What exactly do you think notations like 0.999... and 0.333... mean?
myslsl
joined 1 year ago
Yes, informally in the sense that the error between the two numbers is "arbitrarily small". Sometimes in introductory real analysis courses you see an exercise like: "prove if x, y are real numbers such that x=y, then for any real epsilon > 0 we have |x - y| < epsilon." Which is a more rigorous way to say roughly the same thing. Going back to informality, if you give any required degree of accuracy (epsilon), then the error between x and y (which are the same number), is less than your required degree of accuracy
You are just wrong.
The rigorous explanation for why 0.999...=1 is that 0.999... represents a geometric series of the form 9/10+9/10^2+... by definition, i.e. this is what that notation literally means. The sum of this series follows by taking the limit of the corresponding partial sums of this series (see here) which happens to evaluate to 1 in the particular case of 0.999... this step is by definition of a convergent infinite series.
He is right. 1 approximates 1 to any accuracy you like.
Given that music boxes are very very old it is plausible that beethoven could have made a remark sharing his opinion on this exact issue. I don't mean to agree/disagree with your point, I just find that kind of interesting.
You're getting downvoted but you are right. Stuff like this is a super cool example of exactly the type of thing you are talking about imo.
There's a lot of AI generated art that sucks. But that does not imply that in skilled hands an artist can't use those tools in creative/interesting ways.
Arguably a lot of these tools are designed specifically to reduce the effort a human has to put in to create the art they want to make too.
Eigenvectors, values, spaces etc are all pretty simple as basic definitions. They just turn out to be essential for the proofs of a lot of nice results in my opinion. Stuff like matrix diagonalization, gram schmidt orthogonalization, polar decomposition, singular value decomposition, pseudoinverses, the spectral theorem, jordan canonical form, rational canonical form, sylvesters law of inertia, a bunch of nice facts about orthogonal and normal operators, some nifty eigenvalue based formulas for the determinant and trace etc.
My experience with eigenstuff has been kind of a slow burn. At first it feels like "that's it?", then you do a bunch of tedious calculations that just kind of suck to do... But as you keep going they keep popping up in ways that lead to some really nice results in my opinion.
Not sure what "numerical oscillations in 2d" means? The picture is a 3d graph?
On the bright side, you are now the proud owner of a hip designer bean plate.
Some software can be pretty resilient. I ended up watching this video here recently about running doom using different values for the constant pi that was pretty nifty.