Is "College Algebra" really just high school "Algebra II"?

[sbrothy]

I also find the US notation unintuitive.

 

[symbolipoint]

I also find the US notation unintuitive.

When it is taught, it very quickly makes sense and becomes quickly understood when seen written and easily producible when writing.

 

[gmax137]

[2, ∞[

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

 

[Muu9]

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

[2,3] is a closed set - near 3, there's a point (namely 3) which is in the set but all of it's neighboring points to the right are not.

[2, 3[ (or [2,3)) is an open set - near 3, even though the points to the left of three are in the set, three itself is not, therefore there is no right-most boundary point where you can say "this is in the set but any numbers to the right of this are not".

Personally, I don't like the bracket-only notation, as it seems to suggest that ]2, 4[ represents the complement to [2,4] or the complement to (2,4) (i.e all the number outside that range) rather than just the open set (2,4).

 

[gmax137]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

 

[Muu9]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

No of course not, it's just notation, like x "approaching infinity" in the case of some limits.

 

[symbolipoint]

No of course not, it's just notation, like x "approaching infinity" in the case of some limits.

If infinity is one of the limits for an interval, this is not a clearly understood precise number or value, so that end of the interval remains "open"; so this is why the parenth is used to show that side of the interval (the inner curve of the parenth facing the infinity symbol).

 

[bhobba]

I'm again on thin ice here but from what I've heard from college mathematics introductions some start courses simply assume that pupils didn't understand or pay sufficient attention during high school and thus start from scratch quickly going through the basics just in case.

That happened in my degree in Australia. To get admitted you must have done Math B and C roughly equivalent to UK A Level Math which is Calculus BC plus a bit more. Yet we had Calculus A, which was just a boring rehash of HS calculus. The real shock came with Calculus B which was real analysis. I loved it, but most detested it. The latest I heard is they don't do that any more, starting with Probability and Stochastic Modelling, Abstract Mathematical Reasoning, Linear Algebra, and a Second Major Elective (all math majors where I went do a second Major, which can be an area of math like Stats, Operations Research or Data Science) first semester. Real Analysis was a 3 credit course when I did it. They replaced it with Abstract Reasoning (4 credits) which includes Real Analysis plus a bit more. They got rid of Analysis entirely for a while, which I found a bit depressing.

As far as HS goes likely math or associated majors such as Actuarial Science, Mathematical Physics etc is accelerated a bit and do (at many schools) the equivalent of Pre Algebra, Algebra 1 and 2, and Geometry starting in year 7 in 3 years instead of 4. Then Math B and C years 10 and 11 and 4 year one university math subjects senior year. Taking two Subjects over the summer means you can complete the degree in 2 years instead of 3.

Then I found something interesting. In the US 7000 students take the Calculus BC exam in year 8 or less, with over 50% passing. I thought what? I suspect we can accelerate calculus even further for better students.

To answer the original question in the Australian context if you have not done Math B and C at HS you do Math B as one 4 credit subject then Math C as another 4 credit subject except they call it by different names like foundational math etc. Many degrees just require Math B, some none at all. The thing I find depressing is that 45 years ago (god I am getting old) everyone, and I mean 100%, did the same subjects in years 11 and 12. English, Math B and C, Physics, Chemistry and either Technical Drawing or Biology. Tech Drawing was for those interested in engineering. Biology for those interested in being a doctor, nurse etc. Mathematics guys like me were advised to take whatever appealed - I did Technical Drawing.

Now, and it is a big worry, less than 8% take even Math B and even fewer take Math C. Sad, very sad.

Thanks
Bill

 

[Muu9]

The Queensland (i.e. not all of Australia) A/B/C curriculum was replaced with General Mathematics, Mathematical Methods, and Specialist Mathematics, respectively, in 2019. Students in New South Wales can take Mathematics Extension 2, which is more advanced than Queensland's Specialist Mathematics (formerly C).

 

[apostolosdt]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

Your remark is correct. I guess, no matter the context or the notation, including infinity limit requires special attention. Thanks for bringing it up.

 

[sbrothy]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

I learned infinity is always written as an open-ended set. Thus [2, ∞] makes no sense. Is the same true with the American notation?

 

[Muu9]

I learned infinity is always written as an open-ended set. Thus [2, ∞] makes no sense. Is the same true with the American notation?

Yes, in American notation it would be written [2, infinity)

 

[WWGD]

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

In theory, there's this Compactification of the Reals; one-point or 2-point compactification that includes one of $\pm \infty$.
https://en.m.wikipedia.org/wiki/Compactification_(mathematics)

 

[sbrothy]

Correct me if I'm wrong but, as I understand it, it's pretty much convention. Like they decided that infinity can never be inclusive. Much like you can never divide by zero. It's just the choice that makes the most sense.