> It may look obvious to you, but someone will interpret “the same value” as literally the same.
But it is literally the same? Numbers are immutable, so there is a performance optimization where you can avoid using pointers internally, but the fact that they are immutable also means there is no way to distinguish between them being the same value and them being "different instances".
If you do `let a = []; let b = a; a = [1]` would your students expect that b equals [1] or would they understand that a and b now contain different arrays? If the latter, then why would think that after `let a = 22; let b = a; a = 50;` b also equals 50?
Sorting your data before searching it will only pay off if you need to search multiple things. If instead you need to search for one specific thing then going through things linearly is O(n) while sorting and searching the sorted result will be O(n log(n)).
Depends on your shell, if you execute `which -a cd` it will show you /usr/bin/cd in addition to the built-in command.
(zsh has which as a built in command, apparently bash doesn't, which causes the different output). It's unclear to me what /usr/bin/cd actually accomplishes though, even in bash.
bc and dc are arbitrary precision. By using -l you are specifying that it should keep track of 20 decimal digits (plus you are importing some extra stuff).
You can try higher precision by setting the scale.
a neat calculator is spigot. It has no trouble getting exactly 0 for this calculation:
$ spigot '(2/3)*3-2'
0
furthermore, when you specify a precision number, this sets the precision of the final result, NOT the precision of intermediaries (which is what bc & dc, and many other arbitrary precision calculators do); every printed digit is correctly rounded. [edited to add: truncated towards zero by default; other rounding modes available as commandline flag]
> spigot can never determine the next correctly rounded digit after "0.1234" with certainty, for reasons elucidated in the documentation.
Because spigot rounds towards zero in that mode, and it can only bound the result as arbitrarily close to 0.12345 - i.e., it can never decide between 0.12344999..99something and 0.12345000..00something because the "something" part that might break the tie never appears. This is a general issue with exact real computation; in more complicated cases, it can even be undecidable whether some expression is exactly zero.
When they tell you "the price is low if A is the case, and high if A is not the case" and you say "I'll take the lower price" and A is not the case then you are deceiving them.
When I said "deceive" I was thinking more along the lines of currency being counterfeit or product not being up to spec. Anything not directly related to the transaction is the private business of each party. I don't ask the people I buy things from to show me their accounting books to see if the price they ask for is fair, so why should I accept any questions regarding how much I'm able to afford?
Imagine the expansion rate starts out at 2. Then 1 day later it's 1.5, after 1 more day it's 1.25 and after another day it's 1.125.
Hopefully you can see that if this series continues then the expansion rate is always dropping, but it's headed towards 1, not 0. (And if expansion rate of 1 is too confusing in this context imagine if it starts out at 3 and goes to 2.5, 2.25, 2.125, ..., it still is always decreasing, but it will never be less than 2 which means the universe keeps expanding).
> What's the mechanism that allows the acceleration to drop, without dropping to zero though?
The expansion rate drops because the energy density goes down due to the expansion. However, the dark energy density aka cosmological constant does not go down, therefore providing a nonzero floor.
> Is the dark matter not expanding with everything else
This has nothing to do with dark matter in particular, and it’s space itself that’s expanding, not matter. Also, the expansion has no center, the universe is expanding at every point. Some types of matter expanding and others not would imply a center.
> The expansion rate drops because the energy density goes down due to the expansion.
I thought the idea was that Dark Energy increased with the amount of empty space; so as the space between galaxies expands, the energy driving the expansion also expands - hence the acceleration.
I wrote “density”, i.e. the amount of dark energy remains constant per unit volume. Since the volume increases due to the expansion, the total amount of dark energy increases accordingly, but its density remains the same. This is as opposed to the matter and radiation density, which decreases.
Thank you for clarifying (you said "energy density goes down due to the expansion").
Can you also clarify why galaxies don't expand, but empty space does? I suppose this is something to do with "vacuum energy", but it's not obvious to me that vacuum energy actually requires a vacuum; I thought it was present everywhere, but was only significant in the absence of other "stuff".
I also understand vacuum energy to be related to Hawking Radiation, which is black-body. But black-body radiation is EM radiation; DE is neither black-body nor electromagnetic. Why does vacuum energy not produce observable EM radiation? Is it just too weak to observe?
> you said "energy density goes down due to the expansion"
Yes, the overall energy density (dark energy plus matter/radiation) goes down, but since the dark-energy part of the density remains constant, it provides a floor.
> Can you also clarify why galaxies don't expand, but empty space does?
Gravity. Within a certain range (the size of small galaxy clusters), gravity dominates.
> I suppose this is something to do with "vacuum energy"
Vacuum energy is predicted by quantum mechanics and contributes to (or entirely constitutes) the cosmological constant, and thus is a candidate explanation for dark energy.
1 januari toegevoegd sounds like you're adding the date. If you're going to write it as a sentence it ought to have something like "Op 1 januari", and in that case it actually doesn't matter whether you put the toegevoegd before or after "op 1 januari". But I agree, nothing wrong with "Toegevoegd: 1 januari".
But it is literally the same? Numbers are immutable, so there is a performance optimization where you can avoid using pointers internally, but the fact that they are immutable also means there is no way to distinguish between them being the same value and them being "different instances".
If you do `let a = []; let b = a; a = [1]` would your students expect that b equals [1] or would they understand that a and b now contain different arrays? If the latter, then why would think that after `let a = 22; let b = a; a = 50;` b also equals 50?