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Intermediate Algebra

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Re: Intermediate Algebra

#71

Post by dboreham »

raegis wrote: post #9
Gehinnn wrote: post #3

> 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 motivatio…

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.

Usually it begins with "consider the number line..."

Re: Intermediate Algebra

#72

Post by [deleted] »

whatshisface wrote: post #65
young_unixer wrote: post #16

Earlier quoted context omitted.

What should I read if I want to learn what a number is?

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…

[deleted]

Re: Intermediate Algebra

#73

Post by adrian_b »

whatshisface wrote: post #65
young_unixer wrote: post #16

Earlier quoted context omitted.

What should I read if I want to learn what a number is?

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.

Re: Intermediate Algebra

#74

Post by mangamadaiyan »

mhh__ wrote: post #30
young_unixer wrote: post #16

Earlier quoted context omitted.

What should I read if I want to learn what a number is?

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 :)

Re: Intermediate Algebra

#75

Post by alexdowad »

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.

Re: Intermediate Algebra

#76

Post by ABeeSea »

Gehinnn wrote: post #6
syops wrote: post #4

Earlier quoted context omitted.

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.

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.

Re: Intermediate Algebra

#77

Post by gxs »

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 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.

Re: Intermediate Algebra

#78

Post by Miiko »

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?

Re: Intermediate Algebra

#79

Post by alexdowad »

gxs wrote: post #77
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 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.

Funny 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!

Re: Intermediate Algebra

#80

Post by red_trumpet »

young_unixer wrote: post #16
syops wrote: post #4

Earlier quoted context omitted.

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.

What should I read if I want to learn what a number is?

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.

[1] https://www.amazon.com/Numbers-Graduate-Mathematics-Heinz-Di...

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