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Intermediate Algebra
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- Gehinnn 5y ago> Irrational numbers are defined as any numbers that cannot be written as a ratio of two integers. > Finally, the set of real numbers, denoted R, is defined as the set of all rational numbers combined with the set of all irrational numbers. I'm sorry, this is not how math works. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivation of how to deal with ka/kb = a/b).
- syops 5y agoI think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.
- Gehinnn 5y agoYou are right, I didn't teach low level math courses, but this brain washing is also precisely why I didn't understand math in high school. You cannot argue with this kind of definitions. Everything feels as if it was randomly defined by the teacher. This "intuition" simplifies teaching, but makes understanding harder. It is like a game where you invent rules as you play. No student can win this game.
- JeremyBanks 5y agoA problem is that lots of lower-education math instructors don't understand these concepts deeply themselves. I think it could be okay if these things were clearly framed as "true for the problems we're looking at, but not universal", but they were typically presented as universal by teachers who themselves don't know any better, and that really caught me up too.
- syops 5y agoHere’s the definition of 2 using the standard construction with the Peano axioms. It’s the set containing 0 and 1. The number 1 is the set containing 0 and 0 exists by one of the axioms. It’s not something a person in intermediate algebra can understand. For one, the natural question then is, “what is a set?”. Whatever one does there has to be some brain washing in order to get started. This is unavoidable unless one thinks Principia Mathematica should be the starting point.
- Gehinnn 5y agoWell, the peano arithmetic can be described directly as first order logic without set theory ;) I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.
- threatofrain 5y agoThis volume does not confuse R as the algebraic completion of Q. It is completely reasonable for an Algebra 1/2 teacher to wait for a Calculus or Analysis teacher to discuss the metric completion of Q. Describing R as rational + irrational numbers is a completely solid description.
- Gehinnn 5y agoIt is only solid as long as you don't define irrational numbers literally as everything that is not rational.
- CamperBob2 5y agoWhat would be an example of a real number that's neither rational nor irrational? (I'm not a math guy, in case it's not obvious)
- alisonkisk 5y agoThere aren't any, but it's circular reasoning. It's exactly the same as saying "real numbers are the union of rational numbers and gargoyle numbers". What's a gargoyle number? "It's any number that's not a rational numbers." What's a number? "It's anything that's a rational numbers or a gargoyle number"
- cubano 5y agoYou "win at the game" by learning what is taught, getting the "A", and then doing you own in-depth research about what interests you on your own time. K-12 was, of course, invented by the Germans in order to create good little factory workers that would get up early and work all day and not complain too much. The fact we still use the word Kindergarten is a nod to this origin story. They weren't at all interested in the students gaining any "understanding" and most certainly not in them "winning" in any sense of the word.
- alisonkisk 5y agoWhy on earth would a factory worker go to high school? This argument makes no sense.
- zsmi 5y ago> Everything feels as if it was randomly defined by the teacher. I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.
- ookdatnog 5y agoEuclid's postulates would now be called axioms, not definitions.
- galaxyLogic 5y agoA student may be confused about why just these axions and not some others? It can then be explained that there in fact be alternate set of axions. But still most teachers give us the same standard set of axions. Why? What would happen if they dropped some of them or replaced them with others?
- ABeeSea 5y agoOne of the standard construction of the real numbers is the set of equivalence classes of rational Cauchy sequences. This definition is equivalent of the handwavey definition above (irrationals are the Cauchy sequences that don’t converge to a rational number and the reals are the rationals plus the irrationals.) However almost any construction of the real numbers is challenging to give a simple explanation for. Even leading 19th century mathematicians didn’t truly understand the real numbers until Cantor.
- young_unixer 5y agoWhat should I read if I want to learn what a number is?
- whatshisface 5y agoAssuming you already know what a rational number is, the next step is to tell you what a real number is. A real number is defined as the equivalence class of all sequences of rational numbers that converge to the same value. For example, every sequence of rational numbers that gets arbitrarily close to the square root of two as you go to higher terms is considered "the square root of two." If you don't know what a rational number is, it's the equivalence class of every pair of integers that can be simplified to the same fraction. For example, (4,6) and (2,3) are both rational numbers, and in fact are the same rational number: two thirds. If you don't know what an integer is, it's the natural numbers, but with negative numbers. If you don't know what a natural number is, it's either zero, or a number that follows a natural number. For example one is the number that follows the natural number zero, and two is the number that follows the natural number that is the natural number that follows zero.
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- adrian_b 5y agoInstead of assuming an understanding of what natural numbers are, you could have continued to define all of them as equivalence classes, as that is what they are. The integers are the equivalence classes of differences of natural numbers, while the natural numbers are the equivalence classes of finite sets having the same number of elements (i.e. which may have a bijection between themselves), including the empty set.
- abdullahkhalids 5y ago> having the same number of elements How do you define the _number_ of elements of a finite set without defining natural numbers first?
- ookdatnog 5y agoI don't necessarily disagree with your point that for the given audience it's not appropriate to rigorously define the different sets of numbers. However, I absolutely detest it when teachers just "sweep it under the rug", when they pretend that they just provided a definition when they evidently did not. Like the commenter you replied to, this sort of stuff genuinely threw me off in high school and made me feel like I didn't understand mathematics.
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- raegis 5y agoI'm completely serious here. Could you give us your definition of "number" which you would teach at this level? Please, please, I'm not trying to be an ass or anything. We could start a great conversation here.
- threatofrain 5y agoA number is an informal term within mathematics.
- Aicy 5y agoNot exactly true. You can formally construct the set of real numbers with set theory and then say a number is an element of that set Wikipedia page on different approaches: https://en.m.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...
- threatofrain 5y agoIf we're talking about "number" in general, yes, it is true that this is an informal term among mathematicians.
- Aicy 5y agoWhat mathematicians are you talking about? Set theorists and number theoriests have very precise definitions of numbers. Maybe by "number" in general do you mean something that encapsulates both "real numbers" as in lengths, and "integers" as in the discrete counting numbers? In which you can quite easily do so by defining real numbers, either formally or saying that something like the Wikipedia definition that they are a distance along a line, and then saying integers are a subset of those numbers.
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- sunneversets 5y agoFrom the TOC, it looks like an elementary algebra book. Students usually don't see rigorous construction of integers and rationals till classes titled something to the effect of "intro to proofs 101", "intro to discrete analysis", "abstract algebra 101", "elementary general algebra", "elementary set theory", "basics of elementary number theory"... And the construction of reals would have to wait till something like "elementary real analysis 101". Usually, after elementary algebra come 3 semesters of calculus. Subject as vague and, uhh, as un-mathematical as it contains neither concrete definitions(other than the one for derivative), nor any theorems. Nothing substantial to grasp at when drowning. Traditionally, actual learnable definition-lemma-theorem-corollary-examples style math starts after the calculus sequence. So the audience for the linked book is at least 2 classes removed from the time they get to see number construction. While elementary algebra is both useful and unavoidable, I don't think the same about the calculus. IMHO, the latter is just a waste of time. The sequence elementary algebra -> intro to math proofs -> elementary real analysis -> Lebesgue Integral by way of Daniell-Riesz -> complex analysis -> non-Euclidean/abstract topology -> measure theory -> probability theory -> differential geometry would give anyone world- class education into the nature of functions which the calculus sequence is kind-sorta supposed to give you a tiny and very distant taste of.
- galaxyLogic 5y ago> into the nature of functions which the calculus sequence is kind-sorta supposed to give you a tiny and very distant taste of How about instead starting with Lambda Calculus? It's all about functions and how to define them
- Aicy 5y agoI don't think this is a fair critisim - they've just said it reverse order. They should start with the set of real numbers, denoted R, and then this can be split into two disjoint subsets - the rational numbers and the irrational numbers. From there the natural numbers are a subet of the rational numbers.
- glitchc 5y agoAny given irrational number may not be possible to write down. a/b is a rational number. The set of real numbers is the union of rational and irrational number sets i.e. all possible rational and irrational numbers combined form the real number set.
- Aicy 5y ago> Any given irrational number may not be possible to write down. There's no "may" about it! By definition an irrational number has an infinitely long decimal expantion that does not repeat, so none of them can be written down (other than in symbolic representations like pi)
- seoaeu 5y agoIn fact, since there are only a countably infinite number of symbolic representations but an uncountable number of irrational numbers, nearly all of them cannot be written down at all!
- Koshkin 5y agoIf you mean representations using finite sequences, sure. But the set of all infinite sequences (e.g., of digits) is uncountable.
- seoaeu 5y agoRepresenting a number as an infinite sequence of digits isn't helpful if you are trying to write down the number. You can't write an infinite number of digits after all
- Koshkin 5y agoBut then your point is the identical to the parent’s.
- Aicy 5y agoThat's the point. There is uncountably many infinite sequences, but we only have countably many symbolic representations, so we can't write most real numbers down even with infinite time.
- coderinsg 5y agoReally helpful way to start understanding the symbols. Clearly written.
- alexdowad 5y agoWhen a book is titled "Algebra", it's always a bit ambiguous whether that means "high school algebra" or "abstract algebra". I clicked on this link thinking it was about group/field/ring/etc theory and wondering what an 'intermediate' treatment of those topics would look like.
- gxs 5y agoI have Artin’s Algebra on my bookshelf, most people who see it always tell me they took it in high school. I’m always impressed.
- alexdowad 5y agoFunny stuff. I remember reading a comment by Professor Terry Tao, where he mentioned that one of his weaker areas of mathematics is "algebra". I guess he should have paid more attention in 9th grade math classes!
- nicklaf 5y agoYou know for a few confused seconds I didn't see the joke about the title and took your statement literally, assuming you were talking about some precocious student, who perhaps had been lucky enough to be tutored or concurrently enrolled while still in high school. Funny story, though: this is the upper division algebra text we used at UC Davis, and one amusing anecdote that our professor relayed was that this book was in fact written for MIT freshmen. Supposedly, this is why the book introduces linear algebra from scratch rather than assuming it as a prerequisite....
- gxs 5y agoYep, this is the text the professor used in my abstract algebra class at UC Berkeley. I know they use a bunch of other ones. I definitely wouldn’t have been ready to take this as a freshman - my high school math education was definitely lacking.
- azalemeth 5y agoI thought this too -- in fact the contents of this site are about the level of what good 16-year-olds learn in my country. I'd love to read an accessible 'intermediate' introduction to abstract algebra. Something I've inferred -- and would love to be corrected on -- is that in the US there's a relatively well-defined course implied by the words "calculus" or "algebra". I can guess what they are, but I'm not certain, and it seems to change!
- Miiko 5y agoNot sure what's the point of section 4.3 Factoring Trinomials introducing "trial and error (or guess and check) method", when this task can be solved easily without any trial and errors, by using 6.2 Quadratic Formula. Shouldn't section 4.3 at least mention that possibility? Is there something I do not understand here?
- throaway46546 5y agoGuess and check pissed me off so much back when I was in school.
- xphos 5y agoI think guess and check pissed you off because its taught as guess a random number and check they don't really teach you how to guess smartly and then you waste your time guessing in the wrong direction mindlessly. Its like teaching stands they should checkout every array slot when there are bits of knowledge that you can teach to do a binary search and get to an answer faster. And the math intuition to build the pattern of guessing is much less especially for high school level problems.
- throaway46546 5y agoI just wrote a TI83 program to solve the equation and then spit out a couple randomized "guesses" so I could "show my work".
- xphos 5y agoTry factoring this with the quadratic formula x^5+5x^4+10x^3+10x^2+5x+1. Or try guess and check with -1,0,1. The issue with formula's is that they constrain the space of the problems and ones mind especially. the problem above is factoring is (x+1)^5 but there exists no (and there cannot exist) formula to that can tell you that from the equation. Algebra is taught poorly not because kids don't learn the steps but because it stunts people into viewing math as if you only had the formula its easier. I'm not saying you should know that there cannot exist a 5'th or greater degree integer formula for factoring equations because to say that is very hard. But when it leaves students with very little scaffolding to use when things go wrong / don't fit the formula neatly
- mieses 5y agoWhy are the author (John Redden) and acknowledgements removed from the github.io version? https://scholar.flatworldknowledge.com/books/4372/fwk-redden-ata https://scholar.flatworldknowledge.com/books/4372/fwk-redden... https://open.umn.edu/opentextbooks/textbooks/134 https://open.umn.edu/opentextbooks/textbooks/134
- kwhitefoot 5y agoThe Licensing page says: "This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor. " They seem to be saying that Redden didn't want his name on the Saylor version.
- wly_cdgr 5y agoHide the answers by default, yo. It's so easy to do with an e-book, come onnnnnn
- kwhitefoot 5y agoWhy does it need a calculator?
- xphos 5y agoSorry for the rant but as a math lover this is terrible. In my experience "algebra 2" is taught for multiple years and is in my opinion the most boring soul crushing math I was taught. And I say taught because its forced into you with very few interesting observations and thinking. It focuses on memorizing formula's a procedures, which while these things have there applications I don't consider someone who can say y=mx+b or the a^2+b^2=c^2 knows much about math or more importantly engages with things in a mathematical way. I think there is this focus to produce calculators when you want thinkers and maybe and I hope this has changed in the last 10 years (age 24) but I feel like the American middle school level produces bad math habits and skills wrapped around solving test questions instead of thinking.
- dqv 5y agoI'm definitely feeling the effects of this now trying to get back into math. It's frustrating because I can tell exactly what's happening: I get the same kinds of problems wrong over and over again and exhaust the question bank. The next time around I recognize the problem and basically just solve it from memory. Do you have any recommendations for learning math the right way? Or a better way at least.
- xphos 5y agoThis got long and a little ranty sorry :/ TLDR: Take a proofs course and get perspective outside the problems you are struggling on and return with the ability to ask questions about what your actually doing and how to verify you did it right. I am not blaming you here but this mentality "The next time around I recognize the problem and basically just solve it from memory." Raises alarm bells for me. To me this says you are not engaging critically with a problem and arguable your pervious sentence tells me why. You are going over and over a problem which is fundamentally the same which is a disengaging process of learning and reasoning. My suggestion is to focus less on you got the problem wrong and rather diagnosis why you got it wrong. Which is a very difficult thing to do alone. If you share a problem and the mistake you got I could try and help you. In general thought for me there are 2 major types of errors I run into when doing math. Yours may vary, but mine are Arithmetic errors and not understanding the theories. Arithmetic -- i.e i said something like 1+1=3 somewhere and everything is wrong now. This happens a surprisingly a lot but the usually go back and check all your calculations almost never finds this issue for me. (if possible) I do have a lot more success with plug your answer back into the question and calculating what your input should be. This leads to the second point Understanding the theories (and question) - If you understand what your solution is saying you should be able to find a different question to ask about your answer to be sure its right. If you can't you don't understand what your doing well enough. I have been picking on factoring because its easy to have hard problems that can be solved in many many different ways. But if you factor something essentially you found a zero so you should be able to do 2 things. One is plug your answer in and have the whole expression equal zero after (zero is always easiest to find because math is easy when you can set things to zero.) The other though is divide the the expression by (x-a). This reduce out perfectly and was a major way I checked my errors when I took a cryptography and number theory course. Understanding something deeply is hard because there is no easy testable why to say that you do get it. But if you can ask questions like how do I know this answer is right its a good sign you have a lot more to learn what your actually doing. My final recommendation if you are trying to learn algebra is take a break from that. Do a intro to proofs class or lesson plan if you've never done that before. Those type of courses are suppose to teach systematic approaches to math that scale beyond do you know a formula that works. They also give you more intuition into what your doing because they teach you to think about the results of what you are saying and give perspective that the "algebra 2" courses you take in high school don't bother to even think about because they are focused on doing problems for problems sake which most people have enough problems they don't need more to solve for fun they want tools to handle a more diverse set better.