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> 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
by 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.