Math with the passion of a THOUSAND SUNS

[Don Coyote]

Now how the fuck am I supposed to do inverse trig functions in my head and get an exact answer? I mean my old graphing calculator will spit out exacts, but my scientific calculator only spits out approximations for inverse trig functions. Evidently I am the only in my class who doesn't have a bunch of trig values memorized.

[Doktor Howl]

Now how the fuck am I supposed to do inverse trig functions in my head and get an exact answer? I mean my old graphing calculator will spit out exacts, but my scientific calculator only spits out approximations for inverse trig functions. Evidently I am the only in my class who doesn't have a bunch of trig values memorized.

Um, use the tables.

[Don Coyote]

Now how the fuck am I supposed to do inverse trig functions in my head and get an exact answer? I mean my old graphing calculator will spit out exacts, but my scientific calculator only spits out approximations for inverse trig functions. Evidently I am the only in my class who doesn't have a bunch of trig values memorized.

Um, use the tables.

We don't get to.

[Doktor Howl]

Now how the fuck am I supposed to do inverse trig functions in my head and get an exact answer? I mean my old graphing calculator will spit out exacts, but my scientific calculator only spits out approximations for inverse trig functions. Evidently I am the only in my class who doesn't have a bunch of trig values memorized.

Um, use the tables.

We don't get to.

Your teacher is defective.

[Don Coyote]

Now how the fuck am I supposed to do inverse trig functions in my head and get an exact answer? I mean my old graphing calculator will spit out exacts, but my scientific calculator only spits out approximations for inverse trig functions. Evidently I am the only in my class who doesn't have a bunch of trig values memorized.

Um, use the tables.

We don't get to.

Your teacher is defective.

I suspect this to be true. I won't be taking any further math classes with her instructing.

In other news, I found the button to give me exact values on my calculator. Why the default isn't to spit out exact values for everything I will never know.

[P3nT4gR4m]

I'm doomed.

Don't worry! There is fun math that's a lot less complicated. Complicated math is actually just a bunch of thin layers of simple math, except humans can't think that many layers deep at a time. The trick is that if you practice each layer until it becomes mental muscle memory - like the way you know that 3 + 7 = 10 without having to count - you can easily learn the next layer at a "competent" level. But if you only know a layer at the "competent" level instead of the "automatic" level, you can only learn the next layer at the "basic" level and the layer after that will be basically impossible. If you've ever read math and thought that it made sense to you at the time, but then couldn't explain it two days later, you've had the experience of trying to read at three levels above your automatic zone.

It's especially depressing if you're trying to learn math to do one specific thing, like quantum or stats or engineering, because you think you can build a thin tall tower that reaches to exactly the point you want to get to and save some time / effort, but that doesn't work. Towers of math knowledge collapse if they're more than two levels above the foundation. If you want to build your math up to where you can do quantum or stats or whatever, you have to lay thin layer after thin layer of foundation until you're almost there and then stick one layer of building on top. People think they suck at math because they can't build skyscrapers, but the truth is that the guys doing high level math cant build mental skyscrapers either - they're sitting on a step ladder that just happens to be perched on a foundation that's 40 stories tall.

The natural tendency is to practice one layer until it becomes easy and then move on to the next, but that's wrong. You can work at the easy level and even work ahead a level or two if you have enough timeand whiteboards, but you can't advance until the "easy" part becomes mind numbiingly boring.

This is an awesome explanation of learning anything. Not just math

[Q. G. Pennyworth]

Math thread.

Ask about:
analysis
calculus
topology
graph theory
game theory
abstract algebra (math over fields of functions or other weird objects that aren't normally thought of as numbers.)
vector spaces
predicate logic
lambda calculus
regular expressions
computability
set theory <-- actually probably the best starting point for learning math, even easier than algebra
number theory

This one, please!

[GrannySmith]

Teaching math is a bit of a fetish for me and a lot of other math people. The idea of a "pons asinorum" (bridge of fools) has been around since the Academy. The original Pons Asinorum was a basic geometry proof that a lot of students struggled with. It became a barrier to entry, a "You must be this good at math to learn geometry" marker. If you could cross the Bridge of Fools, you were intelligentsia material; if not, you were a Fool and should probably go back to farming pigs. As we've opened up more fields of mathematics we've found more humps that portions of the population apparently just can't get over. A doctor once told me that there are two kinds of smart people: those who can do calculus and those who can't. Those who can make good scientists and engineers, and those who can't go into medicine or law or some other prestige field that requires lots of intelligence but no advanced math. My grandfather is an example of that - he wanted to be a civil engineer, but after failing calculus three times he gave up and got a PhD in medicine instead. The Greeks didn't have a problem with the idea that some people are just fated to suck at math, but that's deeply offensive to a modern egalitarian. The only other explanation is that we - over two thousand years of mathematicians - suck at teaching math.

I also love teaching mathematics - though i don't work as a tutor anymore, i can't refuse when I'm asked to help somebody out - be it a friend or their kids at school. Last weekend I was having a conversation about how shit most maths teachers are (in my opinion, they are the reason so many people hate maths!) and a friend of mine had the idea that school teachers should take a year off for learning, every two years of working as a teacher. They could do any university course they liked, then apart of satisfying their need for new knowledge (i noticed a lot of good maths teachers stay at the university instead of working in a school, to keep enough time for learning), they could also have a fresh memory of how it feels to learn something new. In this way the teacher would be able to understand how the students might feel when they are learning something that for the teacher is "obvious".

I'm not sure I agree with that doctor on the calculus thing, i think that even the "worst" students could become excellent at calculus, given enough time and a good teacher. So yes, I guess we suck at teaching maths, and again, i suspect this is because we don't give enough incentive to the people who are good at it, to do it.

Definitely. I think part of the problem is that we teach math in order, which is stupid. You don't need algebra or even arithmetic to learn set theory and you don't need to derivatives to learn second-order functions, but we for some reason we refuse to teach math except in arithmetic > algebra > geometry > trigonometry > calculus > formal logic > everything else order.

I woud call that not "in order" Though for me this order also worked; during my undergraduate i started getting lost through all the different maths and then I found set theory - cantor's proof, what a revelation!  - and to top that, I got to learn everything from scratch - layer by layer.
Hmm, now that i see what i wrote it seems like my early undergraduate years would have been much easier if i got to learn set theory first 

[GrannySmith]

Right, I should have described it/thought about it better before i asked, i need theorems that talk not about the homogeneous (one colour) subsets, but the complete opposite, the subsets that contain elements of pairwise different colours. And "colourings of sets of natural numbers" is enough i guess.

I guess I don't understand enough about what you're doing to see what's interesting about it. You're coloring some numbers and then interested in rainbow subsets - those that don't repeat any colors. But a singleton counts as a one-colored rainbow, and those are super boring. Do you require that a rainbow subset on a n-coloring exhibit all n colors? Then the set of rainbow subsets given a given coloring ends up being the Cartesian product of all the colored partitions, so each coloring implies a specific set of n-dimensional vectors. Then you could ask about the structure of the rainbow vectors. You'll never get a nice vector space because you don't have enough zeroes to go around, but maybe there's something interesting there?

Hm, I guess I'm too caught up in another topic lately to properly formulate my question on this one - sorry for that, I try again. Say f is a colouring of N (or of a large subset of N, say about 7 billion ) and say f has n many available colours. For a subset A of N,  is there a relationship between the size of the subset B⊆A that contains only the colours that appear once in A and the number n of available f-colours? Or, what is the distribution of the sizes of B, for a randomly chosen A? (what about if A is "large enough"?) But then we go to statistics which is out of the thread topic i guess

What you wrote i find also very interesting, i didn't see the set X of those n-sized rainbow coloured subsets as the Cartesian product of the one coloured partitions before! Don't we get enough zeroes for interesting structures if we don't restrict the size of rainbow coloured subsets? I mean, X is not exactly the cartesian product - it's isomorphic to it (which of course can be thought of as "equal to"). If we think of the set Y of all rainbow coloured subsets (of any size), it's isomorphic (equal to) the cartesian product of each one coloured partition union the empty set-singleton. So we allow the empty set ø (for those sometimes missing elements) to become our cartesian product's --> vector space's 0, maybe that would give a nicer vector space? But then how to define addition and multiplication? Interesting stuff!

[this feels a bit like cheating: i guess one would have to distinguish between 0 and ø, or just not colour 0? ]

[GrannySmith]

Well, that depends on the axioms you assume! In some fields of maths the definition of 2 is 1+1, for (Peano) Arithmetic it's a theorem of two axioms, they are:
(1)  ∀x∀y(x+S(y))=S(x+y)
(2)  ∀x(x+0)=x
Where 0 is our only constant^*, + is a binary function^* (intended for addition), S( ) is a unary function^* (intended to signify the successor of something), x and y are variables^*, and for a variable x, ∀x means* 'for every x'.

So 1 is defined as S(0), that is, the successor of 0, and 2 is defined as S(S(0)), that is, the successor of 1.
And we want to prove that S(0)+S(0)=S(S(0)) from axioms (1) and (2):
proof^*:
By substituting^* x=S(0) and y=0 to (1) we get:
(3)   S(0)+S(0)=S(S(0)+0)
By substituting^* x=S(0) to (2) we get:
(4)   S(0)+0=S(0)
Because S( ) is a function^*, from (4) we get:
(5) S(S(0)+0)=S(S(0))
And by deduction^* from (3) and (5) we get:
S(0)+S(0)=S(S(0))

Thanks for reminding me of that

^* of course we should have started from predicate logic, languages and theories, defined what a variable and a quantifier '(for all)' is, defined what a formula is, defined the rules that we make deductions with, defined what a proof is, defined substitution to formulas, and defined what a function is, so including all that the proof would be much longer!!

I believe you have proved S(0)+S(0)=S(S(0)), but it is not a proof that 1+1=2 until 1 and 2 are defined as S(0) and S(S(0)), respectively.  This is a bit of a different perspective for me, though.  Thanks

I meant, this all depends on what axioms you assume and how you define 1, 2, +, and = actually, not only these, but one should really start from the very beginning, and define what is the underlying logic that will be used, and how the formulas are built, for example, in this case the basic (elementary) formulas are the ones that include an equality sign, and left and right from it are numbers, which are defined here as: "0 is a number", "for every number n, s(n) is a number", and "if n and m are numbers then "n+m" is also a number. Then one can define ways of combining them (using "and", "or", "not") and a way of making formulas with variables and quantifiers ("for every x", and then a formula which has the variable x in the place of a number).

Of course this is just one logic defined in order to work with arithmetic (actually in order to formally define arithmetic too), probably because it's the most fitting to sombunall's understanding and way of arguing about arithmetic intuitively. If one thinks about arithmetic differenty, or about any different subject, they could define a completely different logic to do it!

Mathematics is all about definitions, and all mathematical statements are "if ... then..." statements!

[Doktor Howl]

I'm all about yanking the hood open and seeing how the engine runs, but I can't (personally) abide this area of maths. 

[Mesozoic Mister Nigel]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

[Don Coyote]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

They accidentally the elder gods?

[Doktor Howl]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

They accidentally the elder gods?

They accidentally the fry cook.

[Mesozoic Mister Nigel]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

They accidentally the elder gods?

They accidentally the fry cook.

No, worse. They end up financial or other quantitative analysts. Once you crack that door to Hell, there's no getting back into the juicier end of research.

The pay is great, which is why it's a trap. Once you get in, you can't get out.