Substitute Form W 9 This Must Be Pdf: Fill & Download for Free

Pre Script: I now have a blog that may be better organized than this:Understanding Bernoulli's Principle.…..Bernoulli's principle is not flawed. The way it is interpreted by many amateurs is flawed. Don't take just my word, ref below.Because of this, it is applied in a flawed manner to explain the lift of an airplane wing. In fact, the manner in which it is commonly used to explain the lift of an airplane wing violates Newton's Laws.Bernoulli's principle, when correctly stated, only says that as a fluid flows along a single path, the velocity and static, or internal pressure are inversely related. What it does NOT do as a assign a cause and effect.It comes directly from Bernoulli's Equation which describes pressures and velocities in the flow inside a pipe. As the flow moves along the pipe, the velocity changes and the pressure changes as the result of the change in diameter.Given the pressure and velocity at one location in the pipe, we can use Bernoulli's Equation to CALCULATE one of those quantities given the other one and another location in the pipe. If carefully applied, can be applied along streamlines when there is no pipe.I must stress that being able to calculate quantities of a natural phenomenon does not necessarily explain how or why the phenomenon occurs. Mathematics is a model that we humans find which mimics the natural phenomenon.What Bernoulli's Principle actually says, as it is most correctly stated, is that an increase in speed (acceleration), along a path is accompanied by a reduction in pressure. Many people assume that this tells us that the higher speed of the fluid creates the lower pressure, which is a false assumption.What this should tell us is that the most common statement that says “fast air has a lower pressure” is false. I call this the “Bad Bernoulli's Principle” and this fast air thing is not a new fundamental law of physics that must be memorized.EDIT: A stream of air from a blower, directed into otherwise still air has the very same pressure as the ambient air around it. This is easily measured in the same way aero engineers measure it as shown in Figure 3 of the following Reference.Please compare Figs 3 & 9Weltner in PDF - "Misinterpretations of Bernoulli's Law":http://user.uni-frankfurt.de/~weltner/Misinterpretations%20of%20Bernoullis%20Law%202011%20internet.pdf…As I explain next, we will see that there is a difference in pressure providing the force that speeds up the fluid.The real science is best understood by keeping Newton's Laws in mind.We know that a fluid has mass and that it takes a force to accelerate a mass. Fluids, such as water and air, have mass and, therefore, there must be a force that changes their velocity. This Force comes in the form of a pressure difference between two locations within the fluid. Whenever there is a difference in pressure between two locations in a fluid, the fluid in between those two locations will be pushed or accelerated by the high pressure towards the lower pressure region. That is why it is moving faster when it gets to the lower pressure region. It is not simply the speed that is the focus here, but it is the change in speed, which is an acceleration.When this “Bad Bernoulli's principle” is applied incorrectly to explain lift, it is assumed that there is some magical reason, usually incorrectly compared to some other concept, which is used to accelerate the air. Then, noting that the air has sped up, it is assumed that this creates the lower pressure.In reality, the pressure is changed by the relative motion of wing in air. It is these pressure differences that are created around the wing which constitute all of the lift. Then, it can be very easily shown that these very same pressure differences cause all, repeat all accelerations which means movement of the air in the vicinity of the wing.The bottom line is that using Bernoulli's principle, even correctly, to explain lift is a poor substitute for the real science. It doesn't correctly explain why the pressure changes occur, but it can be used to calculate values..…I also point out, for those of you still reading, that the common demonstrations of blowing between two balloons or two soda cans, or blowing over the top of a curved piece of paper falsely interpret what is happening. The fact that the Coanda effect is causing a pressure reduction is not understood and, therefore, provides a false interpretation of the demonstration. Another fact which can be very easily measured by anyone who goes to the trouble is that a fast stream of air coming out of a hair dryer leaf blower or soda straw has the same pressure as the surrounding , stationary atmosphere.I also point out that this Bad Bernoulli Principle has been taught for so long to so many people, that amateurs will fight to the death that it is true because some teacher or Professor told them.… EDIT:The Bernoulli Principle lift- myth goes way back... The Cyber History of Bernoulli’s Principle:Thomas M. Keating, James G. MaKinster, Jonathon W. Mills, & Jeffrey A. Nowak February 1, 1999Indiana Universityhttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.88.6970&rep=rep1&type=pdf

YesThe.Whole.Bloody.Grapheme.System.Needs.A.Bloody.ReformIn Hindi too,“box” is baksa,we did not make a whole new alphabet for it.Why did the Romans do it,I ain’t got an explanation.Okay John Katt,I’m copying pasting your most famous answer here.the English spelling system can and should be reformed because it is flawed (half its lexicon is) and what is flawed is very costly and/or takes/wastes a lot more time/money, which takes a lot of time and effort, away from other more crucial subjects. Who pays? Children, parents, teachers, and society do. We all do. You do. To preserve the history of words? 250 years of insanity must stop. Oil lamps had their time. Paradigms have shifted in 250 years. A lingua franca must be better. Is it the spelling system that is rotten or the learners? The teachers? Hundreds of thousands of words in the English lexicon ARE badly spelled. It is not the learners’ fault or the teachers’ fault. It is the system’s faultS.Words hurt and in more ways than … one! Where is the /w/ sound in one? Why an “o” in “words? Hundreds of thousands of English words are misspelled like so in the lexicon! Is the history of words more important than the health of people?A UNIQUE, FAIR, WISE, and EFFICIENT REFORMLearning from research and past reforms and using new technology, this reform de-risks everything, maximizes the opportunity and minimizes ALL issues. First, current users will NOT need to learn a new spelling system because it will be phased-in for 12 years in schools, starting at age 6. (This is unlike many other spelling reforms. That is a game-changer.) Second, it uses a set of diaphonemes (the most popular phonemes of most English dialects) to prevent all polemics. (No one has ever thought about this before.) As a result, it allows for the adoption of an extremely robust, systematic, and phonemic system that will be the envy of all languages. It is a unique, fair, wise, and efficient proposal. MANY paradigms have shifted in 250 years. The English spelling system can and should be re-engineered. English will be finally a worthy lingua franca.MAIN SINE QUA NON CONDITIONS for a reformThat no current users be required to learn the new spellings. It will not be necessary.That the new system be introduced in whole to new students in level 1 of primary classes called cohort 1 (C1) and phased-in, one year at a time.That C1 students and all future cohorts be given bilingual, bicodal courses in the old system. They become a transitional cohort.That other (English 1.0) students get some instruction with the new system, but increasingly so for cohorts that are closer to C1 .That diaphonemes (average of phonemic variations of main dialects) be used. It is an algorithm. It cannot be fairer.That an extremely systematic and phonemic scheme with virtually no exceptions be used. No compromise.That local, dialectal spoken/speech be maintained. C1 will be bicodal.That computer technology be used to instantly transcode material. Writers will see their work published in two varieties or as desired by the customer using a free transcoding program.That no loss of jobs take place. Translators/interpreters will still be needed for the current population and C1.This is the short version. Please do read the full version below. It not only gives the reasons why this should happen, but explains how it should happen (and why this way) in detail. All reasons I know of to not do it are debunked. I have been studying this issue for 5 years. I have heard them all.Why and how?The “how” part offers credible solutions on how to reduce dramatically all of the incredible high costs and/or problems children. parents, and children must face and all of the arguments used by all the skeptics or vested parties to prevent a reform. ALL! For one thing, paradigms have shifted since Samuel Johnson (250 years ago) “dictated” how he thought we (kids and adults) should decode/read, spell/pronounce words. Since Johnson’ time and since some of these arguments were made, a lot has changed. Odd! For another, some reformists expected that current populations change how they spell words and learn how to read in a new way. This was a major error. This issue demanded a new approach that sought outside-of-the-box solutions. There is nothing more idiotic and insane to keep doing something that has failed over and over again and expect different outcomes. Times have changed. Lastly, English has become a lingua franca, spoken by 1.5 billion of people, in some ways or another. This is the short answer. LOL(I would like to recognize and thank Dana Smith and Roman Huczok for their excellent comments and suggestions, which helped make this answer better.)WHY?Consider that as the most used language in the world (spoken by billions of people), it has a spelling system that is an absolute embarrassment for a language and, a fortiori, for a lingua franca, a world language, a language that should be the epitome of all languages. It isn’t. It has hundreds of thousands of words misspelled in its lexicon (see below for evidence), which might explain why you must use a spell checker or can’t pronounce or could not decode words when you learned it, if you learned it, depending on your language (see the end for explanation). With the rapid use of computers and spell checkers, it is my contention that the issue is no more a spelling issue, as it is a decoding/reading and pronouncing issue. This does not take a way from the idea at all though because the spelling issue was never considered by many to be the major issue.Did you know that it takes 3 more years for the average learner to learn just a few thousands words whereas any Finnish or Spanish kid after Grade 1 can decode ANY word in their language’s lexicon in just a few months? (Maybe I am exaggerating a bit, but the differences are startling.) Think teaching the extra time is free? Sorry taxpayers! Think it is easy to learn? Kids labelled reading disabled thank you and so 1/2 of Britons who apparently cannot spell. Illiteracy rates are unusually high in Commonwealth countries (unless you throw lots of money at mitigating the problem like in Canada). But, we are told that it would be impossible to fix (Did they not say this for climate change or women equality,…), but many professors of linguistics like Dr. Yule and Dr. Betts now and Shaw, Roosevelt, Carnegie, Websters, Twain,… think that when there is a will there is a way. Are it tutoring agencies, publishes of English learning material, even teachers, psycholinguists who are claiming it cannot be done? The status quo is lucrative. Who is right? Here is the case.Exhibit A:This is how it could look like.Exhibit B:Now compare that with something that is familiar.Exhibit C:Exhibit D:Which is a mess to deal with? And which is an intelligent, engineered system?ANSWERd bExhibit E:10.1.1.92.1830.pdfBut, why it should be done and how could it be done?1. WHY it should be done?Since Finnish kids start school at age 7 and most English-speaking kids at age 5.5. Given that Finnish is a highly phonemic spelling system, it would allow Commonwealth countries to reduce educational budgets by removing the need for many teachers and many literacy teachers as well. This could be billions in the US. Alternatively, countries could replace these years by the teaching of more crucial matters and include more subject areas in the curriculum. Ethics 101? Financial literacy 102? Programming 103?Billions of foreign learners will finally be able to pronounce words as they are pronounced.There will be less of a chance for Chinese to dethrone English as a lingua franca if English were easier to learn.A country with a better educated population with higher literacy levels might have lower crime rates, but higher employment.More happier children will want to read.Some linguists claim that it is not really worth it. But, their jobs depends largely on this mess (especially psycholinguists). Tutors/tutoring agencies depend on the mess too. Many benefit from the chaos. They get grant money to study our kids who become their little rats. Are they really disabled or is it the system that is? Which is it?Exhibit F:(http://www.elemedu.upatras.gr/en...)So, is it worth it when most kids in English countries can decode words at a 35% level and most others do TWICE as well? This study was done with normal kids. This was a major study too. Are English-speaking kids inherently dumber than all the other kids? Are English-speaking teachers inherently more incompetent than all others? Or is it that the spelling system is … inherently dumber, disabled, disabling,… that if it were a car, it would be recalled in a second? You bet.Exhibit G:If we use Masha Bell’s research on just 7000 words and extrapolate this to the whole English lexicon, it is safe to say that there are hundreds of thousands of words that are “misspelled”. /ə/ AKA a schwa is one of the most frequent phonemes/sounds in English and it can be spelled 13 ways. How do you expect people to learn this? Memorize the pronunciation or the spelling of 1 million words? Most polysyllabic words (mostly not part of the 7000 words) have at least one schwa and often many.Exhibit H:Here is that diagram again.It shows that there are 13 ways of spelling ONE phoneme (the schwa), 12 ways to spell the /ei/ sound or 11 ways the /ɛ/ sound. Hey! I see a pattern! THAT is easy to remember! (Find the list of spellings in the addendum.) Hundred of thousands of misspelled words! NOT! How insane is that? No wonder the native speakers cannot spell/decode words and foreign learners cannot pronounce words (and spell them, if they do not speak a European language). As to teachers, they do the best they can with a messy tool. That it costs a lot of money for taxpayers and parents does not seem to be an issue, but many people don’t know how expensive it is.You want more evidence?Exhibit I:http://files.eric.ed.gov/fulltex...(This booklet focuses on the reading literacy test scores of students in the grade levels where most 9- and 14-year-olds were to te found in 32 national systems of education. Data were collected from 9,073 schools, 10,518 teachers, and 210,059 students. In 1990-1991 thirty-two school systems were involved in the LEA Reading Literacy Study. Participating in the study were: Belgium (French) Netherlands Greece New Zealand Spain s Sweden Nigeria , Norvay Switzerland Thailand Demnark Iceland Philippines Trinidad & Tobago Finland Indonesia Portugal United States Ireland Singapore Venezuela)There is more. From Wikipedia. Exhibit J:“Alphabetic writing systems vary significantly in the depth of their orthography. Englishand French are considered deep orthographies in comparison to Spanish and Italian which are shallow orthographies. A deep orthography like English has letters or letter combinations that do not reliably map to specific phonemes/sound units, and so are ambiguous in terms of the sounds that they represent whereas a transparent or shallow orthography has symbols that (more) uniquely map to sounds, ideally in a one-to-one correspondence or at least with limited or clearly signified (as with accent marks or other distinguishing features) variation. Literacy studies have shown that even for children without reading difficulties like dyslexia, a more transparent orthography is learned more quickly and more easily; this is true across language systems (syllabic, alphabetic, and logographic), and between shallow and deep alphabetic languages. [20] WikipediaThis person’s explains it well:Exhibit K:From The children of the code website.Exhibit LSorry, Chomsky: English spelling is hardly “close to optimal”I rest my case.2) How?Many languages have had reforms (check it out) and many were successful. The better ones did not expect current learners to learn the new system. Moreover, today, we have computers, smart phones, AI,… the paradigm has shifted. Are the naysayers living in the Dark Ages? Why are they so reticent? No one who loves to read this will be bothered. Why would they want to subject kids to mental torture because the MAJORITY of kids struggle, the MAJORITY of citizens struggle, the MAJORITY of foreigners struggle. While the naysayers love exceptions, the evidence is there. No matter how you look at it, it is a mess. (This neurological research might explain why some people are reluctant. It is not so much the people as it is how some brain works! To be sure, people whose identity is shaped because they know English and can earn a living might have an even stronger reaction, but I will prove that there is nothing to be feared about. Change can be a win-win proposition if done with respect and intelligence.)But, how could we do it? There have been a few attempts to try to reform the English spelling system. The most serious one came to an abrupt end 100 years ago. President Roosevelt had initiated. His friend, Carnegie, who had been delegated the task of looking into it, believed that we should not force people to spell and read differently as his board had decided. He preferred a more informal and timid reform where people could decide to adopt the changes or not. (Simplified Spelling Board - Wikipedia) Purists, vested interest groups (publishers), and political games did the rest. (German and French reforms of recent times have suffered similar fates. By trying to appeal to the general population, by making minimal changes, they opened themselves to criticisms for being superfluous or a lot of trouble for nothing.) The main issue, however, is the idea of forcing people to adopt the changes or hoping that people will do. All of this would make a holder of an MBA laugh as both approaches are ineffective. In the case of English, there might be a cultural aspect to this as well. The British culture is know for their ability to manage adversity by keeping a stiff upper lip and avoiding reality by going to bars, as this question demonstrated: What are some dark sides of the UK? Not sure if having to learn to spell and read such a complex system is one of the reasons for these types of behaviours, but it might make them less interested in changing matters.They are proud of their culture and their language, like most people are. That is to be admired, of course. But, this is not about culture. It is not about language per se. It is about a spelling system. Times have changed. Times are changing. There is Quora and we can discuss things. Today, we also know more about management of changes. People are texting now. Computers were introduced 30 years ago and everyone seems to have a smartphone in their hands now. One billion Chinese are soon to be a force to be reckoned with. Which international language are they going to speak? Chinese? Many paradigms have shifted. Not sure what is going to be the tipping point. But, things have evolved.One of the ideas (which is closer to Carnegie's thinking) is that we should NOT try to “force” people who know the current system to learn the new one. However, we have pushed that idea to its extreme. It is our contention that no one, unless they wanted to, should learn to spell using the new system unless they desire it. It is our contention that the key to making a reform work should be to introduce it methodically and slowly in schools first and only in schools. That does not mean that we would introduce bits of the new spelling system to all grades. It will be ALL of the new spelling system starting with the Grade 1 kids, as a wave. Of course, this plan would need to be approved by the government and the people. There will be a congress next year during which a group of linguists and professors at the English Spelling Society will decide which is the system(s) that they recommend. I believe we should use a system that is based on a general dialect that has some, but not all the features of any of the dialects. I am talking about the vowel diaphonemes found on this page. I am including a reformatted sample to give an idea. The multiple dialects that the original chart has have been removed because the idea is that the diaphonemes will be used as the pronunciation guide to be used with whatever spelling system will be chosen. So, in Iezy Ignglish, in all Commonwealth countries “trap” words will now be pronounced with the /ae/ and spelled with an “a” as “trap” and “bath”/”father” words (if the /a:/ is used) will not be spelled baeth/faether (faedher)), and so on and so forth. Btw, this is one diaphoneme that would require an agreement on as it has 2 possible phonemic representations. All others are straightforward. The match with the new spelling system (whatever it is) will be 1:1.Beyond that, it is our view that a reform should take place in all schools once teachers have been trained. It should be starting with a group that has not learned to read and write: 6 year old kids. The rest of the school children would be taught the old system. It might be wise to start teaching these children bits of the new system. Again, the government will look at the recommendations and decide what they feel is best. The next year, the second cohort of new grade 1 kids would start school learning the new system while the older Grade 1 would move into Grade 2, continuing to learn to read and write using the new system. Tablets will be given/lend to all students (school and home) to access information from the internet or other sources, except that this information would be instantly transcoded when they need it, like it happens with Google translate. I think that by the time this happens, most tablets will be very inexpensive for schools so that all students will have one. (Dana Smith felt free tablets should be given to needy families, but I think these should be loaned like textbooks are/were.) They will be like those textbooks that were given to us at the beginning of the year. Btw, transcoding is much faster and more accurate than translating. Eventually, after a few years, some of these cohorts will be taught the basics of English 2.0: how to read street signs, store signs,… They would not learn how to spell using English 1.0, but they will learn to decode a basic set of words and, especially towards the end of their schooling, how to read English 1.0 words of their trade. There might be a need for them to have a slightly different accent depending on how standardized English 2.0 will be, which might depend on which countries decide to participate in the reform. This reform will take 12 years to works its way out, but it will take years to make it occur. Convincing the population will take years and then politicians more years. But, if and when it is approved, the 12 years will give even more time for society to get ready. Free transcoders (programs that can transcode between English 1.0 and English 2.0) will be available for all. This will be very simple to do. In fact, some reformists have made some of them. When these cohorts exit the school system, they will try to find work like all students or they will go to university. Books and manuals should be available in both codes. This should not be so hard for publishing companies and digital copies of these should available for download into tablets. We would hope that by that time students’ books would all be the digital type. Again, these are recommendations.Will a reform be perfect? Is anything perfect? Is the English spelling system now perfect? Why are those lovers of perfection in love with imperfection? There are thousands of imperfections now and they are fine with those? Be coherent! The system will no doubt be much easier to learn and to teach. That is self-evident, as demonstrated earlier. Something is simpler to learn than something that is complex.Will some dialects need to re-align some of the pronunciation of a few words? Probably unless they want to develop their own English 2.0 or unless their kids can be bilingual in both Singapore English which pronounces some words that have a short /a/ as /E/ and the new English 2.0 (an international version). I believe kids have that capacity. Up to them to pronounce it the way they want. We would suggest they teach in school the right way and enforce the change in the media. When there is 250 years of laissez-faire, it is necessary to take extraordinary measures, but it will be up to Singapore and other dialects to align themselves with the diaphonemic spelling that is as impartial as one can make it. Accents will be preserved in many instances though, but they will less … pronounced in some cases. But, bad should not be spelled “bed”. Multiple spellings of words could be allowed, but whenever it is feasible. IN some sense, a re-alignment will nt be bad for those “rogue” dialectal words and speakers. It will allow to be understood by more people in the Commonwealth. Is that bed? I mean “bad?Let me address the most common objections that are often used to prevent any change.1. There are too many accents (AKA dialectal variations in pronunciation).Dialectal accents are started to be “learned” or “perceived” by the age of 2, BEFORE children can link phonemes, allophones, with any spelling, phonemic or not. Here is the research.We know that children (their brain, really) have the capacity to learn many languages, many accents. In Italy, for instance, it is common to hear people know a dialect (usually oral) and speak/read/write the standard Italian as well. We suggest that the only reasonable way to deal with this issue is to make all Commonwealth children start to learn another standard dialect by Grade 1 which will —finally— be the lingua franca that all people around the world have long been awaiting for. (Thanks Roman Huczok for reminding me that keeping a dialect and learning a standard is very doable.)To avoid political issues and help make English a true lingua franca, it would be wise to use the diaphonemes used on the International Phonetic Alphabet chart for English dialects - Wikipedia or some other agreed form. If some populations of certain countries or region were not interested in this new standard, they would have the option of staying with the current system or reform their dialect as they please.This would not be the Armageddon, the end of English as we know it, an incredible loss of culture,… This is about spelling, not language.The internet, public education for all, social media,... are helping standardizing many accents and, if it were to be reformed in this manner, it will be much easier.2. I do not want to learn a new system.You won't have to. I repeat … you will not have to. That is our pledge. I do not want to either. This reform is not for me, you, but for the next generation.The change will occur in schools, starting with as many Grade 1 classes as it is possible. Opting out will be possible. In year 2, another group of Grade 1 will start to learn the new system. The first group will go in Grade 2 and will keep learning the new system (or rather learn using the new system since they will have it mastered decoding and spelling already).3. There will be a need for some people to learn the new system.The 20 to 40 will need to be familiar with the new system, but free programs will be able to transcode from the current system to the other and vice versa, seamlessly and fast. Transcoding is much faster than translating. It is also much more accurate.The cohort that will go into the labour force after 12 to 16 years will speak the same language. Speech recognition software and transcoding programs will do the rest.4. Street signs and vendor signs will need to be respelled/respelt.No. The new spellers will be able to decipher the old system.5. ALL documents will need to be reprinted.No. Digital documents will be transcodable. It is much easier to do so.Should a citizen be interested or be in need to read printed documents that are not in a digital format, I am sure we can figure out ways to efficiently recode these (text-to-speech recognition software to deal with that issue) or have someone read the text to him or her or transcode it.6. Will translators lose their job?No. A good segment of the population will still function in the current system.No. The new spellers will need translation as much as the older generation.There will be a need for some transcoding too.7. Will teachers lose their job?If a Grade 1 teacher were incapable or unwilling to teach the new system, they could be given the task to teach those children who are opting out or be asked to teach the old system (as a second language) or teach older grades. Substantial accommodations should be given to older teachers wanting to plan (prepare material) and/or learn the new code.There will be a 4 or 5 year preparatory period to start the transition (Year 1/Grade 1) which should give people plenty of time to shift, should they want to.Unions will be consulted and a system will be put in place to facilitate the transition for allRetirement by attrition would be one of the ways used to replace teachers.Grade 1 teachers are often able to teach other grades.New students will need a few teachers to teach the old system as a second-language mode.8. The language will lose the morphological links between words that will be lost or reduced with a new more phonemic system.Everyone knows the link between language and linguistics or photography and photographer, for instance. These pairs of words resemble each other, but the link is not automatic in the first pair. A more phonemic system will sometimes improve the semantic relation and sometimes obscure it. At the end of the day, some of the words that are linked by how they look, require the learner to remember the pronunciation of the words since they might not be pronounced as they are written and, obviously, their spelling: photographic, but photography: (/fəˈtɒɡ.rə.fi/ VS /ˌfəʊ.təˈɡræf.ɪk/. Which is better? In a reform spelling, these words would be spelled something like this in Iezy Inglgish: fetogrefy VS fotegrafic. Notice that in both, the stressed syllable is the one that does not have the “e” or schwa. Huge advantage for foreign learners where now no one knows where the word stress is put. Is there anyone who canNOT link the two words semantically? A newer system will improve the link between words that are spoken and words that are written/decoded/read. Learning should be faster as a result. The current system obscures the link between words that are spoken (and heard) and words that are written/decoded.Furthermore, yes, there are words that look like they are related and the link will be obscured, but if spelling and misspellings are so important aren’t they a lot of false-positives that a respelling would clarify? Is ready about reading? Arch and archive? Apathy is about taken a path? Ballet is a small ball? Country is about counting? Breakfast is the action of breaking fast? Colonel is a special colon? Lead (the metal) is about leading? Bus and business? Deteriorate and deter are related? (They are not.) Cancel is about cans and cells? Gig (the performance) and gigantic are related ? Have and haven are related? Ache and achieved? Reinvent and rein (vent)? All and allow? Inventory and invent are linked? Reached and ache? Resent is about sent/sending? Is a beldam a belle dame (It is an ugly woman)? Is noisome about noise (It is not!) Is nowhere about now and here? How many more do I need to prove the point that there are a lot of false positives currently?There are words in the current system that appear to be linked, but aren’t. No one seems to be confused. Invest is about a vest that’s in a coat? Numb and numbers are related? Legal is about leg? Assertive about ass? Acting and actual are related? Deli and deliver? Heaven and heavy? Man and many? Add and address? Earl and early? Pet and petty? How about bigot? Is Nonplussed about being not plus (It means confused!)? Is Disabuse about abuse (it is not!)? Is specious about species or special? Crudités are crude? Terrific and terrible mean the complete opposite. How about restive and restful? How about condone and condemn? Disinterested does not mean not being interested! Is humility and humiliation connected semantically? NOPE! Is sublime less than good like subway, substitute, subtract? Prodigy and prodigal are not related? Awesome and aweful are opposite. Is being “broke” about being broken? Dulcet is about being sweet, not dull. Is ribald about badness? It doesn’t! There are lots of false positives in that sense in the lexicon too.9. Is it worth it?Suppose we make English as regular as Finnish. Now consider, Finnish kids start school at age 7. Most English-speaking kids start at school at age 5.5. How much does it cost to teach all of those kids for an extra 1.5 years. Teachers are expensive. Daycare? Less so. Imagine the possibilities. Also, there is quite a bit of data that indicates that maybe kids do not need to go to go to school at age 5.5. Again, daycare or universal childcare could make the life of millions, dare I say billions of learners, that much better. THAT is not worth it? What is?Illiteracy rates in the 30% levels in most Commonwealth countries will drop with a simpler system.A simpler system will be MUCH cheaper to teach (fewer specialist teachers will be needed).Learning will happen faster. As students HEAR a new word, they will be able to link it to its ONE possible spelling and when they read a new word, they will be able to link it to a word that they heard. The brain connections will be reinforced more efficiently. Lets take a word that you have never seen printed before: “tuleafashouhe”? Are you sure of it pronunciation? Where is that word stress? And then, a few weeks later, you hear on TV “tlayfaychor”? Would you be able to connect the 2? Most likely not, but if it had been spelled as it is pronounced, then the connections would have been made, with more certainty. It is self-evident that more coherence between systems would make learning faster and easier.Fewer kids will be pulled out and shamed as reading disabled.Less crime as more people will be able to read and write. (Robots will do the menial work that illiterate people sometimes must do).Happier labour force.Better educated/literate labour force.Better economy.More people around the world should be able to learn an easier system.Easier travelling and understanding between people.More people will be able to read books written in the new code. Higher profits for English-speakers.10. Which industry will lose?Tutoring agencies and tutors could lose out. Still, we could make the first generation that will learn the new code, bicodal. If this is so, they will surely need help to learn the old code, just like pasts generatiosn did.We need to make this a win-win situation. Anyone displaced will be given a choice of work that is related to what they were doing beforeTeachers (attrition and re-assignment will need to be addressed), but those who cannot cope will be re-assigned.Publishing houses will benefit. Some of the old material will need to be digitized, but a lot has been (Gutenberg project, Google,…)Psychologists who assess students’ reading and writing abilities/intelligence will lose out, but I suspect that this is a small number, seeing how many of these evaluations took place in my 25 years of teaching.11. What do these new spelling systems look like?Some are using most of the spelling rules that exist now. They are just regularizing many of the patterns. (Masha Bell has one system.)A reform would not mean spelling using a phonetic system like IPA. There is no cursive writing (although this could be created I suppose). Cursive writing is faster than printing words, but aren’t more and more people going to use technology to avoid writing all together? Even in rare instances where people are asked to cursive write, a recorder with speech recognition software could do the work of transcribing much more efficiently than any one could, even with short-hand.Other systems attempt to maximize the opportunity as taking a second shot at this will prove unlikely. Iezy Ignglish is such a system. It systematizes the easiest pattern of English: the vowel+e pattern found in many words (piece, clue, foe, reggae,…) and it echoes the long vowel+Consonant+e pattern found in a lot more words, which is more contrived than the first pattern and which makes decoding a much harder tasks than it should (late, cute, core, mite, mere). The simpler pattern would do away with the cumbersome doubling of the consonant rule to change the vowel value: pat/patting, mat/matting VS mate/mating/.Others can be found on the English spelling society website.12. Will communication between the ones who know the new system and the ones that don’t be affected?The language/speech/conversations will be the same.The only communication mode that will be affected is the written mode, but is there anyone who thinks that most people will not have smart phones or tablets or computers to allow this?The internet will need transcoding work, but programs can easily be created I am told by programmers. These programs will be able to transcode tons of material and will do it faster that any translation program (and much better).13. There will be many homophones/heterographs. Will they not make communication harder? (Thanks to Tomas Murphy for that one.)You/ewe, read/reed, ad/add,…No one when speaking and listening is confused. There are very few instances where this is a problem in real life. Many cannot be confused as many are not even the same type of words: check (verb)/cheque (noun), ad/add, it’s/its, their/there/they’re,… Let’s respell them. tchèc, ad, its, dhèr: I went to cash the tchèc. I tchèc the tchèc.Homonyms: bark (dog) and bark (tree). Many words have multiple meaning and one spelling. Is there anyone who is confused.The great majority of these words are disambiguated by the types of words they are. You is a pronoun and ewe is a noun. Read is a verb and reed is a noun. Ad is a noun and add is a verb. The linguistic context helps clarify matters already.The great majority of these words are disambiguated by the context.Hundreds of thousands of misspelling are okay, but 500 homophones will cause issues?There are just 500 homophones and there is close to 1/2 a million words misspelled. We cannot change the spelling of words because 500 words (which upon analysis would not cause comprehension issues if they were respelled.14. Accents will vanish?For the last 250 years (and more) they have NOT vanished even with an extremely POOR system representing them.15. Language changesThe printing press, spell checking, public schooling,.. have cemented the spelling of words for centuries now and will. If one were to re-adjust spelling to be more phonemic/regular, spell checking and printing will operate in the same manner, cementing the new spelling of words. I doubt that vowel shifts will occur again.With a more phonemic system, there will be fewer chances of deviations. Deviations occurred mainly because few people were schooled, could write, could read in the past. The system will be much more stable.16. The current spelling helps me.If you are French, Spanish, Italian (any Romance language), about 50% of words will be spelled more or less like they are in these languages and reading will be easy for these words, obviously. However, the spelling will interfere with the pronunciation of these words. Everybody knows how these speakers have a real tough time pronouncing English words. Why? Because they are not pronounced like they are in these languages. Why? Because there is very little guidance in the spelling to indicate how these words are pronounced. Again, there are hundreds of thousands of words whose pronunciation needs to be memorized.If you are Greek, there are only 6% of the time where spelling will be easier for you, Pronunciation again is going to be a huge problem for hundreds of thousands of words as it with everyone.People who speaks Germanic languages are being helped with the spelling 1/4 of the times. Unless they know a Romance language, they will struggle with the pronunciation of those French words and others. The pronunciation of those Germanic-based words will be easier. They are usually shorter and do not contain the invisible “schwa” that one needs to know about for those bigger Latin/French words.But, what about those people who don;t speak any of these languages? Are Asian-speakers persona non grata? How selfish is it for European-speakers to stick it to the Asians, Arabic, African-speakers? Will they forgive you or will they ask you to learn Chinese or Arabic? What goes around, comes around. Latin used to be the lingua franca of the world and how many people speak Latin now.ADDENDUM:The schwa: about, children, pencil, renovate, supply, syringe, luscious, mission, blood, does, cousin, thorough, and especially. (Even “one” or o_e could be included as it is pronounced “wun”.) So, even 14! (I am including stressed syllables/words as schwas like in does, which some phoneticians feel are very different than the “a” in “about”. This linguistics university course 115 (Pennsylvania) seems to indicate I am right: Phonetic symbols and this linguist in speech makes a case for the schwa being just a reduced phoneme of the /ʌ/)./ei/: great, raid, grey, gray, ballet, mate, table, caffe, matine, reggae, vein, vain/ɛ/: bear, care, aerial, their, there, questionnaire, mayor, bury, any, friend, leopard/i:/: be, been, bean, key, mere, elite, people, ski, debris, quayNOTE:Differences in English acquisitionDifferences in English acquisition depends on the language you learned, you know. If you are a person who speaks a Germanic language, it will be easy for you to speak, read/read out loud, and understand English, day to day English. If you speak a Romance language like French, Italian, Spanish,… it will be relatively easy for you to read/decode/understand written English (especially if it is academic or legal), but it is the speaking and the listening parts that will be challenging, especially if it is a day to day task. English is a mix of both these families, but easier words are related to Germanic languages like Dutch, Danish,… , but more difficult words are usually French, Latin-based Italian, Spanish, … Of course, if you are a speaker of Chinese, of a language that is not related to English in any way, then it is going to be an even greater challenge, as even phonemes and of course alphabetic letters will be completely new to you.Read John Katt's answer to What are your most controversial or unpopular opinions?My blog Njú Ingliş has this article and it’s a good new grapheme system.Here are all the letters we’d use.The original letter is followed by the one with a diacritica=/ə/ againá=/aː/ fatherà=/ʌ/cutå=/ɒ/bottleæ=/æ/batb=/b/butterc=/t͡ʃ/chopperd=/d/dogð=/ð/thise=/ɛ/beté=/eɪ/Bravef=/f/phoneg=/g/godĝ=/d͡ʒ/gelh=/h/halli=/ɪ/bití=/iː/beetj=/j/youk=/k/cutl=/l/lochm=/m/mann=/n/noo=/ɔ/boyó=/oʊ/ohp=/p/Potter*q=/q/Qatarr=/ɹ/rivers=/s/saltş=/ʃ/sheltert=/t/tallþ=/θ/thornu=/ʊ/ruthlessú=/u/bootv=/v/valvew=/w/will*x=/x/loch(Scottish English)*y=/y/few(only in some accents)z=/z/zedž=/ʒ/vision*Q,Y and X are to be used only for transliterating other scripts.Só,Ái ges it wiɫ ték às sàm táim tú get júzd tú ðis þing,bàt its fòr ár ón gud!I’m also making a Greek grapheme system which will look sexier.Áim olsó méking a Grík grafím sistam wic wil luk seksiar.

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Thank you. — Jimmy Wales, Wikipedia FounderHow often would you like to donate?Just OnceGive MonthlySelect an amount (USD)$3$5$10$20$30$50$100Select a payment methodSecure transactionMaybe laterNatural numberFrom Wikipedia, the free encyclopediaThis article is about “positive integers” and “non-negative integers”. For all the numbers …, −2, −1, 0, 1, 2, …, see Integer.Natural numbers can be used for counting (one apple, two apples, three apples, …)Real numbers (ℝ) include the rational (ℚ), which include the integers (ℤ), which include the natural numbers (ℕ)In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".Some definitions, including the standard ISO 80000-2,[1]begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….[2][3][4][5]Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).[6]The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products of thereof); and so on.[7][8]These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.In common language, for example in primary school, natural numbers may be called counting numbers[9]both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.Contents1 History 1.1 Ancient roots 1.2 Modern definitions2 Notation3 Properties 3.1 Addition 3.2 Multiplication 3.3 Relationship between addition and multiplication 3.4 Order 3.5 Division 3.6 Algebraic properties satisfied by the natural numbers4 Generalizations5 Formal definitions 5.1 Peano axioms 5.2 Constructions based on set theory 5.2.1 Von Neumann construction 5.2.2 Other constructions6 See also7 Notes8 References9 External linksHistoryAncient rootsThe Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[10][11][12]is believed to have been used 20,000 years ago for natural number arithmetic.The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.[13]A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.[14]The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.[15][16]The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.[17]The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[18]Independent studies also occurred at around the same time in India, China, and Mesoamerica.[19]Modern definitionsThis section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.(October 2014)(Learn how and when to remove this template message)In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school[which?]of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".[20]In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.[21]In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[22]The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[23]With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[24]and logicians.[25]Other mathematicians also include 0[5]although many have kept the older tradition and take 1 to be the first natural number.[26]Computer scientists often start from zero when enumerating items like loop counters and string- or array- elements.[27][28]NotationThe double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).Mathematicians use N or ℕ (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.[29]This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-naught (ℵ0).[30]To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]ℕ0 = ℕ0 = {0, 1, 2, …}ℕ* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}.Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.[31]ℕ = {0, 1, 2, …}.ℤ+= {1, 2, …}.PropertiesAdditionOne can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.MultiplicationAnalogously, given that addition has been defined, a multiplication operator × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (ℕ*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.Relationship between addition and multiplicationAddition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that ℕ is not a ring; instead it is a semiring (also known as a rig).If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a.OrderIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as ω (omega).DivisionIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such thata = bq + r and r < b.The number q is called the quotient and r is called the remainder of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.Algebraic properties satisfied by the natural numbersThe addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).GeneralizationsTwo important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.Many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω.For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.Georges Reeb used to claim provocatively that The naïve integers don't fill up ℕ. Other generalizations are discussed in the article on numbers.Formal definitionsPeano axiomsMain article: Peano axiomsMany properties of the natural numbers can be derived from the Peano axioms.[32][33]Axiom One: 0 is a natural number.Axiom Two: Every natural number has a successor.Axiom Three: 0 is not the successor of any natural number.Axiom Four: If the successor of [math] x {\displaystyle x} [/math] equals the successor of [math] y {\displaystyle y} [/math], then [math] x {\displaystyle x} [/math] equals [math] y {\displaystyle y} [/math].Axiom Five (the axiom of induction): If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of [math] x {\displaystyle x} [/math] is [math] x + 1 {\displaystyle x+1} [/math]. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.Constructions based on set theoryMain article: Set-theoretic definition of natural numbersVon Neumann constructionIn the area of mathematics called set theory, a special case of the von Neumann ordinal construction[34]defines the natural numbers as follows:Set 0 = { }, the empty set,Define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.It follows that each natural number is equal to the set of all natural numbers less than it:0 = { },1 = 0 ∪ {0} = {0} = {{ }},2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},n = n−1 ∪ {n−1} = {0, 1, …, n−1} = {{ }, {{ }}, …, {{ }, {{ }}, …}}, etc.With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m.Also, with this definition, different possible interpretations of notations like ℝn (n-tuples versus mappings of n into ℝ) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.Other constructionsAlthough the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:Set 0 = { }Define S(a) = {a},It then follows that0 = { },1 = {0} = {{ }},2 = {1} = {{{ }}},n = {n−1} = {{{…}}}, etc.Each natural number is then equal to the set containing just the natural number preceding it.See alsoMathematics portalIntegerSet-theoretic definition of natural numbersPeano axiomsCanonical representation of a positive integerCountable setNumber#Classification for other number systems (rational, real, complex etc.)Notes"Standard number sets and intervals". ISO 80000-2:2009. International Organization for Standardization. p. 6.Weisstein, Eric W. "Natural Number". MathWorld."natural number", Dictionary by Merriam-Webster: America's most-trusted online dictionary, Merriam-Webster, retrieved 4 October 2014Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all natural numbers may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15)Jack G. Ganssle & Michael Barr (2003). Embedded Systems Dictionary. p. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 1578201209. integer 1. n. Any whole number.Mendelson (2008) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)Weisstein, Eric W. "Counting Number". MathWorld.Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of AstronomyGeorges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3"A history of Zero". MacTutor History of Mathematics. Retrieved 2013-01-23. … a tablet found at Kish … thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty placeMann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063.Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations", The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN 9781118853979.Michael L. Gorodetsky (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Группа Квантовых и Прецизионных Измерений. Retrieved 2012-02-13.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."Gray, Jeremy (2008), Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, p. 153Weber, Heinrich L. 1891-2. Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 2:5-23. (The quote is on p. 19.)"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)Eves 1990, Chapter 15L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0.This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. Retrieved 19 January 2015.Hui, Roger. "Is Index Origin 0 a Hindrance?". J Home. Retrieved 19 January 2015.Rudin, W. (1976). Principles of Mathematical Analysis (PDF). New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.Weisstein, Eric W. "Cardinal Number". MathWorld.Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston, MA: Addison-Wesley. p. 133. ISBN 978-0201726343.G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics, Springer, in cooperation with the European Mathematical Society, retrieved 8 October 2014Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f)Halmos (1960) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers). (p. 46)Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)Von Neumann 1923ReferencesBluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.), McGraw-Hill ProfessionalCarothers, N.L. (2000), Real analysis, Cambridge University Press, ISBN 0-521-49756-6Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics (Fifth ed.), Oxford University PressDedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0-486-21010-3 Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-XEves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Thomson, ISBN 978-0-03-029558-4Halmos, Paul (1960), Naive Set Theory, Springer Science & Business MediaHamilton, A. G. (1988), Logic for Mathematicians (Revised ed.), Cambridge University PressJames, Robert C.; James, Glenn (1992), Mathematics Dictionary (Fifth ed.), Chapman & HallLandau, Edmund (1966), Foundations of Analysis (Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-XMac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical SocietyMendelson, Elliott (2008) [1973], Number Systems and the Foundations of Analysis, Dover PublicationsMorash, Ronald P. (1991), Bridge to Abstract Mathematics: Mathematical Proof and Structures (Second ed.), Mcgraw-Hill CollegeMusser, Gary L.; Peterson, Blake E.; Burger, William F. (2013), Mathematics for Elementary Teachers: A Contemporary Approach (10th ed.), Wiley Global Education, ISBN 978-1118457443Szczepanski, Amy F.; Kositsky, Andrew P. (2008), The Complete Idiot's Guide to Pre-algebra, Penguin GroupThomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008), Elementary Real Analysis (Second ed.), ClassicalRealAnalysis.info, ISBN 9781434843678Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1: 199–208 Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.External linksHazewinkel, Michiel, ed. (2001) [1994], "Natural number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4Axioms and Construction of Natural NumbersEssays on the Theory of Numbers by Richard Dedekind at Project Gutenberg[show]vteNumber systems[show]vteClasses of natural numbersCategories:Cardinal numbersElementary mathematicsIntegersNumber theoryNumbersNavigation menuNot logged inTalkContributionsCreate accountLog inArticleTalkReadEditView historySearchMain pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikipedia storeInteractionHelpAbout WikipediaCommunity portalRecent changesContact pageToolsWhat links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationWikidata itemCite this pagePrint/exportCreate a bookDownload as PDFPrintable versionIn other projectsWikimedia CommonsLanguagesAfrikaansAlemannischአማርኛالعربيةAragonésঅসমীয়াAzərbaycancaتۆرکجهবাংলাBân-lâm-gúБашҡортсаБеларускаяБеларуская (тарашкевіца)‎Българскиབོད་ཡིགBosanskiBrezhonegCatalàЧӑвашлаČeštinaCymraegDanskDeutschEestiΕλληνικάEmiliàn e rumagnòlEspañolEsperantoEuskaraفارسیFøroysktFrançaisGaeilgeGalego贛語Хальмг한국어Հայերենहिन्दीHornjoserbsceHrvatskiBahasa IndonesiaInterlinguaИронIsiXhosaÍslenskaItalianoעבריתBasa Jawaಕನ್ನಡქართულიҚазақшаKiswahiliKurdîລາວLatinaLatviešuLëtzebuergeschLietuviųLa .lojban.LumbaartMagyarМакедонскиMalagasyമലയാളംमराठीمصرىBahasa MelayuMirandésМонголမြန်မာဘာသာNederlandsनेपालीनेपाल भाषा日本語NordfriiskNorskNorsk nynorskOʻzbekcha/ўзбекчаਪੰਜਾਬੀپنجابیPatoisភាសាខ្មែរPiemontèisPlattdüütschPolskiPortuguêsRomânăРусскийसंस्कृतम्SarduScotsShqipSicilianuසිංහලSimple EnglishSlovenčinaSlovenščinaŚlůnskiSoomaaligaکوردیСрпски / srpskiSrpskohrvatski / српскохрватскиSuomiSvenskaTagalogதமிழ்Татарча/tatarçaతెలుగుไทยТоҷикӣTürkçeTürkmençeУкраїнськаاردوTiếng ViệtVõro文言West-VlamsWinarayייִדישYorùbá粵語Žemaitėška中文Edit linksThis page was last edited on 28 November 2017, at 18:20.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. 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