sequence blows up after a few iterations.