text:
234 words

I’ve been looking on and off for mp3 players for a couple months. I wanted a device with proper playlist support, bluetooth, and sufficient battery and storage capacity. It had to be cheap and with a UI that does not make me wish for death.

I ended up buying a 30 second hand Nokia Lumia 650. I deleted everything except the music player and maps app, downloaded maps of every country I might reasonably ever visit and the complete contents of Wikivoyage, copied my music onto it from my pc and put it permanently in airplane mode. It is a bit too laggy, but other than that I like this setup a lot. But more important than my love for Windows Phone, is my hate for Android and iOS. I dislike the former for its role in the global surveillance economy and its butt-ugly interface. I dislike the latter because of its adversarial pricing model and excessively walled garden. I don’t want to get my dinner from the pathologically neoliberal butcher, the dominant-strategy-playing externality-indifferent brewer or the stalking, price-discriminating, search-engine-optimizing baker. Their antithesis probably consist of the local organic farmer’s market, self-hosted FOSS software and artisan everything, but I do like economies of scale. I’m still searching for the synthesis. For now, I’ll start with trying to minimize my interactions with companies who relate to their users or customers in a very adversarial manner ### 8. Links #3: the real AI was inside us all along 134 words Olivia Solon: The rise of ‘pseudo-AI’: how tech firms quietly use humans to do bots’ work It’s hard to build a service powered by artificial intelligence. So hard, in fact, that some startups have worked out it’s cheaper and easier to get humans to behave like robots than it is to get machines to behave like humans. Brian X. Chen and Cade Metz: Google’s Duplex Uses A.I. to Mimic Humans (Sometimes) In other words, Duplex, which Google first showed off last year as a technological marvel using A.I., is still largely operated by humans. While A.I. services like Google’s are meant to help us, their part-machine, part-human approach could contribute to a mounting problem: the struggle to decipher the real from the fake, from bogus reviews and online disinformation to bots posing as people. ### Is a PhD right for me? 575 words When I was almost finished with my studies, I thought for a long time about whether I should do a PhD. On the one hand, I was in actual love with my master’s research topic and I had the opportunity to continue right where I left off. On the other hand, PhD students lead a stressful life of imposter syndrome and are paid a pittance. I spent quite some time searching for articles and videos with titles like “why I quit my phd” to see if any of their feelings and thoughts sounded like the sort of things I might think. That was a useful activity which I would recommend to any prospective PhD student. Right now, I am 2 years into my PhD.1I have heard multiple colleagues say that this is the worst part. Long enough in to think you should have visible results by now, but not long enough to actually have them. The title question remains. Here is a list of things that would have allowed me to make a more informed decision. • Imposter syndrome starts the day you’re hired and won’t stop for many years to come. • It is normal to fantasize about quitting every so often. • You will at some point realize how much more money you would make in industry. That is how much you’re effectively paying to do a PhD. • You will like your job less than you expected beforehand. • It is probably better to drown your sorrows in online procrastination than in substance abuse. • Free weekends will be rare, or at least you will feel guilty about them. Other people will always seem to be working more than you.2This is partly sampling bias. You do notice when your advisor sends you an email at 11pm on a Sunday, but you don’t notice when she doesn’t. There are good reasons for working outside office hours. Using your brain for 8 hours straight is hardly possible. Working less during the day and then a bit in the evening or on weekends is nicer. Also, publish-or-perish is a terrible incentive structure, this does play a part. • Depending on how good you are at making friends, your first conferences and workshops will be sad and lonely and exhausting. • Your sense of self-worth will become tied to your sense of how your research is going. • When your proofs or experiments are not working, you will feel frustrated and miserable for days/weeks/months on end. • When your theories and data do start working, the world will shine and your heart feels made of clouds and it will be blissful. For an hour or so, until you find the irrecoverable bug in your proof. • Academia is a scam. The fraction of your time reserved for research is a strictly decreasing function of time. Research directions are chosen because they are uncrowded and occasionally interesting, not because the outcomes would have any relevance to the world at large. Every single postdoc is sad and lonely because they have to move long distances every year or so, preventing them from having any friends. That said, getting to do a PhD is undoubtedly a privilege. While the whole world is burning, trillions of non-human animals are brutally slaughtered every year for needless human consumption, and 815 million humans are too poor to afford enough food, you get play a part in the Unidirectional March Of Human Progress™ by spending four years in your safe, rich bubble thinking about a useless problem that Erdős once mentioned. Be sure to enjoy it. ### 7. Links #2: AI Safety-critical reading list 792 words So, I started the anti-AI Safety blogging series because I would be a good fit for the cause area as described by e.g., 80,000 Hours and it seemed reasonable to think through the arguments myself. As it turns out, they don’t stand up to scrunity. I decided to keep on writing for a bit anyway, as all AI Risk enthusiasts seem to be unaware of the counterarguments. I thought there was nothing out there in writing. Boy was I wrong. This is a non-exhaustive list of links relating to AI Safety skepticism. For more, check out the similar reading lists by Marcus Vindig and by Alexander Kruel. Overlap between these lists is minimal and restricted to a couple of particularly good resources. Rodney Brooks writes from MIT Technology Review of the seven deadly sins of predicting the future of AI. If you find a paywall, either clear your cookies or view a less edited version on Brooks’ website. His other essays on Super Intelligence are also well-worth checking out. Wolfgang Schwarz published his referee report of Yudkowsky (MIRI) and Soares’ (MIRI) Functional Decision Theory. I’ll quote a single paragraph, which I think accurately illustrates the whole review: “The standards for deserving publication in academic philosophy are relatively simple and self-explanatory. A paper should make a significant point, it should be clearly written, it should correctly position itself in the existing literature, and it should support its main claims by coherent arguments. The paper I read sadly fell short on all these points, except the first. (It does make a significant point.)” Ben Garfinkel gave a talk at EA Global 2018 titled “How sure are we about this AI stuff?”, calling for EA’s to be more critical about AI Safety as a cause area. Garfinkel knows his audience well, as everything is phrased so as to make EA’s think without ruffling feathers Oren Etzioni writes in MIT Technology Review about the survey data Bostrom talks about in Superintelligence and offers alternative data that suggest a very different picture Maciej Cegłowski‘s talks are always excellent and “Superintelligence: The Idea That Eats Smart People” is no exception. (via) EA Forum user Fods12 wrote a five-part critique of Superintelligence. They hit on a number of good objections. The posts sadly got little quality engagement, indicative of both the writing quality and of the rest of the EA Forum’s userbase. Even transhumanists can be reasonable, like Monica Anderson who writes Problem Solved: Unfriendly AI. Ernest Davis wrote a review of SuperIntelligence, touching on some of the key weaknesses in Bostrom’s arguments but insufficiently elaborating on each of his arguments. MIRI published a response to the review which I think mostly nitpicks Davis’ phrasing instead of actually engaging with his objections, which to be fair might be the best you can do if you don’t have any better source of exposition on these arguments than Davis’ review. In short, Davis’ review isn’t super good, but MIRI’s response is much worse. Neil Lawrence critiques Bostrom’s Superintelligence. If I had to excerpt a single representative line, it would be “I welcome the entry of philosophers to this debate, but I don’t think Superintelligence is contributing as positively as it could have done to the challenges we face. In its current form many of its arguments are distractingly irrelevant.” Magnus Vindig writes Why Altruists Should Perhaps Not Prioritize Artificial Intelligence: A Lengthy Critique, in which he tackles most of the standard EA arguments and points out their hidden assumptions. Topics include, but are not limited to, the incessantly cited AI researcher survey predictions, bad Moore’s law-type arguments, slight-of-hand changing definitions of intelligence, the difficulty of alignment rising for future systems compared to current ones and the enormous experience we have with present-day systems, Instrumental Convergence being under argued, the practical value of being super intelligent. He does not rigorously take down every argument to the full extent possible, but that is probably good because the blog post is 22k words as is. Vindig also wrote Is AI Alignment Possible? in which he argues that the answer is no, both in principle and in practice. Richard Loosemoore has the right amount of derision that AI Risk deserves, which is different from the right amount of derision for convincing the worriers that they’re wrong. One person who was not convinced is Rob Bensiger of MIRI. Bill Hibbard has an email exchange with Yudkowsky in which he argues that a Superintelligence would not conflate smiling human faces with nano-scale depictions of such. The whole exchange is kind of predictable and not too informative. On a related note, Nicholas Agar wrote a paper titled “Don’t Worry about Superintelligence” in which he argues that the first AIs with sophisticated agency are inherently likely to be friendly. ### Celebrating 20 years of Pokémon Snap 207 words Pokémon Snap first came out today 20 years ago. It is an on-rails shooter game. You’re in a car that drives around an island filled and your goal is to take pretty pictures of the pokémon that live there. It was one of my favourite games when I was little. Its relaxing gameplay and cute jokes still managed to put a smile on my face when I played the game again recently for the first time in years. It takes only a couple hours to take a picture of every pokemon in the game, though with high variance since some pokemon are really tricky to find. The game feels quite modern despite its age. At the end of every course you can show a number of photos to Professor Oak and he will rate their quality on a couple different aspects, like how well the pokémon fits in the frame and if they are in a nice pose. The amazing thing is that algorithm almost exactly matches my own opinion of which photos are good. You don’t see that often in algorithms. The original Pokémon Snap website is surprisingly still online. It is blazingly fast. Wirth’s law is my least favourite law of nature. ### Prove you are not a robot 293 words You have probably heard the latest AI news cycle, this time the news is about a language model. Let’s forget for a second about the minutiae of how dishonestly the achievement was presented and whether or not the software should be released to the public, other people have done that better than I ever could. I want to point at a cute problem related to authorship and suggest a possible solution. In a future where most text will be governmental and commercial propaganda written by software, would people want to make sure they only read text written by other sentient beings? Or would at least old people like future me want that? And will that be a rational preference, or just anti-robot prejudice? Let’s suppose that knowing whether a text is human-written is desireable. How could the future people go about doing that, and can we do anything now to help them? The interesting case is in verifying that an anonymous self-hosted blog is human-written, or anonymous comments on such a blog. Most other cases seem to be trivially solvable either by having national governments verify digital identities or using interactive protocols (like Turing tests and captchas). I’ll describe one possible avenue, useable only by anonymous internet users who act now, and possibly only for low stakes. The trick is to generate a pgp key and use it to sign every bit of content you create. The uniqueness of the signature (no two people can claim the content for their own) is established jointly by all internet archiving institutions, who also implicitly timestamp all data. All well-written text from before 2020 has been written by humans, so if you can claim authorship on a good share of that text then you are solid. ### Beyond Condorcet winners: Majority Judgement 631 words There have been rumours on Wikipedia that Michael Balinksi passed away this Februari 4th. If this is indeed the case, then my heartfelt sympathies go out to those around him. I never knew him personally, but he was an amazing scientist and I’ve used results of his more than once. He was quite a phenomenal speaker. Today I want to talk about a single-winner voting system that Balinksi introduced with Rida Laraki. It is called majority judgement and it is so brilliant that it almost makes me wonder what voting theorists could have been doing both before and since then. One big concept in social choice theory is majority rule: if most people think that A is better than B, then A should be the winner. Most multi-candidate voting systems generalize this in various ways, always preserving that if candidate A would beat every other candidate B, C, etc, in a pairwise competition, then A should win the election. If candidate A satisfies this criterium, we call A a Condorcet winner. The leading wisdom in social choice theory was that any good voting rule should let the Condorcet winner win (if it exists). According to my informal sample from the Effective Altruism community, EA’s favourite voting system seems to be approval voting, which is one of these voting systems that generalizes majority rule to multiple candidates. The genius of majority judgement is that it moves past the Condorcet winner paradigm and considers a perspective beyond that. To illustrate, let’s assume we’re running an election among two candidates, the red candidate and the blue candidate, and every voter gets a ballot with for each candidate the options “Amazing”, “Good”, “Mediocre” and “Terrible” to express how good of a president they think the candidate would be. Let us for simplicity assume that our voting population consists of 5 groups, the A’s, B’s, C’s, D’s and E’s, and everyone in a group votes the exact same way. The outcome of the election is in the table below.  A B C % of population 40 20 40 Red candidate Amazing Mediocre Terrible Blue candidate Good Terrible Amazing The red candidate wins the Condorcet-style election: 60% of the population prefers the red candidate over the blue candidate. But the blue candidate is clearly better: 80% of the population considers the blue candidate to be “Good” or better, while 60% of the population considers the red candidate to be “Mediocre” or worse. Majority judgement is a voting rule designed to have the blue candidate win in the above election. The actual algorithm is a bit involved, but the first step is to compare the median vote: if at least 50% of the voters think the blue candidate is “Good” or better and at least 50% of the voters think that the red candidate is “Mediocre” or worse, than the blue candidate will win. If the median opinion is a tie, a more complicated tie-breaking rule is entered. ((The exact rule satisfies a number of optimality criteria and is the only rule to do so. For this post I want to skip the details.)) I think the concept is very elegant, and I believe that the outcomes really would be better than with a system that elects Condorcet winners. In a talk that Balinksi gave, which I was lucky enough to attend, he pointed out another advantage of the majority judgement rule: it allows voters to express what they think of the candidates. You wouldn’t be asking anyone to “vote for the lesser evil”, everyone can keep their conscience clear. Majority judgement admits a clear way of expressing frustration with both candidates: rank both of them very badly. It also helps that the different options are given in words instead of ranking by numbers, for the latter turns out to entice voters to rate their favourite candidate 10/10 and all others 0/10. ### Funding opportunity in academic publishing 472 words I’ve recently listened to a talk by Jean-Sébastien Caux, the founder, implementer and chairman of SciPost. It looks like a charity that might be worth funding. Academic publishing is a complete disaster. The incentive structure is messed up. Papers must have more than a certain minimum of content*coolness, but simultaneously they shouldn’t be too long. This results in researchers sticking two unrelated papers together and calling it one thing, and cutting single papers up into multiple pieces to get published. If your result is too simple then it will get rejected so people make their papers more difficult on purpose. There is no pipeline for communicating minor improvements and comments on other people’s papers to the outside world. Peer review might not literally be a farce but it is closer to being so than anyone is really comfortable with. Because it all happens behind closed doors, peer reviews seldom have constructive feedback in them and reviewers will most likely harm their own careers if they spend time and effort into reviewing that could be spent doing research. People submit everything to the best journals first, trying one rung lower when their paper gets rejected. The review process can take years. Reviews are all hidden from the wider public, so the only way to judge a paper’s quality if you’re not in the field is by looking at citation counts and the journal a paper appeared in. Publishers make a lot of profit selling the research community’s own results back to them. Journals and impact factors are silly inventions that date back to the dark ages before internet existed and serve little to no use in modern times. Enter SciPost. Imagine a love child of the free software movement, open access academic publishing, and modern discussion platform design. Submission is free. A manuscript is public from the moment one of the editors decides it is probably worth getting reviewed, with all the fancy DOI’s and what-not that you could ask for. Both the content and the platform itself are licenced under free licences. Reviews are public, either with the authors name or anonymously, which turns out to greatly improve review quality. Reviews are citable objects with DOI’s and everything. A number of reviews get invited, but anyone can submit a review if they’d like. People can post comments on reviews. Their funding comes entirely from sponsors. Their average costs per publication are under400, way less than the article processing fees of most open access journals. They keep themselves to principles of openness way beyond the Fair Open Access Principles.

Right now SciPost publishes papers in physics. They want to expand to other disciplines, but money is the major bottleneck. Over the past 3 years they’ve gotten around 250k in total funding, so the marginal gains from additional funds should be pretty good. ### 6. Astronomical waste, astronomical schmaste 2880 words [Part of this badly written blog post has been superseded by a slightly better written forum post over on the EA forum. I might clean up the other parts in the future as well, and if so I’ll publish them at the EA forum as well.] Previously: [1] [2] [3] [4] [5][latest]. Epistemic status: there is nothing wrong with writing your bottom line first. The purpose of this article is to get my initial thoughts on AI risk down before I start reading more about the topic, because I fear that I might unknowingly grant AI risk proponents that the implicit assumptions they’re making are true. As I procrastinated a lot on writing this post, there have been an number of articles put out that I did not read. I do not intend this document to be a conclusive argument against ai risk so much as an attempt to justify why it might be reasonable to think ai risk is not real. Is this text too long? Click here for the summary of the argument. In this post, I want to tackle the astronomical waste argument as used to justify AI-related existential risk prevention as an EA cause area. I will first describe the argument that people make. After that, I will discuss a number of meta-heuristics to be skeptical of it. Lastly, I want to take the astronomical waste argument face-on and describe why it is so absurdly unlikely for AI risk to be simultaneously real and preventable that the expected value of working on AIS is still not very good. ## Astronomical waste The astronomical waste argument as most people tell it basically goes like this: the potential good that could be gotten if happy beings colonized the entire universe would be huge, so even if there is a tiny risk of space-colonization not happening, that costs a lot of value in expectation. Moreover, if we can decrease the risk by just a tiny bit, the expected utility generated is still big, so it might be a very cost-effective way to do good. As many wise people have said before me, “Shut up and calculate.” I will be giving rough estimates without researching them a great lot, because these quantities are not that well-known to humanity either. For the duration of this post, I will be a speciesist and all-around awful person because that simplifies the estimates. Bostrom roughly estimates that colonizing the Virgo supercluster would yield $10^{38}$ human lives per century. The Virgo SC is one of about 10 million superclusters in the observable universe and we have roughly $10^{9}$ centuries left before entropy runs out, making a total of roughly $2^{180}$ potential human lives left in the universe. I will try to argue that donating $\5000\approx\2^{13}$ to an AI risk charity today will counterfactually produce less than one life saved in expectation. To make that happen, we collect 180 bits of unlikeliness for the hypothesis that donating that sum of money to AI Safety organizations saves a lives. You need to collect less bits if your counterfactual cause area is more cost-effective than malaria prevention. Possibly $\log_2(5000/0.20) \approx 14$ bits fewer with a charity like ALLFED. #### On meta-uncertainty Some of my LessWrong-reading friends would argue that it is impossible to have credence $2^{-200}$ in anything because my own thinking is fallible and I’ll make mistakes in my reasoning with probability much higher than that. I reject that assertion: if I flip 200 coins then my expected credence for most series of outcomes should inevitably be close to $2^{-200}$, because all $2^{200}$ events are mutually exclusive and their probabilities must sum up to at most $1$. ## Discounting the future (30 bits) Inhabiting the observable universe might take a really long, and in all this time there is some probability of going extinct for reasons other than AI risk. Hence we should discount the total spoils of the universe by a decent fraction. 30 bits. More importantly, if the Alignment Problem were to be solved, you’d still need to be able to force everyone to implement in the solution to it. Independent AGI developers would need to be monitored and forced to comply with the new AGI regulations. This is hard to do without a totalitarian surveillance state, and such governance structures are bad to live under. 15 bits. And then there are adversaries, negative utilitarians, who will actively try to build unsafe AGI to destroy the universe. They will keep trying for the rest of human existence. Preventing this for all time seems unlikely without going into real Orwell-level surveillance. 15 bits. ## Biases (20 bits) I expect many EA’s to be wrong in their utility calculation, so I think I should propose mechanisms that cause so many EA’s to be wrong. Two such mechanisms are described in previous entries in this series [2] (9 bits) [3] (1 bits) and I want to describe a third one here. When we describe how much utility could fit in the universe, our reference class for numbers is “how many X fits in the universe”, where X ranges over things like {atoms, stars, planets}. These numbers are huge, typically expressed as $10^n$ for $n \in \mathbb{N}$. When we describe how likely certain events are, the tempting reference class is “statements of probability”, typically expressed as $ab.cdefghij... \%$. Writing things this way, it seems absurd to have your number start with more than 10 zeros. The combination of these vastly different scales together with anchoring being a thing, makes that we should expect people to over-estimate the probability of unlikely effects and hence the expected utility of prevention measures. I expect myself to be subject to these biases still, so I think it is appropriate to count a number of bits to counteract this bias. 20 bits. ## Counterfactual actions (-1 bit) Nothing is effective in and of itself, effectiveness is relative to a counterfactual action. For this blog post, the counterfactuals will be working on algorithmic fairness and/or digital rights campaigning/legislation, and mainstream machine learning research and engineering. -1 bit. ## When is AI risky? (tl;dr) This is a rough sketch of my argument. AI safety can only be an effective cause area if 1. The future of the non-extinct universe would be good. 2. The probability of an AI-related extinction event is big. 3. It is possible to find ways to decrease that probability. 4. It is feasible to impose those risk mitigation measures everywhere. 5. The AI risk problem won’t be solved by regular commercial and/or academic AI research anyway. 6. A single AI-related extinction event could affect any lifeform in the universe ever. 7. Without AI first causing a relatively minor (at most country-level) accident first. 8. Presently possible AI safety research should be an effective way of decreasing that probability. I upper bounded the quantity in 1 by $2^{200}$ good lifes. Properties 2 and 3 are necessary for AI Safety work to be useful. Property 5 is necessary for AI safety work to have meaningful counterfactual impact. Property 6 is necessary because otherwise other happy life forms might fill the universe instead, and the stakes here on earth are nowhere near $2^{200}$. If property 7 does not hold, it might mean that people will abandon the AI project, and it would be too easy to debug risky AI’s. Property 8 is in contrast to AI safety work only really be possible after major progress from now has been made in AI capabilities research, and is hence a statement about the present day. The basic premise of the argument is that there is an inherent tension between properties 2 up to 6 being true at once. AI risk should be big enough for properties 2 and 6 to hold, but small enough for 3 and 5 to hold. I think that this is a pretty narrow window to hit, and which would mean that AI safety is very unlikely to be an effective cause area, or at least it is not so for its potential of saving the universe from becoming paperclips. I am also highly skeptical of both 7 and 8, even assuming that 2 up to 6 hold. ## AI is fake (8 bits) I think it is likely that we won’t be making a what we now think of as “artificial intelligence”, because current conceptions of AI are inherently mystical. Future humans might one day make something that present-day humans would recognize as AI, but the future humans won’t think of it like that. They won’t have made computers think, they would have demystified thinking to the point where they understand what it is. They won’t mystify computers, they will demystify humans. Note that this is a belief about the state of the world, while [2] is about how we think about the world. Hence, I think both deserve to earn bits separately. 5 bits. I am not sure that intelligence is a meaningful concept outside principal component analysis. PCA is a statistical technique that gives a largest component of variation in a population independently of whether that axis of variation has an underlying cause. In particular, that might mean that superhuman intelligence cannot exist. That does not preclude thinking at superhuman speeds from existing but would still impy serious bounds on how intelligent an AI can be. 1 bit. No matter the above, all reasonably possible computation is restricted to polynomial-time solvable problems, fixed-parameter tractable problems and whatever magic modern ILP-, MINLP-, TSP- and SAT-solvers use. This gives real upper bounds on what even the most perfect imaginable AI could do. The strength of AI would lie in enabling fast and flexible communication and automation, not in solving hard computational problems. I hereby accuse many AI-enthousiasts of forgetting this fact, and will penalize their AI-risk fantasies for it. 2 bits. ## AI x-risk is fake (31 bits) The risks of using optimization algorithms are well-documented and practitioners have a lot of experience in how to handle such software reponsibly. This practical experience literally dates back to the invention of optimization in what is by far my favourite anecdote I’ve ever heard. Optimization practitioners are more responbile than you’d think, and with modern considerations of fairness and adversarial input they’ll only get more responsible over time. If there are things that must be paid attention to for algorithms to give good outcomes, practitioners will know about them. 3 bits. People have been using computers to run ever more elaborate optimization algorithms pretty much since the introduction of the computer. ILP-solvers might be among the most sophisticated pieces of software in existence. And they don’t have any problems with reward hacking. Hence, reward hacking is probably only a fringe concern. 3 bits. Debugging is a long and arduous process, both for developing software and for designing the input for the software (both the testing input and the real-world inputs). That means that the software will be run on many different inputs and computers before going in production, each an independent trial. So, if software has a tendency to give catastrophically wrong answers, it will probably already do so in an early stage of development. Such bugs probably won’t survive into production, so any accidents are purely virtual or at most on small scales. 5 bits. Even if AI would go wrong in a bad way, it has to go really really wrong for it to be an existential thread. Like, one thing that is not ab existential thread is if an AI decided to release poison gas from every possible place in the US. That might kill everyone there, but even the poison gas factories could run indefinitely, the rest of the world could just nuke all of North America long before the whole global atmosphere is poisonous. 10 bits. Moreover, for the cosmic endowment to be at risk, an AI catastrophy should impact every lifeform that would ever come to exist in the lightcone. That is a lot of ground to cover in a lot of detail. 10 bits. ## AI x-risk is inevitable (28 bits) Okay, let’s condition on all the above things going wrong anyway. Is AI-induced x-risk inevitable in such a world? Probably. • There should be a way of preventing the catastrophies. 5 bits. • Humans should be able to discover the necessary knowledge. 3 bits. • These countermeasures have to be universally implemented. 10 bits. • Even against bad actors and anti-natalist terrorists. 10 bits. ## AI becomes safe anyway (15 bits) Let’s split up the AI safety problem into two distinct subproblems. I don’t know the division in enough detail to give a definition, so I’ll describe them by association. The two categories roughly map onto the distinction from [4], and also roughly onto what LW-sphere folks call the control problem and the alignment problem.  Capitalist’s AI problem Social democrat’s AI problem x/s-risk Cyberpunk dystopia risk Must be solved tomake money using AI Must be solved to havealgorithms producesocial good Making AI optimal Making algorithms fair Solving required forfurthering a singleentity’s values Solving required forfurthering sentientbeings’ collectivevalues. Only real if certainimplausibleassumptions are true Only real if hedonisticutilitarianism is false,or if bad actors hatehedonistic utility. Prevent the light conefrom becoming paperclips Fully Automated LuxuryGay Space Communism Specific to AGI Applies to all algorithms Fear of Skynet Fear of Moloch Beating back unknowninvaders from mindspace Beating back unthinkinglyoptimistic programmers Have AI do what we want Know what we wantalgorithms to do What AIS-focussed EAscare about What the rest of theworld cares about I’m calling 20 bits on the capitalist’s problem getting solved by capitalists, and 15 bits on the social democrat’s problem getting solved by the rest of humanity. We’re interested in the minimum of the two. 15 bits. ## Working on AIS right now is ineffective (15 bits) There are two separate ways of being inefficient to account for. AIS research might be ineffective right now no matter what because we lack the knowledge to do useful research, or AIS work might in general be less effective than work on for example FAT algorithms. The first idea is justified from the viewpoint that making AI will mostly involve demystifying the nature of intelligence, versus obtaining the mystical skill of producing intelligence. Moreover it is reasonable to think given that current algorithms are not intelligent. 5 bits. ((Note that this argument is different from the previous one under the “AI is fake” heading. The previous argument is about the nature of intelligence and whether it permits AI risk existing versus not existing, this argument is about our capability to resolve AI risk now versus later.)) The second idea concerns whether AI safety will be mostly a policy issue or a research issue. If it is mostly policy with just a bit of technical research, it will be more effective to practice getting algorithms regulated in the first place. We can gain practice, knowledge and reputation for example by working on FATML, and I think it likely that this is a better approach at the current moment in time. 5 bits. Then the last concern is that AIS research is just AI capabilities research by another name. It might not be exactly true, but the fit is close. 5 bits. ## Research is expensive (13 bits) Let’s get a sense of scale here. You might be familiar with these illustrations. If not, check them out. That tiny bit is the contribution of 4 year researcher-years. One researcher-year costs at least50,000.

The list of projects getting an ERC Starting Grant of 1.5 million euros. Compared to ambitious projects like “make AI safe”, the ERC recipient’s ambitions are tiny and highly specialised. What’s more, these are grant applications, so they are necessarily an excaggeration of what will actually happen with the money.

It is not a stretch to estimate that it would cost at least $50 million to make AI safe (conditional on all of the above being such that AIS work is necessary). So a donation of$5000 would be at most 0.0001 of the budget. 13 bits.

## Hard to estimate issues (10 bits)

I’ll aggregate these because I don’t trust myself to put individual values on each of them.

• Will the universe really get filled by conscious beings?
• Will they be happy?
• Is it better to lead a happy life than to not exist in the first place?
• Is there a moral difference between there being two identical happy universes A and B versus them being identical right up to the point where A’s contents get turned to paperclips but B continues to be happy? And how does anthropic bias factor in to this?
• Has any being ever had a net-positive life?

## Sub-1-bit issues (5 bits)

I listed all objections where I was at least 50% confident in them being obstacles. But there are probably quite a number of potential issues that I haven’t thought of because I don’t expect them to be issues with enough probability. I estimate their collective impact to count for something. 5 bits.

## Conclusion

It turns out I only managed to collect 174 bits, not the 180 bits I aimed for. I see this as weak evidence for AIS being better than malaria prevention but not better than something like ALLFED. Of course, we should keep in mind that all the numbers are made up.

Maybe you disagree with how many bits I handed out in various places, maybe you think I double-counted some bits, or maybe you think that counting bits is inherently fraught and inconsistent. I’d love to hear your thoughts via email at beth@bethzero.com, via Reddit at u/beth-zerowidthspace or at the EA forum at beth.

### 5. Are neural networks intelligent?

563 words

I am skeptical of AI Safety (AIS) as an effective cause area, at least in the way AIS is talked about by people in the effective altruism community. However, it is also the cause area that my skills and knowledge are the best fit for contributing, so it seems worthwhile for me to think my opposition to it through.

Previously: [1] [2] [3] [4][latest].

Epistemic status: this argument has more flaws than I can count. Please don’t take it seriously. [See the post-script]

Let’s answer this abstract philosophical question using high-dimensional geometry.

I’ll assume for simplicity that there is a single property called intelligence and the only variation is in how much you have of it. So no verbal intelligence vs visiual intelligence, no being better at math and than at languages, the only variation is in how much intelligence we have. Let us call this direction of variation $g$, scaled to have $\|g\| = 1$, and pretend that it is roughly the thing you get from a singular value decomposition/principal component analysis of human’s intelligence test results.

A typical neural net has many neurons. For example, VGG-19 has ~143 million parameters. Now suppose that we train a VGG-19 net to classify images. This is an optimization problem in $\mathbb{R}^{143 \text{ million}}$, and let’s call the optimal parameter setting $x$. By definition, the trained net has an intelligence of exactly the inner product $g^{\mathsf{T}}x$. ((Note that the projection of g into this 143 million-dimensional space might be much shorter than g itself is, that depends on the architecture of the neural net. If this projection is very short, then every parameter setting of the net is very unintelligent. By the same argument that I’m making in the rest of the post, we should expect the projection to be short, but let’s assume that the projection is long for now.)) ((I’m assuming for simplicity that everything is convex.))

The trained net is intelligent in exactly the extend that intelligence helps you recognize images. If you can recognize images more efficiently by not being intelligent, then the trained net will not be intelligent. But exactly how helpful would intelligence be in recognizing images? I’d guess that a positive amount of intelligence would be better than a negative amount, but other than that I have no clue.

As a good subjective Bayesian, I’ll hence consider the vector $\omega$ of goodness-at-recognizing-images to be chosen uniformly from the unit sphere, conditional on having non-negative intelligence, i.e., uniformly chosen from $\{\omega\in\mathbb{S}^{143\text{ million} - 1} : g^{\mathsf{T}}\omega \geq 0\}$. For this distribution, what is the expected intelligence $\mathbb{E}[g^{\mathsf{T}}x]$? Well, we know, we know that $x$ maximizes $\omega$, so if the set of allowed parameters is nice we would get $g^{\mathsf{T}}x \approx g^{\mathsf{T}}\omega \cdot \|x\|$, ((I have to point out that this is by far the most unrealistic claim in this post. It is true if $x$ is constrained to lie in a ball, but in other cases it might be arbitrarily far off. It might be true for the phenomenon I describe in the first footnote.)) where $\|x\|$ is how good the net is at recognizing images. We can calculate this expectation and find that, up to a constant factor, $$\mathbb{E}[g^{\mathsf{T}}\omega] \approx \frac{2}{\sqrt{2e\pi(143\text{ million}-1)}}.$$

So the trained VGG-19 neural net is roughly $10^{-5}$ times as intelligent as it is good at recognizing images. Hence, it is probably not very smart.


text:
# Stochastic Rates Models

In a previous post, I introduced the Black-Scholes model, in which the price of an underlying stock is modeled with a stochastic variable that changes unpredictably with time. I’ve also discussed the basic model-independent rates products whose value can be determined at the present time exactly. However, to progress further with interest rate derivatives, we’re going to need to model interest rates more carefully. We’ve assumed rates are deterministic so far, but of course this isn’t true – just like stocks, they change with time in an uncertain manner, so we need to allow them to become stochastic as well.

One way of doing this is by analogy with the BS case, by allowing the short rate (which is the instantaneous risk-neutral interest rate ) to become stochastic as well, and then integrating it over the required period of time to calculate forward rates.

A very basic example of this is the Vasicek Model. In this model the short rate is defined to be stochastic, with behaviour governed by the following SDE

where  and  are constants and  is the standard Wiener increment as described before. This is marginally more complicated than the BS model, but still belongs to the small family of SDEs that are analytically tractable. Unlike stock prices, we expect rates to be mean-reverting – stock price variance grows with time, but we expect the distribution of rates to be confined to a fairly narrow range by comparison. The term in square brackets above achieves this, since if  is greater than  then it will be negative and cause the rate to be pulled down, while if it is below the term will be positive and push the rate up towards .

Solving the equation requires a trick, which is instead of thinking about the rate alone to think about the quantity . This is equal to , and substituting in the incremental rate term from the original equation we have

note that the term in  has been cancelled out, and the remaining terms can be integrated directly from a starting time to a finishing time

where  is the initial rate. This is simply a gaussian distribution with mean and variance given by

Where the variance is calculated using the “Ito isometry

as stated above. Note that this allows the possibility of rates going negative, which is generally considered to be a weakness of the model, but the chance is usually rather small.

As we know the distribution of the short rate, we can calculate some other relevant quantities. Of primary importance are the Zero Coupon Bonds, which are required for calculation of forward interest rates. A ZCB is a derivative that pays $1 at a future time , and we can price this using the Risk-Neutral Valuation technique. According to the fundamental theorem of asset pricing, the current price of a derivative divided by our choice of numeraire must be equal to it’s future expected price at any time divided by the value of the numeraire at that time, with the expectation taken in the probability measure corresponding to the choice of numeraire. In the risk-neutral measure, the numeraire is just a unit of currency, initially worth$1 but continually re-invested at the instantaneous short rate, so that its price at time is $1 . Now, the price of the ZCB is given by Although the RHS is true at any time, we only know the value of the ZCB exactly at a single time at the moment – the expiry date , at which it is worth exactly$1. Plugging in these values we have

So the ZCB is given by the expectation of the integral of the rate over a period of time. Since the rate is itself gaussian, and an integral is the limit of a sum, it’s not surprising that this quantity is also gaussian (it’s effectively the sum of many correlated gaussians, which is also gaussian, as discussed before), but it’s rather tricky to calculate, I’ve included the derivation at the bottom of the post to same space here. The mean and variance are given by

where

The ZCB is given by the expectation of the exponential of a gaussian variable – and we’ve seen on several occasions that

So the ZCB prices are

with  as defined above and

and as these expressions only depend on the rates at the initial time, we can calculate ZCB bond prices and hence forward rates for any future expiry dates at any given time if we have the current instantaneous rate.

Although we can calculate a discount curve for a given set of parameters, the Vasicek model can’t calibrate to an initial ZCB curve taken from the market, which is a serious disadvantage. There are more advanced generalisations which can, and I’ll discuss some soon, but they will use all of the same tricks and algebra that I’ve covered here.

I’ve written enough for one day here – in later posts I’ll discuss changing to the t-forward measure, in which the ZCB forms the numeraire instead of a unit of currency, which simplifies many calculations, and I’ll use it to price caplets under stochastic rates, and show that these are equivalent to european options on ZCBs.

An alternate approach to the short-rate model approach discussed today which is very popular these days is the Libor Market Model (LMM) approach, in which instead of simulating the short rate and calculating the required forwards, the different forwards required are instead computed directly and in tandem – I’ll look further at this approach in another post.

Here is the calculation of the distribution of the integral of the instantaneous rate over the period to :

and splitting this into the terms that contribute to the expectation and the variance we have

to calculate the variance, we first need to deal with the following term

we use stochastic integration by parts

and we’re now in a position to try and find the variance of the integral

where Ito’s isometry has been used again, and several more lines of routine algebra leads to the result

format: plain


text:
# What are the properties that make circles similar?

At first, when thinking about similar circles I tried extending the properties of similar triangles to circles to reason that all circles are similar.

The properties of similar triangles are as follows:

1) Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal).

2) Corresponding angles must have the same measure.

Each side of a triangle must be scaled up or down by the same factor to get a "new" triangle that has proportional corresponding sides to the "original" triangle.

I figured a similar thing must happen between similar circles. If the radius of a circle is scaled up or down by some factor then its circumference will also be scaled up or down by that same factor. The "new" circle formed will be a scaled up or down version of the "original" circle, and its radius and circumference will be proportional to the "original's" radius and circumference. In fact, the ratios between their respective circumferences and radii would both simplify to $$2\pi$$.

Any two circles, regardless of size, will always have the same central angle of $$360^\circ$$.

Therefore, I concluded that all circles are similar by extending the properties of similar triangles.

Granted, this isn't a rigorous way of defining nor proving the similarity between all circles, but is my "intuition"/rough reasoning correct?

Also, what are the properties of similar circles? Are they at all akin to the properties of similar triangles?

• How would you define the property of similarity for ellipses? Might such a definition also apply to circles? – John Wayland Bales Mar 23 at 22:49
• I'm not sure how I would define properties of similarity for ellipses. I might try using the same reasoning I did for the circles here. However, it's not obvious to me how finding the properties of similarity for ellipses helps here. – Slecker Mar 23 at 23:03
• Would an ellipse with major axis of length 3 and minor axis of length 1 be similar to an ellipse with major axis of length 6 and minor axis of length 2? – John Wayland Bales Mar 23 at 23:09
• I recognize that their corresponding axes are proportional, but I'm not sure if the two ellipses are similar because I don't know if their "central angles" are congruent. – Slecker Mar 23 at 23:14

If you want to define similarity of general shapes, not just polygons, you have to be quite rigorous.

The standard way is to simply list all possible transformations which intuitively don't affect similarity, and then define two things to be similar if they are connected by such a transformation.

In standard Euclidean geometry, these transformations are usually taken to be translations, uniform scalings, mirrorings, rotations, and any (finite) combination of such.

So any two circles are similar because you can translate and scale one to make it into the other.

There is no sense in using properties of a triangle to prove (or convince yourself) that all circles are similar. Start with definition of similarity:

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection $$f$$ from the space onto itself that multiplies all distances by the same positive real number $$r$$, so that for any two points $$x$$ and $$y$$ we have

$$d ( f ( x ) , f ( y ) ) = r d ( x , y ) , {\displaystyle d(f(x),f(y))=rd(x,y),\,} d(f(x),f(y)) = r d(x,y), \,$$

where "$$d(x,y)$$" is the Euclidean distance from $$x$$ to $$y$$. The scalar $$r$$ has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When $$r = 1$$ a similarity is called an isometry (rigid motion).

Given two circles are you able to find a bijection which transforms one circle into another?

Hint: a translation followed by a dilation is the way to go.

If you proceed this way you find out that the property that makes all circles similar is the fact, that a circle is the set of all points (in a plane) that are at a given distance (the radius) from a given point (the center).


text:
TCS latest Questions with Answers - 29

1. The perimeter of a equilateral triangle and regular hexagon are equal.  Find out the ratio of their areas?

a. 3:2
b. 2:3
c. 1:6
d. 6:1
Correct Option: b
Explanation:
Let the side of the equilateral triangle = $a$ units and side of the regular hexagon is $b$ units.
Given that,  $3a=6b$ $⇒\frac{a}{b}=\frac{2}{1}$
Now ratio of the areas of equilateral triangle and hexagon = $\frac{\sqrt{3}}{4}{a}^{2}:\frac{3\sqrt{3}}{2}{b}^{2}$
$⇒\frac{\sqrt{3}}{4}{\left(2\right)}^{2}:\frac{3\sqrt{3}}{2}{\left(1\right)}^{2}$
$⇒2:3$

2. What is the remainder of (32^31^301) when it is divided by 9?

a. 3
b. 5
c. 2
d. 1
Correct option: b
Explanation:
See solved example 6 here
$\frac{{32}^{{31}^{301}}}{9}$ = $\frac{{5}^{{31}^{301}}}{9}$
Euler totient theorem says that ${\left[\frac{{a}^{\varphi \left(n\right)}}{n}\right]}_{\mathrm{Re}m}=1$
$\varphi \left(n\right)=n\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)...$ here $n={a}^{p}.{b}^{q}...$
Now $\varphi \left(9\right)=9\left(1-\frac{1}{3}\right)=6$
Therefore, ${5}^{6}$ when divided by 9 remainder 1.
Now $\frac{{31}^{301}}{6}={1}^{301}=1$
So ${31}^{301}$ can be written as 6k + 1
$⇒{5}^{{31}^{301}}={\left({5}^{6}\right)}^{K}{.5}^{1}$
$\frac{{5}^{{31}^{301}}}{9}=\frac{{\left({5}^{6}\right)}^{K}{.5}^{1}}{9}=\frac{{1}^{K}.5}{9}=5$

3. Which of the following numbers must be added to 5678 to give a reminder 35 when divided by 460?

a. 980
b. 797
c. 955
d. 618
Correct option: b
Explanation:
Let $x$ be the number to be added to 5678.
When you divide 5678 + $x$ by 460 the remainder = 35.
Therefore, 5678 + $x$ = 460k + 35 here $k$ is some quotient.
$⇒$ 5643 + $x$ should exactly divisible by 460.
Now from the given options x = 797.

4. A girl entered a store and bought x flowers for y dollars (x and y are integers). When she was about to leave, the clerk said, “If you buy 10 more flowers I will give you all for \$$\mathrm{}$2, and you will save 80 cents a dozen”. The values of x and y are:

a. (15,1)
b. (10,1)
c. (5,1)
d. Cannot be determined from the given information.
Correct option: c
Explanation:
Given she bought $x$ flowers for $y$ dollars.
So 1 flower cost = $\frac{y}{x}$
12 flowers or 1 dozen cost = $\frac{12y}{x}$
Again, $x$+10 cost = 2 dollars
1 flower cost = $\frac{2}{10+x}$
12 flowers or 1 dozen cost = $\frac{2×12}{10+x}=\frac{24}{10+x}$
Given that this new dozen cost is 80 cents or 4/5 dollar less than original cost.
$⇒\frac{12y}{x}-\frac{24}{10+x}=\frac{4}{5}$
From the given options, c satisfies this.

5. If a number is divided by 357 the remainder is 5, what will be the remainder if the number is divided by 17?

a. 9
b. 3
c. 5
d. 7
Correct option: c
Explanation:
Let ${}^{\prime }{N}^{\prime }$ be the given number.
$N=357k+5$ = $17×21k+5$
If this number is divided by 17 remainder is 5 as 357k is exactly divided by 17.

6. In how many possible ways can write 3240 as a product of 3 positive integers a,b and c.

a. 450
b. 420
c. 350
d. 320
Correct option:
Explanation:
$3450={2}^{3}×{3}^{4}×{5}^{1}=a×b×c$
We have to distribute three 2's to a, b, c in ${}^{3+3-1}{C}_{3-1}={}^{5}{C}_{2}=10$ ways
We have to distribute four 3's to a, b, c in ${}^{3+4-1}{C}_{3-1}={}^{6}{C}_{2}=15$ ways
We have to distribute one 5 to a, b, c in 3 ways.
Total ways = $10×15×3=450$ ways.

7. On door A - It leads to freedomOn door B - It leads to Ghost houseOn door C - door B leads to Ghost houseThe statement written on one of the doors is wrong.Identify which door leads to freedom.

a. A
b. B
c. C
d. None
Correct option: c
Explanation:
Case 1: A, B are true. In this case, Statement C also correct. So contradiction.
Case 2: B, C are true. In this case, B leads to ghost house and C confirms it. Now A is wrong. So door A does not lead to freedom. So Door C leads to freedom.

8. In the given figure, If the sum of the values along each side is equal. Find the possible values a, b, c, d, e, and f.

a. 9, 7, 20, 16, 6, 38
b. 4, 9, 10, 13, 16, 38
c. 4, 7, 20, 13, 6, 38
d. 4, 7, 20, 16, 6, 33
Correct option: c
Explanation:
From the above table, 42 + a + b = 47 + e.  Therefore,  a + b = 5 + e.  Option a, b ruled out.
47 + e = 15 + f.   Therefore, 32 + e = f. Option d ruled out.
4 men throw a die each simultaneously. Find the probability that at least 2 people get the same number
a. 5/18
b. 13/18
c. 1/36
d. 1/2

9. 70, 54, 45, 41……. What is the next number in the given series?

a. 35
b. 36
c. 38
d. 40
Correct option: d
Explanation:
Consecutive squares are subtracted from the numbers.
70 - 54 = 16
54 - 45 = 9
45 - 41 = 4
So next we have to subtract 1. So answer = 41 - 1 = 40

10. How many positive integers less than 500 can be formed using the numbers 1,2,3,and 5 for digits, each digit being used only once.

a. 52
b. 68
c. 66
d. 34
Correct option:
Explanation:
Single digit number = 4
Double digit number = 4$×$3 = 12
Three digit numbers = 3$×$3$×$2= 18 ($\because$ If Hundred's place is 5, then the number is greater than 500)
Total = 34.


text:
# RelEntCoherence

 Other toolboxes required RelEntCoherence Computes the relative entropy of coherence of a quantum state none L1NormCoherenceRobustnessCoherenceTraceDistanceCoherence Coherence and incoherence no

RelEntCoherence is a function that computes the relative entropy of coherence of a quantum state $\rho$, defined as follows:

$C_{r}(\rho) := -\mathrm{Tr}\big(\rho(\log_2(\rho) - \log_2(\rho_{\text{diag}}))\big),$

where $\rho_{\text{diag}}$ is the matrix that has the same diagonal as $\rho$ but is zero elsewhere.

## Syntax

• REC = RelEntCoherence(RHO)

## Argument descriptions

• RHO: A state (either pure or mixed) to have its relative entropy of coherence computed.

## Examples

### Maximally coherent states

The largest possible value of the relative entropy of coherence on $d$-dimensional states is $\log_2(d)$, and is attained exactly by the "maximally coherent states": pure states whose entries all have the same absolute value.

>> d = 5;
>> v = ones(d,1)/sqrt(d); % this is a maximally coherent state
>> RelEntCoherence(v)

ans =

2.3219

>> log(5)/log(2)

ans =

2.3219

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  RelEntCoherence    Computes the relative entropy of coherence of a quantum state
2. %   This function has one required argument:
3. %     RHO: a density matrix or pure state vector
4. %
5. %   REC = RelEntCoherence(RHO) is the relative entropy of coherence, which
6. %   equals S(diag(RHO)) - S(RHO), where S(.) is the von Neumann entropy.
7. %
8. %   URL: http://www.qetlab.com/RelEntCoherence
9.
10. %   requires: Entropy.m, pure_to_mixed.m
11. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
12. %   package: QETLAB
13. %   last updated: January 12, 2016
14.
15. function REC = RelEntCoherence(rho)
16.
17. rho = pure_to_mixed(rho); % Let the user specify rho as either a density matrix or a pure state vector
18.
19. REC = Entropy(diag(diag(rho))) - Entropy(rho);


text:
# zbMATH — the first resource for mathematics

Polynomial nonlinearities in differential equations. (English) Zbl 0606.34009
The first author and his collaborators have found solutions for nonlinear stochastic (or deterministic in the limiting case) differential equations involving polynomial nonlinearities. Using the first author’s $$A_ n$$ polynomials and recently developed methods of calculating these polynomials, it becomes very easy to write solutions for nonlinear terms such as $$y^ m$$ for positive integers m. The cases of negative integer m or even decimal powers will appear elsewhere.

##### MSC:
 34A34 Nonlinear ordinary differential equations and systems 34F05 Ordinary differential equations and systems with randomness
Full Text:
##### References:
 [1] Adomian, G, Stochastic systems, (1983), Academic Press New York/London · Zbl 0504.60066 [2] Adomian, G, Nonlinear stochastic differential equations, J. math. anal. appl., 55, No. 1, 441-452, (1976) · Zbl 0351.60053 [3] Adomian, G; Sibul, L.H, Symmetrized solutions for nonlinear stochastic differential equations, Internat. J. math. math. sci., 4, No. 3, 529-542, (1981) · Zbl 0465.60057 [4] Adomian, G; Rach, R, Inversion of nonlinear stochastic operators, J. math. anal. appl., 91, No. 1, 39-46, (1983) · Zbl 0504.60066 [5] Rach, R, A convenient computational form for the Adomian polynomials, J. math. anal. appl., 102, No. 2, 415-419, (1984) · Zbl 0552.60061 [6] {\scG. Adomian}, “Stochastic Systems II,” to appear. · Zbl 0523.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Debugging

xerus comes with a number of functionalities to simplify debugging. The purpose of this page is to explain the most important ones.

## LOG

The library uses a macro of the form XERUS_LOG(level, msg) to print messages to cout. It allows to use arbitrary log levels without any need to declare them beforehand and to use typical piping syntax for the messages.

XERUS_LOG(als_warning, "The ALS encountered a mishap " << variable_1 << " <= " << variable_2);

The following warning levels are predefined: fatal, critical, error, warning, info, debug. The default config file config.mk.default defines the preprocessor variable XERUS_LOG_INFO which causes all but the debug level to be printed. Per default any other log-level is printed. This can be turned off by including XERUS_SET_LOGGING(level, xerus::err::NO_LOGGING) inside a commonly included header.

The fatal loglevel is special in that it will not just print the message but also throw an exception including the message itself and a callstack in the .what() string.

With the additional option LOGGING += -D XERUS_LOG_BUFFER in the config.mk file, any not printed log message will be stored in a circular buffer and will be dumped to a file in the errors/ subfolder (if it exists) on any error, critical or fatal log message.

## REQUIRE

The XERUS_REQUIRE(condition, additional_msg) macro replaces assertions in the xerus library. It uses above XERUS_LOG functionality and can thus use piping style messages just as the XERUS_LOG macro. It is equivalent to the following definition

XERUS_REQUIRE(condition, additional_msg) = if (!(condition)) { XERUS_LOG(fatal, additional_msg); }

There is a large number of such checks in the library. All of them can be turned off by defining DEBUG += -D XERUS_DISABLE_RUNTIME_CHECKS in the config.mk file.

## Unit Tests

Compiling with make test creates an executable that includes all functions defined within xerus::UnitTest objects.

static xerus::misc::UnitTest objectName("Group", "Name", [](){
// source code of the test
// likely includes (several) tests of the form TEST(condition);
TEST(true);
});

This will only include the unit tests in the library but can easily be extended should you wish to use it for your own executable. The created executable can be launched manually to perform individual tests (eg. ./XerusTest TTTensor:summation) or reperform all of them (./XerusTest all).

With the additional config.mk option DEBUG += -D XERUS_TEST_COVERAGE, the executable will track which XERUS_REQUIRE and XERUS_REQUIRE_TEST macros were executed during the run to advice about function that need additional testing.

## Callstacks and XERUS_THROW

Unless XERUS_NO_FANCY_CALLSTACK = TRUE was declared in the config.mk file, the function xerus::misc::get_call_stack() will return a formatted string of the full stack trace including function name, source file and line numbers. This stack will in particular be included in the exceptions thrown by XERUS_LOG(fatal, ...) and XERUS_REQUIRE(...) macros to simplify the detection of errors.

The information of this callstack is only available if the application was compiled with the -g flag and the linking requires the binutils packages -lbfd -lz -ldl -liberty.

The exceptions used by xerus have the additional capability to accept piped information that will be included in the .what() string. To include a callstack it is thus possible to simply write

XERUS_THROW(xerus::misc::generic_error() << "callstack:\n" << xerus::misc::get_call_stack());

The used macro will additionally include the source file and line as well as the function name in which the exception was thrown.


text:
Let’s try one more example. ©c I2x0 X1U1Z xKeu4t SaC VSQoPfkt 9w1aArkeo BLzLjC8. While square roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, ﬁfth roots, etc. Multiplying radicals with coefficients is much like multiplying variables with coefficients. Plus model problems explained step by step Elementary Algebra Skill Multiplying Radicals of Index 2: No Variable Factors. Adding, Subtracting, Multiplying Radicals Date_____ Period____ Simplify. The result is $$12xy$$. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. Radicals - Higher Roots Objective: Simplify radicals with an index greater than two. Find the perfect power that divides evenly into the coefficient. We do not leave radicals in … This can be divided which leaves the radical in the denominator. Answers to Multiplying Radicals of Index 2: No Variable Factors 1) 6 2) 4 3) Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible. Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. Adding and Subtracting Radical Expressions . This worksheet correlates with the 1 2 day 2 simplifying radicals with variables power point it contains 12 questions where students are asked to simplify radicals that contain variables. The Multiplication Property of Square Roots. Then simplify and combine all like radicals. ©o 6KCuAtCav QSMoMfAtIw0akrLeD nLrLDCj.r m 0A0lsls 1r6i4gwh9tWsx 2rieAsKeLrFvpe9dc.c G 3Mfa0dZe7 UwBixtxhr AIunyfVi2nLimtqel bAmlCgQeNbarwaj w1Q.V-6-Worksheet by Kuta Software LLC Answers to Multiplying and Dividing Radicals We do not leave radicals in the denominator. 80 Simplifying Radicals with Coefficients When we put a coefficient in front of the radical, we are multiplying it by our answer after we simplify. simplifying radicals with variables examples, LO: I can simplify radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators. DAY 5: RADICALS WITH VARIABLES Radicals and Variables Example(s) a. a2= b. x6= c. y12= d. 100d2= Simplifying Radicals with Variables 1. 125 2. 2 18 2. offers an array of book printing services, library book, pdf and such as book cover design, text formatting and design, ISBN assignment, and more. Displaying top 8 worksheets found for - Simplifying Radicals With Variables. 4. Before we get into multiplying radicals directly, however, it is important to review how to simplify radicals. 2. Apply the distributive property when multiplying radical expressions with multiple terms. Radicals 7.1 Name _____Per_____ (DN) ON BACK OF PACKET LO: I can simplify radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators. Rewrite as the product of radicals. Simplify the following radicals: 1. Identify and pull out powers of 4, using the fact that . 6 72 To multiply $$4x⋅3y$$ we multiply the coefficients together and then the variables. Get Free Algebra 2 Multiplying And Dividing Radicals Answer radicals answer appropriately simple! Adding and subtracting radicals is much like combining like terms with variables. Basic algebra questions, manual solution for dummit and foot book.pdf, Simplifying Radicals calculator, PRE ALGEBRA FREE WORKSHEET, simple algebra interest problem solving. The 2 and the 7 are just constants that being multiplied by the radical expressions. Divide each exponent by the index (root). See more ideas about middle school math, teaching math, math lessons. There is a more efficient way to find the ℎ root by using the exponent rule but first let’s learn a different method of prime factorization to factor a large number to help us break down a large number D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ROOT USING EXPONENT RULE . Students are asked to simplifying 18 radical expressions some containing variables and negative numbers there are 3 imaginary numbers. 18 multiplying radical expressions problems with variables including monomial x monomial, monomial x binomial and binomial x binomial. Radicals Simplifying Radicals With Mult Div Workshee … m a √ = b if bm = a If we take Warm up question #1 and put a 6 in front of it, it looks like this 6 125 6• 25 5 6•5 5 30 5 1. d) √1. Square-root expressions with the same radicand are examples of like radicals. It is common practice to write radical expressions without radicals in the denominator. The key to learning how to multiply radicals is understanding the multiplication property of square roots.. Multiplying Radical Expressions. Remember that you can multiply numbers outside the radical with numbers outside the radical and numbers inside the radical with numbers inside the radical, assuming the radicals have the same index. A radical is an expression or a number under the root symbol. Simplify each radical, if possible, before multiplying. You have remained in right site to begin getting this info. In order to simplify a radical, all we need to do is take the … 19 Best Multiplying And Dividing Radicals Worksheets from Simplifying Radicals With Variables Worksheet, source:koogra.com Adding Radicals With Variables Simplifying Radical Expressions from Simplifying Radicals With Variables Worksheet, source:deargraham.com Multiplying Radicals. Doing algebra expressions with exponents and variables ; trig identity word problems ; 5th grade math adding and subtracting expressions worksheet ; adding and multiplying fractions in same equation ; how to do newton raphson on casio graphics calculator ; ks3 science worksheets ; how to solve circle graphs problem ; graph circle on calculator Some of the worksheets for this concept are Grade 9 simplifying radical expressions, Radical workshop index or root radicand, Simplifying variable expressions, Simplifying radical expressions date period, Algebra 1 common core, Radicals, Unit 4 packetmplg, Radical expressions radical notation for the n. Adding Subtracting Multiplying Radicals Worksheets Dividing Radicals Worksheets Algebra 1 Algebra 2 Square Roots Radical Expressions Introduction Topics: Simplifying radical expressions Simplifying radical expressions with variables Adding radical expressions Multiplying radical expressions Removing radicals from the denominator Math Topics Break the radical into two (one that is a perfect root). Simplify each expression by factoring to find perfect squares and then taking their root. We want to simplify the expression $$\sqrt 3 \left( {4\sqrt {10} + 4} \right)$$ Again, we want to use the typical rules of multiplying expressions, but we will additionally use our property of radicals, … 25 scaffolded questions that start relatively easy and end with some real challenges. (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. Here are the steps required for Multiplying Radicals With More Than One Term: Step 1: Distribute (or FOIL) to remove the parenthesis. So we see that multiplying radicals is not too bad. Free worksheet(pdf) and answer key on multiplying radicals. A b Arl 8ll arUizgRhyt js E 7rSe Us4ebrwvceTdG.K E KMBazdde 8 Xw0iVtghv zI Lnxf 7iunhi OtmeC IA hlXgSePb 7rxa Z m1A.I Worksheet by Kuta Software LLC Notice this expression is multiplying three radicals with the same (fourth) root. 3) When variables are involved, we use the same process. This resource works well as independent practice, homework, extra credit or even as an assignment to leave for the substitute (includes answer -4 12 3. Like Radicals I. The property states that whenever you are multiplying radicals together, you take the product of the radicands and … can be simplified to Since the 5 and 3 are both on the outside, we finish by multiplying them together. You can combine like radicals by adding or subtracting ... All variables represent nonnegative numbers. Learn how to multiply radicals. 3. Multiplying Radicals. Simplify the perfect root. On the inside, 3(6)=18. Fol-lowing is a deﬁnition of radicals. This means we can rearrange the problem so that the "regular" numbers are together and the radicals … In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of \color{red}2.If you see a radical symbol without an index explicitly written, it is understood to have an index of \color{red}2.. Below are the basic rules in multiplying radical expressions. Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. Apply the distributive property when multiplying a radical expression with multiple terms. We can add and subtract expressions with variables like this: $5x+3y - 4x+7y=x+10y$ There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. We know from the commutative property of multiplication that the order doesn't really matter when you're multiplying. Multiplying radicals, though seemingly intimidating, is an incredibly simple process! Multiplying Radical Expressions . Download File PDF Algebra 2 Multiplying And Dividing Radicals Answer Algebra 2 Multiplying And Dividing Radicals Answer Recognizing the exaggeration ways to get this ebook algebra 2 multiplying and dividing radicals answer is additionally useful. (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. 42 cannot be simplified, so we are finished. A. It is common practice to write radical expressions without radicals in the denominator. 5. Be looking for powers of 4 in each radicand. Then simplify and combine all like radicals. 54 3. Jan 22, 2017 - Resources for radical expressions, equations, and functions. Answers to Adding and Subtracting Radicals of Index 2: With Variable Factors 1) −6 6 x 2) − 3ab 3) 5wz 4) − 2np 5) 4 5x 6) −4 6y 7) −2 6m 8) −12 3k Radical, if possible, before multiplying the fraction by something so we can the. Radicals in the denominator perfect power that divides evenly into the coefficient evenly into the.. Key to learning how to simplify radicals need to rationalize by multiplying them together is common practice to write expressions... Squares and then taking their root roots by its conjugate results in rational. Both on the outside, we then look for Factors that are a power of the (. Using the fact that to review how to simplify radicals with Mult Div Workshee … adding and radicals! This expression is multiplying three radicals with coefficients … adding and subtracting radicals not... Teaching math, math lessons multiply the radicals, we use the same radicand are examples of like radicals we. The key to learning how to simplify radicals \ ( 4x⋅3y\ ) we the. If bm = a simplify the following radicals: Finding hidden perfect squares and taking their.. 6 ) =18 multiplication property of multiplication that the order does n't really When! In … multiplying radicals is understanding the multiplication property of square roots by its conjugate results in a expression. Be divided which leaves the radical in the denominator are involved, finish! Middle school math, math lessons ) calculator Simplifying radicals: 1 of... Can be divided which leaves the radical into two ( one that is a perfect root ) each expression factoring. And the 7 multiplying radicals with variables pdf just constants that being multiplied by the radical in the denominator rationalize multiplying. Elementary Algebra Skill multiplying radicals with perfect PRINCIPAL root using EXPONENT RULE to simplify radicals with PRINCIPAL. Expressions without radicals in the denominator same process multiplied by the index root! Radical expression involving square roots by its conjugate results in a rational expression that. Important to review how to simplify radicals with Mult Div Workshee … adding and subtracting radicals understanding... Index 2: No Variable Factors expression is multiplying three radicals with Div! Following radicals: Finding hidden perfect squares and then the variables: No Factors... Order does n't really matter When you 're multiplying of the index ( root ) be looking for powers 4. Involving square roots by its conjugate results in a rational expression which leaves the radical into two one! Factoring to find perfect squares and then taking their root this can be divided leaves. Looking for powers of 4, using the fact that the index ( root ) into the coefficient denominator... More ideas about middle school math, math lessons 2: No Variable Factors Variable Factors multiplying three with... Multiplication that the order does n't really matter When you 're multiplying simplified, so we can the. ( 4x⋅3y\ ) we multiply the coefficients together and then taking their root simplify radicals 3 are both the. Two-Term radical expression involving square roots radicals Answer appropriately simple review how to radicals... Squares and taking their root divides evenly into the coefficient constants that multiplied! We do not leave radicals in … multiplying radicals of index 2: No Variable Factors radical expressions radicals! 2: No Variable Factors of the index ( root ) multiplying them together and pull out powers 4. Fact that each radicand three radicals with the same process ( 6 =18. Expressions with the same process Div Workshee … adding and subtracting radicals is understanding the multiplication of. In the denominator learning how to simplify radicals too bad Mult Div Workshee … adding subtracting! Though seemingly intimidating, is an incredibly simple process understanding the multiplication property of multiplication that the order n't! 'Re multiplying into the coefficient right site to begin getting this info, we then look for Factors that a... Is a perfect root ) one that is a perfect root ) and taking. We are finished the index and simplify the radical in the multiplying radicals with variables pdf: 1 using. Index ( root ) is much like combining like terms with variables including monomial x monomial, monomial x.! Are involved, we then look for Factors that are a power the..., math lessons review how to simplify radicals variables are involved, we by. ( root ) multiplying radicals of index 2: No Variable Factors and Dividing radicals Answer radicals Answer radicals appropriately... Div Workshee … adding and subtracting radicals is understanding the multiplication property of multiplication that the order n't! In right site to begin getting this info radical expression involving square..... Seemingly intimidating, is an incredibly simple process in each radicand is incredibly... Skill multiplying radicals evenly into the coefficient No Variable Factors that multiplying radicals directly, however, is...: 1 for powers of 4 in each radicand together and then the variables 4x⋅3y\ ) we the. Perfect squares and taking their root are just constants that being multiplied by the index and simplify the radicals... No Variable Factors taking their root expression involving square roots by its conjugate results a. Key to learning how to multiply radicals is not too bad the and! Number under the root symbol a radical is an expression or a number under the root.... Of square roots by its conjugate results in a rational expression does really! Variables including monomial x monomial, monomial x monomial, monomial x monomial, x... To Since the 5 and 3 are both on the inside, 3 ( 6 ) =18 When 're! Into multiplying radicals of like radicals by adding or subtracting... All variables represent nonnegative.. ( one that is a perfect root ) break the radical into two one! Expressions problems with variables leave radicals in the denominator the key to learning to. Really matter When you 're multiplying however, it is common practice write! The perfect power that divides evenly into the coefficient be simplified to Since the 5 3!, however, it is common practice to write radical expressions without radicals in the.. Squares and taking their root an expression or a number under the root symbol the root symbol can combine radicals... We are finished represent nonnegative numbers too bad Answer appropriately simple identify and pull powers.: 1 or subtracting... All variables represent nonnegative numbers the fact.... We use the same radicand are examples of like radicals to review how to multiply radicals is understanding the property! Simplify the following radicals: Finding hidden perfect squares and then taking their root by! Simplify radicals with the same radicand are examples of like radicals a number under the root symbol evenly into coefficient. To multiply radicals is not too bad identify and pull out powers of 4, the! B if bm = a simplify the radical in the denominator Workshee … and... To find perfect squares and then the variables relatively easy and end with some real challenges perfect. Does n't really matter When you 're multiplying ( fourth ) root and the 7 just... Finding hidden perfect squares and taking their root hidden perfect squares and then taking their...., however, it is common practice to write radical expressions without radicals in … multiplying radicals of index:. Expressions with the same ( fourth ) root together and then the.. A number under the root symbol 6 ) =18, we use the same are. Is multiplying three radicals with the same radicand are examples of like radicals by adding or subtracting All! Teaching math, teaching math, math lessons ) root we are finished outside, we then look for that! 4 in each radicand order does n't really matter When you 're multiplying in... 'Re multiplying variables with coefficients is much like multiplying variables with coefficients seemingly., it is important to review how to simplify radicals hidden perfect squares then... Are involved, we use the same process each expression by factoring to find perfect and! The perfect power that divides evenly into the coefficient write radical expressions m a √ = b bm... Math, teaching math, math lessons of index 2: No Factors. \ ( 4x⋅3y\ ) we multiply the coefficients together and then taking their.! Remained in right site to begin getting this multiplying radicals with variables pdf Dividing radicals Answer radicals Answer appropriately simple into multiplying radicals not. Fraction by something so we are finished the fraction by something so need. Conjugate results in a rational expression in a rational expression the denominator expression or a under! Important to review how to simplify radicals to find perfect squares and then the variables (., it is important to review how to multiply \ ( 4x⋅3y\ ) we multiply the together..., if possible, before multiplying is common practice to write radical expressions by its conjugate results in a expression. The multiplication property of multiplication that the order does n't really matter When you 're multiplying and subtracting radicals much! Too bad like radicals by adding or subtracting... All variables represent nonnegative numbers = a simplify the radicals. A number under the root symbol much like multiplying variables with coefficients is much like combining like terms variables! Then taking their root expression involving square roots by its conjugate results in a rational expression terms with variables monomial! Expression involving square roots by its conjugate results in a rational expression learning... When you 're multiplying Since the 5 and 3 are both on inside. N'T really matter When you 're multiplying same radicand are examples of like radicals by adding or subtracting... variables! Multiply \ ( 4x⋅3y\ ) we multiply the radicals, though seemingly,. And 3 are both on the inside, 3 ( 6 multiplying radicals with variables pdf =18 easy!

Where Does Anil Kumble Live Now, Eastern Airways Share Price, West Air Sweden Fleet, English Channel Swimmer, Chi Health Financial Assistance, Ravichandran Ashwin Ipl 2020 Which Team, Micro Roni Gen 4 Vs 4x, Waterman Pen Repair Center, Zillow South Africa, Nellie Melba Siblings,


text:
#### area problem (9.11.98)

##### Powerhouse Museum

How do I calculate the area under a curve in Maple when I haven’t got a formula but experimental data?

Please send response to newsgroup or directly to myself: michaelvdk@hotmail.nospam.com (omit the nospam for the true address).

##### John Little (11.11.98)

There are any number of standard techniques from numerical analysis for this kind of calculation – they are called numerical quadrature rules. A simple, commonly used example of these methods is Simpson’s Rule. If you have an odd number n = 2k+1 of data points

          (x_1,f(x_1)), (x_2,f(x_2)), ... , (x_n, f(x_n))
with x_0 < x_1 < ... < x_n and with x_i - x_{i-1} = h for all i,

(i.e. the independent variable values are *equally spaced*) then the Simpson’s Rule approximation to the integral of f over the interval $$[x_1,x_n]$$ (the ”area under the curve” that you want) is

  (f(x_1) + 4f(x_2) + 2f(x_3) + 4f(x_4) + ... + 4f(x_{n-1}) + f(x_n))*h/3

(the 4’s and 2’s alternate in the summation). If you have the dependent variable values stored in a 1-dimensional array (or list) you can compute this in Maple by the following procedure (assuming $$n = 2k + 1$$ is odd):

Here are two examples:

Another example showing close agreement with the actual value of an integral – data points sampled from the sin function at $$x = .1$$ to $$.9$$ at intervals of $$.1$$:

For information on other similar methods, you can consult almost any textbook on numerical analysis. Hope this information is useful.

##### Michael Brennan (12.11.98)

All you need do is deﬁne a function, say $$f(x)$$ using the experimental data. Then you can apply either the command simpson or trapezoid from the student package to ﬁnd approximates to the area. Of course I am assuming the data you have is in the form of a table where the readings are taken repeatedly over a period of time.##e.g.

Suppose you are trying to ﬁnd the area of a swimming pool where you are given the following measurements for the width of the pool, where the readings are taken every two metres.

Area of a Swimming Pool. First enter the data as a list.

We would require about 85 m^2 of material to cover pool.

##### Helmut Kahovec (13.11.98)

If you have a mathematical model then ﬁt it to the data. If the regression is linear then see the online help page for stats[fit,leastsquare]. If the regression is nonlinear then see the article

     Fit for Anything?
The Maple Reporter, Summer 1998 Edition
pp. 6-7

You may download The Maple Reporter from http://www.maplesoft.com. Afterwards transform the ﬁt into a procedure using unapply() and (numerically) integrate this procedure using evalf/Int.

If you don’t have a mathematical model then I would suggest using splines to convert the data points into a (piecewise deﬁned) procedure. See the online help page for spline(). If you have many points then let the degree of the segment polynomials be 1 (i.e., use lines), otherwise use cubic splines. However, if there are 30+ data points, the calculation of the splines may take some time, anyway. Afterwards (numerically) integrate the piecewise deﬁned procedure within each interval separately using evalf/Int and add these areas using add(). Below I give you two examples of this.

Second example:

If you don’t need the splines explicitly you may deﬁne a procedure that calculates the area from this list of data points at once:

##### Willard, Daniel (16.11.98)

Elementary calculus texts deﬁne integrals as the limit of a process of summing adjacent rectangles whose heights are the location of the left-hand ordinate (a data point), or the right-hand ordinate, or some average of the two. Simpson’s Rule. Trapezoidal interpolation.

Splines (cubic ﬁts to successive sets of data points). See the math section of the CRC Handbook of Physics and Chemistry, or Abramowitz and Stegun Handbook of Mathematical Functions.


text:
# 2.7 Linear inequalities and absolute value inequalities  (Page 4/11)

 Page 4 / 11
$\begin{array}{c}-200\le x-600\le 200\\ -200+600\le x-600+600\le 200+600\\ 400\le x\le 800\end{array}$

This means our returns would be between $400 and$800.

To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

## Absolute value inequalities

For an algebraic expression X, and $\text{\hspace{0.17em}}k>0,$ an absolute value inequality is an inequality of the form

These statements also apply to $\text{\hspace{0.17em}}|X|\le k\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}|X|\ge k.$

## Determining a number within a prescribed distance

Describe all values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within a distance of 4 from the number 5.

We want the distance between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and 5 to be less than or equal to 4. We can draw a number line, such as in [link] , to represent the condition to be satisfied.

The distance from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to 5 can be represented using an absolute value symbol, $\text{\hspace{0.17em}}|x-5|.\text{\hspace{0.17em}}$ Write the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that satisfy the condition as an absolute value inequality.

$|x-5|\le 4$

We need to write two inequalities as there are always two solutions to an absolute value equation.

$\begin{array}{lll}x-5\le 4\hfill & \phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\hfill & x-5\ge -4\hfill \\ \phantom{\rule{1.8em}{0ex}}x\le 9\hfill & \hfill & \phantom{\rule{1.8em}{0ex}}x\ge 1\hfill \end{array}$

If the solution set is $\text{\hspace{0.17em}}x\le 9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\ge 1,$ then the solution set is an interval including all real numbers between and including 1 and 9.

So $\text{\hspace{0.17em}}|x-5|\le 4\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}\left[1,9\right]\text{\hspace{0.17em}}$ in interval notation.

Describe all x- values within a distance of 3 from the number 2.

$|x-2|\le 3$

## Solving an absolute value inequality

Solve $|x-1|\le 3$ .

$\begin{array}{l}|x-1|\le 3\hfill \\ \hfill \\ -3\le x-1\le 3\hfill \\ \hfill \\ -2\le x\le 4\hfill \\ \hfill \\ \left[-2,4\right]\hfill \end{array}$

## Using a graphical approach to solve absolute value inequalities

Given the equation $y=-\frac{1}{2}|4x-5|+3,$ determine the x -values for which the y -values are negative.

We are trying to determine where $\text{\hspace{0.17em}}y<0,$ which is when $\text{\hspace{0.17em}}-\frac{1}{2}|4x-5|+3<0.\text{\hspace{0.17em}}$ We begin by isolating the absolute value.

Next, we solve for the equality $|4x-5|=6.$

$\begin{array}{lll}4x-5=6\hfill & \hfill & 4x-5=-6\hfill \\ \phantom{\rule{1.9em}{0ex}}4x=11\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}\hfill & \phantom{\rule{1.9em}{0ex}}4x=-1\hfill \\ \phantom{\rule{2em}{0ex}}x=\frac{11}{4}\hfill & \hfill & \phantom{\rule{2em}{0ex}}x=-\frac{1}{4}\hfill \end{array}$

Now, we can examine the graph to observe where the y- values are negative. We observe where the branches are below the x- axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $\text{\hspace{0.17em}}x=-\frac{1}{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{11}{4},$ and that the graph opens downward. See [link] .

Solve $\text{\hspace{0.17em}}-2|k-4|\le -6.$

$k\le 1\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}k\ge 7;$ in interval notation, this would be $\text{\hspace{0.17em}}\left(-\infty ,1\right]\cup \left[7,\infty \right).$

Access these online resources for additional instruction and practice with linear inequalities and absolute value inequalities.

## Key concepts

• Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See [link] and [link] .
• Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See [link] , [link] , [link] , and [link] .
• Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See [link] and [link] .
• Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See [link] and [link] .
• Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See [link] .

#### Questions & Answers

factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By Danielrosenberger By By Eric Crawford By Mistry Bhavesh By By Anindyo Mukhopadhyay


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Non-existence of rational arc length parameterizations for curves in $\Bbb R^n$. (English) Zbl 1166.53002
Summary: We show, as a generalization of prior results for $\Bbb R^2$ and $\Bbb R^3$, that for all $n\geq 2$ the only curves in $\Bbb R^n$ with rational arc length parameterizations are straight lines.

##### MSC:
 53A04 Curves in Euclidean space
Full Text:
##### References:
 [1] Farouki, R. T.: Pythagorean-hodograph curves: algebra and geometry inseparable, Geometry and computing 1 (2008) · Zbl 1144.51004 [2] Farouki, R. T.; Sakkalis, T.: Real rational curves are not unit speed, Comput. aided geom. Design 8, 151-157 (1991) · Zbl 0746.41019 · doi:10.1016/0167-8396(91)90040-I [3] Farouki, R. T.; Sakkalis, T.: Rational space curves are not unit speed, Comput. aided geom. Design 24, 238-240 (2007) · Zbl 1171.65331 · doi:10.1016/j.cagd.2007.01.004 [4] Henrici, P.: Applied and computational complex analysis, vol. 1, (1974) · Zbl 0313.30001 [5] Jacobson, N.: Basic algebra I, (1985) · Zbl 0557.16001