Good place to learn assembly?

[helios]

In geometry, we'd need to know the names of theorems and such.

In my logics class, you may find a question worded like this, "could ____ be derived using the ___ rule?" So if you don't know the rule by name, you're screwed. Thankfully it was an online course.

You're complaining about remembering names of theorems? The final for Analysis I at my school (it's the course all physics, mathematics, and CS students take) includes remembering proofs. You have no idea what pain is.

A mathematical proof is an argument for a proposition. If you use a theorem without proving it and without naming it (so the reader can't look the proof up), the reader is entirely within their right to disregard your work on the basis that you might be making shit up. If you use a theorem you need to at least be able to name it, otherwise you're expecting the reader to know the statements of all known theorems.

EDIT: But in your defense, IMO all mathematics exams should be open-book. The point should be to test how well you understand the material, not how good you are at memorizing trivia. The point of paper is not relying on fallible human memory.

[zapshe]

You're complaining about remembering names of theorems? The final for Analysis I at my school (it's the course all physics, mathematics, and CS students take) includes remembering proofs. You have no idea what pain is.

Trust me, I may never know what true pain is in academics. I breeze through all my classes. Even when I complain, it's because I had to do some modicum of work. I assume colleges elsewhere (like where you are) may be significantly tougher. I'll gladly take the lazy way and use to free time to learn more languages and such myself.

Moreover, I complained about remembering the names of the theorems because I already knew what the theorems were saying. In fact, whenever I forgot the name I'd just write down basically what the theorem was about instead for points in geometry. In the logics course, I didn't have to memorize anything, half the answers were online. I did have to know how to use the theorems "practically" for the lab portions though.

If you use a theorem without proving it and without naming it (so the reader can't look the proof up), the reader is entirely within their right to disregard your work on the basis that you might be making shit up.

Never had to prove a theorem before, the whole point of a theorem is that it's already proven.

If you use a theorem you need to at least be able to name it, otherwise you're expecting the reader to know the statements of all known theorems.

If they test me on these theorems, they better be able to identify which theorem I'm talking about if I end up writing it out. I hate it most when even the professor doesn't know the things they test on by heart, yet expect us to know it.

EDIT: But in your defense, IMO all mathematics exams should be open-book. The point should be to test how well you understand the material, not how good you are at memorizing trivia. The point of paper is not relying on fallible human memory.

My thoughts as well. If it was open book it would make students try to actually learn the application of the material and understand it rather than trying to memorize and eventually forget. In fact, for my Calculus classes I'd only study a day or two before the test and then right before the test - then it all disappears pretty quickly after I'm done. I should thank my stage acting days for giving me practice on how to memorize quickly.

[jonnin]

early geometry with a few proofs, like triangle and angle proofs (this angle equals that angle because intersecting parallel lines blah blah or that triangle equals the other one via side angle side) are nothing like the formal stuff that a heavy proofs course will throw at you. Neither are the occasional 'prove it' problems in a normal math class. Those are just to introduce you to the ideas.

one of the first, and easy ones that I recall from my first proofs class was this:
show that there are more numbers between 0 and 1 than there are integers.
which you eventually show that 1/1 and 1/2 .. 1/3 1/4 1/5 missed a bunch of numbers like 1/1.137 and 1/4.62 but stating that formally takes a solid half page. Even if you had all the 1/ns to infinity, there is still at least one more that was missed, in other words. they got harder, of course. Any average person over the age of 10 can see the answer, but making the professor happy with how you state it is a whole new ballgame.

[zapshe]

are nothing like the formal stuff that a heavy proofs course will throw at you

I figured, lots of friends struggled and some failed. I do hear some people find it a breeze though.

Any average person over the age of 10 can see the answer, but making the professor happy with how you state it is a whole new ballgame.

This is a life lesson for all classes !

show that there are more numbers between 0 and 1 than there are integers.

I don't think this is true though. While there are an infinite amount of number that can be made from values between 0 and 1, there are infinite values that can be made from integers. Whenever you find another value between 0 and 1, you just add another 1 to your integer.

It would make more sense if the proof was to prove that there are as many numbers between 0 and 1 as there are integers - which would force you to prove that both values are infinity.

[helios]

I don't think this is true though.

Yes. The naturals, the integers, and the rationals are all countably infinite. The set of all reals and all continuous subsets of reals (e.g. [0;1], [0;inf), etc.) are all uncountably infinite.
You can comfortably fit all the integers in [0;1]:

...
-2 => 1/4
-1 => 1/3
0 => 1/2
1 => 2/3
2 => 3/4
...

but [0;1] doesn't fit in any countable set.

Consider all the possible strings you could make with the Latin alphabet, including repetitions ("aa" is valid) and reorderings ("ab" and "ba" are different strings). That set is countably infinite.
An uncountable set would be the set of all possible infinite strings that use that same alphabet.

The reals are pretty wacky. Not only are continuous subsets of them the same size as the entire set, almost all reals are irrational, almost all reals are transcendental, and almost all reals are incomputable.

[zapshe]

Countable and uncountable infinity refer to a concept really - but both still go to infinity.

While I understand the concept, that you can have an even smaller subset of [0;1] like [.1;.2] and have another completely infinite set - therefore there are (possibly) an infinitely many number of sets within the infinite set [0;1]. This means there are more infinite sets within [0;1] than the single infinite set of integer numbers.

However, both [0;1] and all integer numbers will converge to infinity, meaning there should be a 1:1 ratio. To say either set has more numbers than the other would be saying one infinity was larger than another.

[Ganado]

Just wanted to say, I find it funny that a discussion about assembly (pretty down to earth, what-you-see-is-what-you-get) turned into a discussion about uncountable sets and other far more abstract ideas.

Yes, one infinity is larger than the other.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

[helios]

While I understand the concept, that you can have an even smaller subset of [0;1] like [.1;.2] and have another completely infinite set - therefore there are (possibly) an infinitely many number of sets within the infinite set [0;1]. This means there are more infinite sets within [0;1] than the single infinite set of integer numbers.

This is irrelevant, because Q can also be subdivided into sets with the same number of elements as the whole. Q∩[0;1] and Q∩[0.1;0.9] have the same number of elements as Q, yet Q is still countable.

However, both [0;1] and all integer numbers will converge to infinity

This is a meaningless statement.

meaning there should be a 1:1 ratio

Nope. You can map the integers into the reals, but not the reals into the integers.

To say either set has more numbers than the other would be saying one infinity was larger than another.

This is correct. Some infinite sets are larger than others. In particular, the set of integers is said to have cardinality א₀, while the set of reals is said to have cardinality א₁, where א₀ < א₁. Look up Cantor's diagonal argument for the proof of this inequality.
Whether there can exist sets larger than the integers and smaller than the reals is an unsolved problem.

EDIT: It's possible your browser is rendering the above incorrectly (it does for me). It should be Aleph_0, not 0_Aleph.

[jonnin]

If you are going to take this class, the concept of different infinitys needs to be grasped soon. Have you had that annoying math class/section on convergence and divergence of series? It ties back to that somewhat.

[zapshe]

Have you had that annoying math class/section on convergence and divergence of series? It ties back to that somewhat.

Yes, the issue was that none of it was explained in terms of why it was true. Remembered it long enough to use it on the problem and none of it was explained or proved.

This is correct. Some infinite sets are larger than others.

Nice, infinite sets are something I just heard of in this topic, so I suppose I had the wrong impression.

If you are going to take this class, the concept of different infinitys needs to be grasped soon.

Won't be an issue I imagine.