Both ways are just notation. There’s nothing more real about 3/10 compared to 0.3.
Telling you otherwise might have worked as a educational “shorthand”, but there are no mathematical difficulties as long as you use good definitions of what you mean when you write them down.
The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits.
I agree that ultimately both are just notations. I do think the fractional notation has some definite advantages and few disadvantages, so I think it's better to regard it as more canonical.
I disagree though that it's necessary or even useful to think of 0.99... or 0.33... as sequences or limits. It's of course possible, but it complicates a very simple concept, in my opinion, and muddies the waters of how we should be using notions such as equality or inifinity.
For example, it's generally seen as a bad idea to declare that some infinite sum is actually equal to its limit, because that only applies when the series of partial sums converges. It's more rigorous to say that sigma(1/n) for n from 2 going to infinity converges to 1, not that it is equal to 1; or to say that lim(sigma(1/n)) for n from 2 to infinity = 1.
So, to say that 0.xxx... = sigma(x/10^n) for n from 1 to infinity, and to show that this is equal to 1 for x = 9, muddies the waters a bit. It still gives this impression that you need to do an infinite addition to show that 0.999... is equal to 1, when it's in fact just a notation for 9/9 = 1.
It's better in my opinion to show how to calculate the repeating decimal expansion of a fraction, and to show that there exists no fraction whose decimal expansion is 0.9... repeating.
> The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits.
Also a possible third thing: not enjoying working in a Base that makes factors of 3 hard to write. Thirds seem like common enough fractions "naturally" but decimal (Base-10) makes them hard to write. It's one of the reasons there are a lot of proponents of Base-12 as a better base for people, especially children, because it has a factor of 3 and thirds have nice clean duodecimal representations. (Base-60 is another fun option; it's also Babylonian approved and how we got 60 minutes and 60 seconds as common unit sizes.)
You get the same problem with 0.44... + 0.55... - I don't think that makes it any easier to anyone who is confused. It's more likely just that 0.33... and 0.66... are very common and simple repeating fractions that lead to this issue.
Sure, I was just pointing out that Base you use for your math does affect how common repeating digits are, based on the available factors in that base.
In Base-12 math, 1/3 = 0.4 and 2/3 = 0.8. With the tradeoff that 1/5 is 0.2947 repeating (the entire 2947 has the repeating over-bar).
Base-10 only has the two main factors 2 and 5, so repeating fractions are much more common in decimal representation, making this overall problem much more common, than compared to duodecimal/dozenal/Base-12 (or even hexadecimal/Base-16). It's interesting that this is a trade-off directly related to the base number of digits we want to express rational numbers in.
Whether you multiply by 10 or 2, the same "counter" argument from the article stands. Only now you don't have a trailing zero after infinite nines, you have a trailing 8.
I don't understand how you can even have a trailing zero after an infinite number of nines. Surely any place that someone would want to put the zero can be refuted by correctly stating that a nine goes there (it's an infinite number of them, after all) and there is literally no "last" place.
I’ve seen videos of actual mathematicians complaining to each other about how the general public thinks like GP. There is no last digit. Every time you reach the horizon there’s another horizon.
Technically you don't have an '8', you keep doing a carried sum forever, think about it. The last eight will be set to 9 forever and appended a new one to it. Thus, you are getting a periodical 1.9_ in practice.
There is no eight. This is something I’ve heard actual mathematicians complain about to other actual mathematicians: the non math public misunderstands infinite series as “imagine a number so big you can’t fathom it and add 1 more number to it. That’s not how things work.
Going as far as you can imagine and a little farther is an infinitesimal of the real infinite.
This comment exemplifies the puritanical view that nakedness is somehow bad or impure.
Notice that the men in the paintings of Vallejo are also almost completely naked, hacking away at monsters with large weapons. Yet you did not point to them and say they were indecent.
I really hope that the rest of the world doesn’t take over the sex/violence sensitivities as are prevalent in the US.
I wish we'd drop it ourselves. It's excruciating living amongst people who will pick up torches and pitchforks when anyone under 18 is remotely exposed to sexual content, but shrug and look stupid when that same cohort goes on a shooting spree.
Attributing it all to puritans is misleading and even anachronistic. A direct link would be possible at the time when the phrase “puritanism sells” made sense, but a number of generations have alreafy been born into the “sex sells” world.
So there's a country with a giant porn industry, but at the same time newspapers are full of stories about “sexual predators” hiding under every bush, and someone's naked breast is considered worthy of being turned into a “national scandal”. Intuition hints that one is intertwined with another.
When people from elsewhere hear about gender neutral bathhouse (formerly “common village/family bathhouse”, or simply “a river”), they picture themselves asking a granny how's steam in the sauna. In certain countries, they immediately think of some kind of orgy instead (based on media descriptions and fantasies). And certain people of limited wit even try to reenact them in reality, with pathetic results.
Specific social convention makes people gasp, roll their eyes, and hide the children when they see someone without clothes, not their strict moral principles. It is highly unlikely that it will disappear by itself. On the contrary, there are forces that benefit from it. The aforementioned porn industry have successfully used commoners' fears to ban non-corporate-produced wanking materials from the biggest websites. Not even nature is allowed to compete with exclusive providers of images of sexual nature to the consumers.
Modesty is a virtue shared by many cultures throughout the world. It is not exclusive to the US or Puritans. And the US is not a cultural monolith. Millions of people in the US do not hold modesty as a virtue.
It is true that modesty is not an exclusively US puritan ideal, but the US has a disproportionately strong cultural influence on other societies.
In my neck of the woods we are much more relaxed about nakedness, and find the scandals in the US around this topic amusing. But for instance the lengths at which for instance facebook goes to to avoid nipples and penises feels more like an overstep in cultural freedom.
Facepook is just another textbook example of bureaucracy breaking loose. Companies that give them money (for ads) declare that “content” (i. e. everything that exists in this world) can be “SFW” and “NSFW”, and they only want the former. Ergo, women don't have breasts. End of story. All the excuses and exceptions corporate human robots invent afterwards are just irrelevant icing on the cake.
It would be fine to just ignore that stupidity, but people who spend time in that hellish environment adapt to it, and actually start to believe that there is some deep meaning to the rituals they have to make, and even invent their own explanations.
Is the misspelling of FB an intentional Russian joke? I haven’t heard that one before, but I can probably not unhear it the next time I hear a Russian speaker say that word.
"Modesty" is culturally defined in the first place. What you would consider "modest" today often would be extremely immodest a century ago. And, conversely, some societies don't consider nudity to be immodest in and of itself.
Libsodium was independently audited by respectable reviewers. OP is spreading FUD for some very weird reason.
Libsodium is also extremely robust. The only crypto project I’ve seen that is as footgunless is google’s tink, and that isn’t available for a JS environment.
What’s great about libsodium is that it’s a single code base that works everywhere. RSA libs I’ve used have subtle differences when it comes to loading keys in different formats and also incompatibilities due to dropping leasing zero bytes for instance. Compared to that, libsodium was a breeze that just worked.
I do work in JavaScript, but I also had no idea what Biome was. Apparently the Rome project rebranded. The main appeal, I think, is that it is written in native code and significantly faster than Prettier.
I had learned of both Biome and Rome, but didn’t know about the rebrand. Makes me think they’ll switch gears to focus on improving the linter after reaching 100% parity.
A contributor is currently working on this feature [0]
We are currently discussing which options to add to the rule on the Discord of Biome.
We expect to ship the feature with the next release (on February).
A nightly will be released earlier to gather feedback.
I agree. If you look at the Swift version of the library, it will generate and prepend the nonce for you automatically (if you use the right overloaded method). It will also strip it and use it during decryption.
My guess is they wanted to maintain compatibility with NaCl.
At my previous job, we used libsodium as the basis for all our crypto on web, iOS and android. This after a myriad of subtle problems and inconsistencies using multiple libraries for RSA/AES.
Libsodium is pretty opinionated, which is good. The only thing that actually came close in dev experience was google’s ~~Think~~ Tink library, but that isn’t mature on web and probably never will be.
I can fully recommend Tink, especially when libsodium isn’t available. Although its documentation isn’t very good, it‘s build to be hard to misuse. You still need some cryptographic knowledge though, as it isn’t that opinionated as libsodium.
