I 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.
You 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.
A 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.
Here’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.
Well, 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.
This 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.
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"
I think by the definition given at the beginning of the thread irrational numbers are numbers that can’t be expressed as a ratio of two natural numbers. i is irrational by that definition, but not a real.
Just to clarify: there are lots of numbers that aren't real numbers (for example, imaginary numbers). Intuitively the real numbers are all the points along the number line, including rationals and irrationals (such as root 2, pi, or e). There are lots of other comments in this thread that give a good explanation of how that works formally.
If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).
i or some hyper real numbers are neither irrational nor rational.
Basically all crazy extensions of R used to solve even more crazy problems. They don't really exist naturally (and you can make up your own field extension of R as you wish), but they are handy if you want to compute stuff.
But by definition there are no real numbers that are neither rational or irrational.
The algebraic closure of the rationals is not the reals. The reals are the completion of the rationals using the standard metric.
The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!
You "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.
> 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.
One 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.
I'll preface this by saying that I got bored and didn't finish it (Axioms? Rubbish, where's my field theory etc.) but Terence Tao's book on Algebra seemed like a somewhat gentle and very thoughtful introduction to the subject. Not necessarily easy by any means but it looks like he has put a lot of work into the pedagogy (whereas some mathematicians just shit out theorem and proof onto the page with no regard whatsoever for the prose, justification or flow - but I (am forced to) digest)
It depends on your mathematical background. I don’t know of any books about this at the level below senior undergraduate mathematics. If you are familiar with sets I can outline the idea behind how to define non negative integers.
We assume the empty set exists and call this 0. We define 1 to be the set containing 0. So 1 = {0}. We define 2 to be the set containing 0 and 1. So 2 = {0, 1}.
Let’s look at this set: {a, b}. I know this set has size 2 and not 1 because I can map {a, b} to {0, 1} in a one-to-fashion. I can’t map {a, b} to {0} in a one-to-one fashion. We say any set has size 2 if it can be mapped to {0, 1} in a one-to-one fashion.
First define natural numbers as sysops did (or you can use peanos axioms).
Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.
The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.
The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut
Numbers start with counting. You can build up more kinds of numbers, but tearing down counting into smaller pieces is surprisingly hard. Personally, I think it's good enough to start with any two distinguishable states. You can then keep combining those states in various ways to form the counting numbers. Note that more complex symbols like "1 2 3" are themselves effectively a form of tally, where you are counting the number of angles. You can combine your method with geometry or a clock and call it "measurement" of space and time, respectively.
(Real numbers are my favorite construction because I was enamored with Cantor's diagonal argument when I first learned it. It's quite clever and hints at the magic mathematicians are capable of. Although most mathematicians (algebra peeps) seem to like the classic "root 2 is irrational" proof more.)
I think that set theory and other analytic can be instructive, but can also obscure what's happening. The exact encoding of numbers using sets is just an "implementation detail", in the sense that there's many ways you can build natural numbers (for example) using set theory and they are all equivalent.
So it's like learning data structures by coding in assembly, which is what Donald Knuth thinks is the right thing to do anyway, but some other teachers would disagree. But if you want to see some high level construction, you could look to eg. Tarski's synthetic construction of reals
Which uses integers rather than sets as the building block, and is simpler than many constructions. And, of course integers themselves can be constructed out of sets, but they can be constructed out of lambda calculus terms as well https://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datat... among many other constructions - but when we finally define integers, we can abstract away the implementation details (and that's really the crux of the question!)
Sounds like a philosophical question. Mathematicians define axiomatisations and definitions that attempt to characterise in a rigorous way our intuition of numbers. But they don't tell us what numbers are.
I don't have any specific resources to recommend, however I'll give my take on the foundations of 'numbers'.
We use the label 'number' to refer to a broad swathe of mathematical objects, objects that are different but also so similar they often appear interchangeable (for example counting numbers and fractions).
In the formal mathematical sense, a specific type of number is a group of objects which have been defined to have specific properties. I'm going to leave object and property as defined in the usual sense, I think most people have a good idea about what those are and not sure I can add anything to them.
There are no rules as to what properties you are allowed to give to objects, nor what group of objects you want to include, but generally if you are learning about some specific thing it's because people find them useful or interesting; the definitions we have for different types of numbers are the ones we have found useful or interesting.
Remember that the different types of number appear very similar. It is common to build a 'hierarchy' of differnt types of numbers, where we start with a simple type of number and then add new properties and objects when we find limitations we don't want.
The first type of number in this hierarchy are the natural numbers (sometimes called counting numbers). The most common properties defined for these today are called the Peano axioms [0]. There are quite a few of them, and the history of how we came to the formalisation is very interesting (a lot of it is about avoiding inconsistencies/contradictions) but the key ideas are:
- there is a natural number called 0
- every natural number has a successor, which is also a natural number - we can write S(n) is the successor of n
Most of the other axioms define what it means for two natural numbers to be equal (=).
Just having these objects isn't particularly useful, we typically want to do things like add, multiply, and compare numbers. To do that we include some operations: addition (+), multiplication (*), and total ordering(<=).
These are defined as, taking a, b, c as natural numbers:
- a + 0 = a
- a + S(b) = S(a + b) (this is recursive, so if we define 1 as 1:=S(0) then we have 1+1 = 1+S(0) = S(1+0) = S(1))
- a * 0 = 0
- a * S(b) = a + (a * b)
- a <= b if (and only if) there exists some c such that a + c = b
Importantly, using these definitions, we can say that the natural numbers are closed under addition and multiplication; whenever you add or multiply two natural numbers together you get another natural number.
To continue building the hierarchy we notice that there are operations we would like to do but are not possible for every natural number (please note I am skipping over the formalisations from hereon and talking about the motivation for different types of numbers).
We notice that if we can add two numbers together we should be able to subtract them again. If a + b = c, then c - b = a. However (for example) 0 - 1 is not a natural number. So we extend the natural numbers to the integers, such that the integers are closed under subtraction.
If we can multiply it makes sense to try and divide, but 2 / 3 is not an integer so we extend integers to the rationals (ratios of integers) which are closed under division. We add an object called 2/3 so that now when we can say 2 / 3 = 2/3.
The next step in the hierarchy is a bit more complex. We notice that we can define a subset of rational numbers that all meet a certain criteria, for example all rational numbers that are less than 2. We call 2 an upper bound of that subset. Notice that 2 is a rational number, and that there are no rational numbers smaller than 2 that are also an upper bound of our subset - 2 is the least upper bound. Define a new subset, where we say a rational number x is in the subset if x * x <...
Assuming 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.
Instead 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.
you can show two sets have the same number of elements without having an intrinsic notion of "number" - find a bijection between them, mapping every member of set A to set B and vice versa, and you know you have two identically-sized sets without doing any counting.
In order to decide that two sets have the same number of elements (for this relationship various names have been used, e.g. equipotence, equipollence, equinumerosity), you do not need numbers or being able to count.
You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.
This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.
This is actually how the numbers originated, for humans and for many other animals.
Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.
When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.
When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.
If you look closely, I didn't assume natural numbers, I stopped at Peano axioms. Stopping at set theory would have been reasonable but it would have added a line at the end about sets that wasn't describing numbers per se.
The book Numbers [1] by Ebbinghaus et. al. is a comprehensive treatment, starting from natural numbers up to complex numbers (and quaternions iirc). It also has lots of historic context, how our number systems evolved.
I 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.
I'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.
What 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.
Mathematicians have precise definitions of "real numbers", "complex numbers", etc. but not of "number." For example, are the hyperreals numbers? Nonstandard integers? Quaternions?
Mathematicians have definitions of all those things. And if you are working in something like the quaternions so do all the previous you mentioned (real numbers, complex numbers) because they are subsets of the well defined quaternions.
I agree in a sense there is no agreed-upon simple non-formal definition which encapsulates all of those together but I think that's unfair to try to do since hyperreals, quaternions, etc are extreme extensions which aren't used in most ordinary mathematics.
That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc
You can go down the same rabbit hole I guess and say "yeah! true! distance has no definition either!" but I don't find that very helpful. At this point you are just saying nothing can be defined, at which point the phrase "we have no precise definition of X" has no meaning as it is true for all X.
Some things are formally defined terms of art and some things aren't. There are some people here who don't merely want to learn a model for the naturals, they want some kind of spiritual essence of a number. It's useful to say that number is an informal term, and that the fuzzy boundaries you think with are adequate for communication.
Similarly, "crazy" is not a term of art for psychiatry. That doesn't mean a lack of definition, and lacking formalism does not mean inadequate for communication. By using a fuzzy term for communication, you imply a taste in curation and abstraction. It means you don't want to get into the details of what crazy means.
> That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc
A metric is a term of art, and if that's the level of specificity you wanted, then it's adequate.
You can make this argument on literally any definition - you have to start with something.
So dictionary.com says the defintion of distance is:
> the extent or amount of space between two things, points, lines, etc.
That works for me.
I think it makes sense to start with something physical, in the real world that people - and especially kids, can intuatively understand. Our brains by default understand what distance is. Maybe that's true for numbers as well- and that's ok too.
I'm sure there are more elegant approaches, but I would start like this:
A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations.
It is not well defined and should not be used as basis for other definitions.
"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.
Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.
The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.
The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.
This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.
I understand very well that the peano axioms for example are logical and not circular.
My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).
So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.
I don’t give a definition. The students know real numbers are the “decimal numbers”. They don’t need to know a definition at this point and giving one more precise than “decimal numbers” will just confuse them. They gain familiarity by practice in the same way a Fahrenheit person like myself can only get intuition with Celsius temperatures is by using them.
At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that students are already expected to think something like this.
I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex numbers as numbers but hasn't encountered the term "real numbers"), but it's wrong to pretend that this is a mathematical definition. Even just removing the words "defined as" would be fine.
> At this level, associating numbers with potentially infinite decimals would not be too bad.
I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.
> I find number with an infinite number of decimals quite an abstract concept.
If students are familiar with the idea that fractions can be represented as repeating decimals and are introduced to π as a non-repeating decimal, then I think the idea of the set of all possibly repeating decimals should make sense to them.
> Another issue with this is that it confuses numbers with their representation.
It's probably worth acknowledging that some numbers have multiple representations just to avoid arguments about whether 0.999.. = 1.
However, is it much worse to say that a real number is an equivalence class containing 1 or 2 infinite decimals than to say that a real number is a non-empty, downwards closed, proper subset of the rationals? I agree it's nicer to define the reals as a field with some extra properties, but I'm not convinced it's wrong to say that they are their representation.
At that level, I'd use a geometric definition. I would postulate the existence of a set R of numbers which represent the distance along a line based on a given unit. From there, you "show" that natural and rational numbers belong to R. You show that some numbers are irrational (easy to prove that sqrt(2) is).
There are some postulate there but I think it provides the right intuition and it's not a obviously circular definition.
From 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.
I 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.
Any 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.
> 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)
In 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!
Representing 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
That'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.
When 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.
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!
You 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....
I 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!
Before College all math courses in the US 'build up' to Calculus 1 & 2 [1,2] presumably to prepare motivated students for engineering. Calculus 2 is the highest you can go in a public High School before a student will start thinking about starting their college career early or doing an independent study. This one track path is not without its side effects. As you can see by skimming this textbook the information on linear systems is lacking - something that might be more interesting to those who want to study CS.
Not 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?
I 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.
Try 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
> Try factoring this with the quadratic formula x^5+5x^4+10x^3+10x^2+5x+1.
But this is not a trinomial and, of course, quadratic formula is not applicable here. My objection was specifically for section 4.3, which does not provide any clues how to extend the described method to more complex polynomials anyway.
> The issue with formula's is that they constrain the space of the problems and ones mind especially.
That I agree with - teaching only to use formulas is bad, - but how to find polynomial roots is not relevant here. The proper way to teach factorization is IMHO to show relations to roots first - which are not necessary to be found by quadratic formula, it may be the same trial and error method, - instead of introducing a special method that works only in very limited cases.
> 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.
Err... learning some steps (without explaining where they come from) is not a proper way to tech math anyway, and I surely do not advocate that. My point was different - instead of teaching some steps to solve particular problem (factoring trinomials), the better approach is to teach the underlying theory and real methods used in modern mathematics.
That is, instead of using factorization as a method of solving polynomial equation, the proper approach is exactly the opposite, using equation roots for the factorization. Well, at least that's how they taught me to do things, and I do believe this is the correct way.
"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.
Sorry 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.
I'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.
This 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.
Thanks for responding. I think I was just discouraged because I was doing a lot of arithmetic errors. It was unclear if the errors I was making were errors of logic or errors of arithmetic. Just like learning a new programming language: you have logical errors and syntax errors (not just ones that get thrown by the compiler, but ones that are just improper use of syntax) and sometimes it's unclear what error you're really making.
I think the bigger issue was that in high school and even college math classes was that I always felt like I was rushing or being rushed. There wasn't really time to internalize concepts or ask why.
>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.
I'll take a look at this. I did proofs in a geometry class and really liked it.
The Art of Problem Solving books are great and really focus on understanding the concepts deeply.
The problem selection varies widely in difficulty and they all have corresponding solution manuals so you can check your answers and problem-solving approach.
If you were to go through all of them, you would come out with an extremely strong foundation in mathematics that would make undergraduate-level material much more accessible.
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[ 4.5 ms ] story [ 200 ms ] thread> 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).
My background on this topic is that I’ve taught intermediate algebra for over 20 years.
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.
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"
But I must admit I haven’t read the whole post.
If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).
But by definition there are no real numbers that are neither rational or irrational.
The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!
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.
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.
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?
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.
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
We assume the empty set exists and call this 0. We define 1 to be the set containing 0. So 1 = {0}. We define 2 to be the set containing 0 and 1. So 2 = {0, 1}.
Let’s look at this set: {a, b}. I know this set has size 2 and not 1 because I can map {a, b} to {0, 1} in a one-to-fashion. I can’t map {a, b} to {0} in a one-to-one fashion. We say any set has size 2 if it can be mapped to {0, 1} in a one-to-one fashion.
First define natural numbers as sysops did (or you can use peanos axioms).
Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.
The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.
The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut
(Real numbers are my favorite construction because I was enamored with Cantor's diagonal argument when I first learned it. It's quite clever and hints at the magic mathematicians are capable of. Although most mathematicians (algebra peeps) seem to like the classic "root 2 is irrational" proof more.)
This vsauce video is actually a really accurate (and entertaining!) introduction to set theory: https://youtu.be/s86-Z-CbaHA
So it's like learning data structures by coding in assembly, which is what Donald Knuth thinks is the right thing to do anyway, but some other teachers would disagree. But if you want to see some high level construction, you could look to eg. Tarski's synthetic construction of reals
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_t...
Which doesn't build reals using other theories as building blocks; real numbers are real numbers.
Anyway, from the analytic constructions of reals, I'm most partial to
https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
Which uses integers rather than sets as the building block, and is simpler than many constructions. And, of course integers themselves can be constructed out of sets, but they can be constructed out of lambda calculus terms as well https://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datat... among many other constructions - but when we finally define integers, we can abstract away the implementation details (and that's really the crux of the question!)
Anyway there's a discussion of analytic vs synthetic mathematics in this post, https://golem.ph.utexas.edu/category/2015/02/introduction_to... (it seems that part 2 wasn't written unfortunately)
We use the label 'number' to refer to a broad swathe of mathematical objects, objects that are different but also so similar they often appear interchangeable (for example counting numbers and fractions).
In the formal mathematical sense, a specific type of number is a group of objects which have been defined to have specific properties. I'm going to leave object and property as defined in the usual sense, I think most people have a good idea about what those are and not sure I can add anything to them.
There are no rules as to what properties you are allowed to give to objects, nor what group of objects you want to include, but generally if you are learning about some specific thing it's because people find them useful or interesting; the definitions we have for different types of numbers are the ones we have found useful or interesting.
Remember that the different types of number appear very similar. It is common to build a 'hierarchy' of differnt types of numbers, where we start with a simple type of number and then add new properties and objects when we find limitations we don't want.
The first type of number in this hierarchy are the natural numbers (sometimes called counting numbers). The most common properties defined for these today are called the Peano axioms [0]. There are quite a few of them, and the history of how we came to the formalisation is very interesting (a lot of it is about avoiding inconsistencies/contradictions) but the key ideas are:
- there is a natural number called 0
- every natural number has a successor, which is also a natural number - we can write S(n) is the successor of n
Most of the other axioms define what it means for two natural numbers to be equal (=).
Just having these objects isn't particularly useful, we typically want to do things like add, multiply, and compare numbers. To do that we include some operations: addition (+), multiplication (*), and total ordering(<=).
These are defined as, taking a, b, c as natural numbers:
- a + 0 = a
- a + S(b) = S(a + b) (this is recursive, so if we define 1 as 1:=S(0) then we have 1+1 = 1+S(0) = S(1+0) = S(1))
- a * 0 = 0
- a * S(b) = a + (a * b)
- a <= b if (and only if) there exists some c such that a + c = b
Importantly, using these definitions, we can say that the natural numbers are closed under addition and multiplication; whenever you add or multiply two natural numbers together you get another natural number.
To continue building the hierarchy we notice that there are operations we would like to do but are not possible for every natural number (please note I am skipping over the formalisations from hereon and talking about the motivation for different types of numbers).
We notice that if we can add two numbers together we should be able to subtract them again. If a + b = c, then c - b = a. However (for example) 0 - 1 is not a natural number. So we extend the natural numbers to the integers, such that the integers are closed under subtraction.
If we can multiply it makes sense to try and divide, but 2 / 3 is not an integer so we extend integers to the rationals (ratios of integers) which are closed under division. We add an object called 2/3 so that now when we can say 2 / 3 = 2/3.
The next step in the hierarchy is a bit more complex. We notice that we can define a subset of rational numbers that all meet a certain criteria, for example all rational numbers that are less than 2. We call 2 an upper bound of that subset. Notice that 2 is a rational number, and that there are no rational numbers smaller than 2 that are also an upper bound of our subset - 2 is the least upper bound. Define a new subset, where we say a rational number x is in the subset if x * x <...
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.
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.
How do you define the _number_ of elements of a finite set without defining natural numbers first?
You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.
This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.
This is actually how the numbers originated, for humans and for many other animals.
Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.
When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.
When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.
[1] https://www.amazon.com/Numbers-Graduate-Mathematics-Heinz-Di...
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.
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.
I agree in a sense there is no agreed-upon simple non-formal definition which encapsulates all of those together but I think that's unfair to try to do since hyperreals, quaternions, etc are extreme extensions which aren't used in most ordinary mathematics.
That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc
You can go down the same rabbit hole I guess and say "yeah! true! distance has no definition either!" but I don't find that very helpful. At this point you are just saying nothing can be defined, at which point the phrase "we have no precise definition of X" has no meaning as it is true for all X.
Similarly, "crazy" is not a term of art for psychiatry. That doesn't mean a lack of definition, and lacking formalism does not mean inadequate for communication. By using a fuzzy term for communication, you imply a taste in curation and abstraction. It means you don't want to get into the details of what crazy means.
> That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc
A metric is a term of art, and if that's the level of specificity you wanted, then it's adequate.
> In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line
That's the intuitive definition that makes the most sense to me.
So dictionary.com says the defintion of distance is:
> the extent or amount of space between two things, points, lines, etc.
That works for me.
I think it makes sense to start with something physical, in the real world that people - and especially kids, can intuatively understand. Our brains by default understand what distance is. Maybe that's true for numbers as well- and that's ok too.
A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations. It is not well defined and should not be used as basis for other definitions.
"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.
Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.
The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.
The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.
This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.
Equations like "a + x = b", "ax = b" and "sin(x) = 0" have no meaning unless you have an understanding of what the terms mean, or could mean.
My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).
So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.
and now the time has come for Intro to Complex Analysis, good luck not leaving everyone completely dumbfounded.
TBH this is the worst explanation of anything I have ever seen.
I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex numbers as numbers but hasn't encountered the term "real numbers"), but it's wrong to pretend that this is a mathematical definition. Even just removing the words "defined as" would be fine.
I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.
If students are familiar with the idea that fractions can be represented as repeating decimals and are introduced to π as a non-repeating decimal, then I think the idea of the set of all possibly repeating decimals should make sense to them.
> Another issue with this is that it confuses numbers with their representation.
It's probably worth acknowledging that some numbers have multiple representations just to avoid arguments about whether 0.999.. = 1.
However, is it much worse to say that a real number is an equivalence class containing 1 or 2 infinite decimals than to say that a real number is a non-empty, downwards closed, proper subset of the rationals? I agree it's nicer to define the reals as a field with some extra properties, but I'm not convinced it's wrong to say that they are their representation.
There are some postulate there but I think it provides the right intuition and it's not a obviously circular definition.
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.
How about instead starting with Lambda Calculus? It's all about functions and how to define them
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.
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.
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)
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!
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....
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.
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!
[1]: https://www.khanacademy.org/math/calculus-1
[2]: https://www.khanacademy.org/math/calculus-2
https://golem.ph.utexas.edu/category/2013/12/the_long_grind_...
Is there something I do not understand here?
But this is not a trinomial and, of course, quadratic formula is not applicable here. My objection was specifically for section 4.3, which does not provide any clues how to extend the described method to more complex polynomials anyway.
> The issue with formula's is that they constrain the space of the problems and ones mind especially.
That I agree with - teaching only to use formulas is bad, - but how to find polynomial roots is not relevant here. The proper way to teach factorization is IMHO to show relations to roots first - which are not necessary to be found by quadratic formula, it may be the same trial and error method, - instead of introducing a special method that works only in very limited cases.
> 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.
Err... learning some steps (without explaining where they come from) is not a proper way to tech math anyway, and I surely do not advocate that. My point was different - instead of teaching some steps to solve particular problem (factoring trinomials), the better approach is to teach the underlying theory and real methods used in modern mathematics.
That is, instead of using factorization as a method of solving polynomial equation, the proper approach is exactly the opposite, using equation roots for the factorization. Well, at least that's how they taught me to do things, and I do believe this is the correct way.
https://scholar.flatworldknowledge.com/books/4372/fwk-redden...
https://open.umn.edu/opentextbooks/textbooks/134
"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.
Do you have any recommendations for learning math the right way? Or a better way at least.
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.
I think the bigger issue was that in high school and even college math classes was that I always felt like I was rushing or being rushed. There wasn't really time to internalize concepts or ask why.
>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.
I'll take a look at this. I did proofs in a geometry class and really liked it.
The problem selection varies widely in difficulty and they all have corresponding solution manuals so you can check your answers and problem-solving approach.
If you were to go through all of them, you would come out with an extremely strong foundation in mathematics that would make undergraduate-level material much more accessible.