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[3man75]

I've been in contact with a private tutor who has gone from 110$ an hour to 35$ an hour.

He claims that if I can give him time in 3 weeks and I work hard enough (which is "really hard") then I can understand mathematical concepts on my own to the point where i'll be able to figure new things out on my own.

It sounds very tempting (the whole things will cost me 350$ though) but i'm skeptical and wanted to ask if this is possible? I don't want to get sucered in by someone trying to make money/playing the "I know how you feel card". Don't get me wrong I WANT to learn math this well but for a large cost of 350$ I want to make sure it's possible first and not some fake magic pill similarity.

Thoughts? Also thanks reelya I actually got a similar question correct on a quiz today

[i2amroy]

Eh, tutoring can be very helpful... or it can be a complete waste of money. A lot of it depends on what you put into it, and the particular person who is tutoring you on a particular subject. That said if you are looking into tutoring I'd definitely look into other options than just a private tutor. Not sure if you are in college right now or not (you mentioned a major), but I know that the vast majority of colleges offer totally free tutoring, especially in lower level classes in things like math, physics, english, and so forth (and Professors also have office hours, and even TA's in classes often have office hours on top of that).

And even if you do decide to stay with the private tutor I'd still probably look into paying as you go instead of in just a single lump sum, it's totally possible that your learning style and your tutor's teaching just won't "mesh well" (I've known plenty of people in tutoring who have gone through a couple different tutors before finding one who works well with them), and you don't want to be out a couple hundred dollars if you end up looking for someone different after the first two lessons.

I am definitely going to agree with Reelya in that a large part of math is simply the memorization of a variety of different formulas and templates, and then being able to use pattern recognition to recognize which particular formula or template (or set of formula or templates) is applicable in your current situation. To steal the earlier example, the first step is just the memorization of the template "y=mx+b", so that we know exactly what each piece of the template (y, m, x, and b) does. Once we've done that, then it's just a matter of being able to recognize when "y=mx+b" is applicable to a line, and fill in the information that line gives you into the template.

I'm not going to lie, for some people the memorization step can be very difficult and for others it isn't. I have some friends who barely have to look at an equation more than once and they can use it perfectly, and others who drill and kill practice problems for hours to memorize it. It's up to you to determine how much practice you need to feel that you have memorized a given formula, and to look into potentially doing "extra" work if you feel you need it. Similarly the "recognization" step ease can be different for different people. I know some people who can look at problems and recognize the basics of how a problem can be broken down in seconds, even if they don't actually remember the formulas they need, and others who need to manually review each of the different formulas to see which one is applicable to their given solution. It's up to you to recognize how much practice you need on a given formula or on the recognization of its use, and to keep practicing until you reach the point where you can do things without putting a huge amount of thought into it. The fields of math are just like any other learned skill, everyone starts out as a beginner, and it's only through constant practice that people get better at it. Math is full of similarities which help the process, just as how learning to drive a car will help you learn how to drive a big truck, but it still comes down to a basic level of simply doing tasks over and over again until the pathways for them become hardcoded into your brain.

[Helgoland]

I am definitely going to agree with Reelya in that a large part of math is simply the memorization of a variety of different formulas and templates, and then being able to use pattern recognition to recognize which particular formula or template (or set of formula or templates) is applicable in your current situation.

That's one way to do it, but not a particularly good one. More important than pure memorization is understanding where the formulas come from, how they are derived - that way you can adapt them if you hit a more complicated situation than you're used to.

[Bauglir]

What I've found is that you need to memorize the notation, as well as any formulas whose derivation is beyond you (for example, the quadratic formula for most people). Otherwise, it's best to treat it as any other kind of reading. I try to build up abstract blobs of properties about, say, variables, in the same way I do characters in a story, only instead of knowledge like "Macbeth kills Duncan" (spoiler alert), it's knowledge like "a = b + 7", and that also implies knowledge about what = and + mean. In the context of a class, often the particular stuff I need to know is readily available in recently-formed blobs got from studying (ones more like the = and + blobs than the a blob). And what isn't can usually be derived on the spot, unless I forgot something like the quadratic formula blob that has to be recalled whole and without reference to others (which is hard).

In my experience, programming tends to work the same way. What's actually important is the information encoded in the written text - but to get at it you need to understand notation, and you also need to have some way of black-boxing the information your brain doesn't know how to deal with. But then, that's just my approach, and I cannot guarantee it works for everybody.

Also, question - I'm looking to encode words as numerical vectors. My current plan is to have each element of the vector represent a character (using ASCII as the map; for now words with non-ASCII characters are discarded), and to store positions by summing powers of 1/2. Having butchered that explanation, perhaps an example:
"Hello" would be represented by a vector of mostly 0s, with the following exceptions:
element 72 = 1/2
element 101 = 1/4
element 108 = 1/8 + 1/16 = 3/16
element 111 = 1/32

The question is, is there any way this can cause me to lose information? In other words, am I guaranteed to be able to reverse the process? In still other words, does each distinct vector uniquely encode a distinct word? I think so, but I'm not actually sure. Also, I wouldn't mind any terminology corrections, cause I am super bad at this stuff right now. Which, given what I said before the question, is a problem >____________>

[frostshotgg]

Bauglir, I don't understand where those vectors are coming from. They seem relatively arbitrary. If they're all arbitrarily chosen, then you need a dictionary to translate them back 1:1 and there's no advantage to having them in that form.

[Reelya]

Another good resource are video tutorials. We live in the best era in history so far for access to free education materials, so take advantage of that. Often you will find that something is easier to understand when you can hear it explained, rather than reading it in a book. Follow this up with doing excercises that match the videos' content, then wait a bit, and read it in text form as well. Then do some more exercises and/or watch more related videos. Having something explained a few different ways can help you get a full grasp of the concepts.

Bauglir, I don't understand where those vectors are coming from. They seem relatively arbitrary. If they're all arbitrarily chosen, then you need a dictionary to translate them back 1:1 and there's no advantage to having them in that form.

That's fractional binary, which is assuming infinite precision since they're theoretical numbers. But if the number is a real number, you can just use regular binary to store the positions. Either fractional or normal binary will encode bits of data exactly the same, in principle.

But you don't even need a vector to encode the data if you assume an infinite precision number. you can any encode 2 real numbers in a single real number by counting every 2nd digit as one number or the other. And you can interleave an arbitrary number of different number's digits together, then decode that to get all the original numbers. So with infinite precision you can store almost any set of numbers you want in a single number. (Cantor's higher infinities not withstanding).

[Bauglir]

Yeah, I suppose fractional binary is how I'd describe what I had in mind. I need fixed-length vectors of numbers between 0 and 1 for what I'm planning to do (which amounts to shoving them into a senseless, thrashing neural net and seeing what happens, if you're curious). I suppose it does break down once I exceed the precision that I can handle; it may be necessary to store them as ordinary binary and only convert to fractions when necessary for scaling reasons.

My first thought was to use ordinal numbers, but I realized that once you sum those you won't be able to distinguish things like 1/4 from 1/6 + 1/12. My next thought was to use sequential primes, but I realized that reversing that encoding would (I think) amount to factoring, and while I don't expect to encounter any words long enough for that to actually pose a computational problem, I want to avoid it on principle because what really happens and what I expect to happen are not necessarily related. So my next thought was this, which I think you can fairly easily handle in time linear in the length of the word? You'd just run down each element in the vector and subtract each power of 2 less than the element's value until it equals 0, and the positions the character appears in are the powers of 2 that you were able to subtract. I think. I'm using very sloppy notation, describing it in natural language like that, but I don't know the mathematical notation off the top of my head. I guess I could put it in pseudocode but this post is already running longer than I'd like it to.

Anyway, thanks!

[frostshotgg]

I'm not sure I follow what you mean with vectors. Those seem like normal numbers you're using there, not lines to something. You could maybe do some shenanigans with literally writing words in coordinate space, then converting to ordered pairs, then doing math and reading them back.

[Bauglir]

I guess it might be easier to think of 'em as single-column (or single-row) matrices. Near as I can tell that's the same thing.

[frostshotgg]

So basically you have a list of numbers you put in. That's all.

Anyways, this is starting to sound more and more like a weirdass comp-sci question, so more appropriate for the programming thread. Probably ask there, the weirdass ways you could compress a word are definitely a comp-sci thing.

[Bauglir]

Yeah, it is ultimately for a weirdass comp-sci thing, but the thing I was wondering about was the mathematics. Now that I'm more awake, I think I can word it better -
"Does the result of a sum

x₁*(1/(2^1))+...+x_n*(1/(2^n)), where each x is either 0 or 1

provide enough information to determine a unique sequence

(x₁, ..., x_n)?"

The question's been answered, I think ("Yes, it does"). Maybe that wording's no better.

[ZetaX]

The answer is indeed "yes" (assuming you are talking about the actual number, not a floating point number on a computer), but depending on the actual situation (what do you want to use it for¿) it might still be suboptimal.

[Bauglir]

Ah, well, I'm using it to encode a word's orthography before shoving it into a recurrent neural net for it to tell me whether or not a tag (as in the sorts of things we use to format text around here) ought to have been present sometime before it in the sequence of words. That's the weirdass comp-sci thing. It's not the only information I'm sending in, and it's probably got some disadvantages - for example, I'm possibly implying that the further you get into a word the less its letters matter for specifying what word it is. Which probably isn't, strictly speaking, true, what with the existence of prefixes and all. But at this point I'm just looking for something that will hopefully work well enough to function; in the unlikely event that I produce something of value, either I or some engineers who actually know what the hell they are doing can optimize it later. And that will probably be in the programming thread.

[Tylui]

I'm currently doing lines and rational expressions. Stuck on two different problems for both and rationals are really kicking my a$$.

For rational expressions i'm stuck on something thats like:

1+1
     x
-----------
1-1
   x

That is 1 plus 1 over x. All of that is over 1 minus 1 over x.

I just have no idea how to deal with fractions this way...

The second one is a line probelem where the problem gives you two points (-3,4) and (2,5). I'm suppose to "express your answer using either general form or slope intercept form.". I imagine getting the slope but that I can do using Y2 - Y1 over X2 - X1.

I'm a little bit late, but I think seeing it written out might help?

Spoiler (click to show/hide)

[Reelya]

I'm not sure I follow what you mean with vectors. Those seem like normal numbers you're using there, not lines to something. You could maybe do some shenanigans with literally writing words in coordinate space, then converting to ordered pairs, then doing math and reading them back.

"vector" in the general sense of an n-vector which is an ordered set of numbers. Vector math doesn't always represent distance and direction. Think general linear algebra where you often multiply a matrix by a vector to get some result. Often the vector isn't representing a direction at all. In those cases the vector can represent solutions to a set of linear equations or the like, which don't have any specific relationship to spatial coordinates, even if they can be interpreted as doing so.

What bauglir is talking about in mathematical geometry terms is an n-dimensional space, where each dimension represents a different character of some code. Each possible word would be represented by a unique point in this space, and for a specific dimension, say the 'k' dimension, you interpret the point in base-2 notation, and the 0's and 1's of that notation indicate where in the word the k's occur. This is all perfectly reasonable and logical. Even if it's not at all practical.