Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Intermediate Algebra

saylordotorg.github.io

Post reply on HN
Print view View original

81–90 of 121 posts

Re: Intermediate Algebra

#81

Post by Arrrlex »

Koshkin wrote: post #55
kenjackson wrote: post #44

Earlier quoted context omitted.

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. But I must admit I haven’t read the whole post.

Looks like the set of reals is simply taken as the number universe (i.e. all numbers are already assumed to be 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).

Re: Intermediate Algebra

#82

Post by throaway46546 »

Miiko wrote: post #78

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? Is there something I do not understand here?

Guess and check pissed me off so much back when I was in school.

Re: Intermediate Algebra

#83

Post by azalemeth »

alexdowad wrote: post #75

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 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!

Re: Intermediate Algebra

#84

Post by abdullahkhalids »

adrian_b wrote: post #73
whatshisface wrote: post #65

Earlier quoted context omitted.

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 ra…

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…

> having the same number of elements

How do you define the _number_ of elements of a finite set without defining natural numbers first?

Re: Intermediate Algebra

#85

Post by lkozma »

alexdowad wrote: post #75

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.

That's true, but "high school algebra" in some countries also includes some group/ring theory basics (or it used to).

Re: Intermediate Algebra

#87

Post by canjobear »

azalemeth wrote: post #83
alexdowad wrote: post #75

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 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,…

You’re right. This book contains what US high schoolers learn in “Algebra 2.”

Re: Intermediate Algebra

#88

Post by mhh__ »

mangamadaiyan wrote: post #74
mhh__ wrote: post #30

Earlier quoted context omitted.

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 whatsoev…

Did you mean Terence Tao's book on Analysis? If not, could you post a reference please? I'd be very interested in a Terry Tao book on Algebra :)

I did mean Analysis, sorry. Too late to edit now I suspect.

Re: Intermediate Algebra

#90

Post by Aicy »

SatvikBeri wrote: post #60
Aicy wrote: post #20

Earlier quoted context omitted.

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…

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.

Post reply on HN