Having Problems with Math

Posted: Sat Jul 04, 2009 2:01 pm

I wrote this to my mentors and fellow students today concerning how I feel about math. have any of you guys felt the same way but still managed to pass your college level math classes? If so.. how?

Quote:

I am very worried myself. None of the Math makes any sense to me at
all. None of the tutorial sites make sense to me either. I never was
good at math and I truly have no idea where to begin. It's like some
alien language to me. I need to know "why" I'm doing something from the
very beginning. Otherwise with Math, I fail at everything. I most
certainly cannot write essays about things of which I don't even
understand. I am provided tons of examples all over the internet but
not one text book I have skimmed is telling me "why". Once I
understand the "why", I can get the "how". Seems like this is skipped
among every single resource I have found. I just don't know what to do.
Are there any "good" resources out there for Math illiterates like me?
For instance, "why" does i = -1 or the square of -1? Some resources out
there even contradict this statement. Some professors say that "i"
should never equal the square of negative 1. All very confusing.

Intern Joined: Feb 13, 2004 Posts: 297 Location: Belfast

Being from the UK I'm not entirely sure the level of college maths in the States. I do understand and sympathize with your situation though, as for many calculus and up can seem a bit impenetrable and terribly abstract; especially that square of -1 guff.

The why on the other hand is an entirely different matter. Perhaps the single biggest unifying factor in the sciences in the application of mathematics, it's the core languaged shared outside of philosophical reasoning. To understand an observed phenomena we need to be able to rationalise it into a model, one that can be universally shared both in time and space from the origin. The only way to do this accurately is to convert it into a graph with the associated algebra, geometry and trig functions. Only with the model in place can we conduct subsequent experiments to determine how the real world differs from the hypothetical model; and in doing so develop more accurate models with a greater level of understanding. In this way the "why" is that Maths presents a concise way to represent these models; or in other words draw accurate lines on paper.

Observations and measurement on the otherhand are ultimately meaningless without models through which to understand them.

For example, we could be describing leaf structure as a relationship of length and width; changes in generational abundance of genotypes in a population; how fast a reaction takes place or develops; how a population sample can differ from the population. I could wordiosely describe all of these things with prose, however it is much simpler and easier to understand when they are presented in a visual format. These examples could be represented on paper by something as simple as a straight line a=xb+c, it could be a parabolic curve ax²+bx+c=f(x), an exponential a=b^e or a series SumA_n=a0+a₁+a₂+a₃ to name but a few. When I'm presented with a mathematical model I can begin to imagine in my head and subsequently reproduce on paper a graphical reconstruction; which is to a certain extent an accurate representation, something that prose could never achieve. For example my window is a four sided shape allows for a plethora of interpretations however by giving you to geometric rule of the height x being equal to the two times the width y (x=2y) you can accurately model its dimensions; subsequently defining the angles of sides at 90^o eliminates any four sided shapes like diamonds, rhombi, parallelograms or shapes drawn on spheres. I can then examine other windows to determine if they fit my model, or if it needs revision. Being able to take real world observations and integrate them into a formula or derive values then becomes a universal language free from subjective interpretation.

In this way Maths in Science is not so much about numbers as it is about model construction, and graphing. The best way to do this of course being algebra, trig and geometry. For me this didn't click until I started really studying the models of Hardy-Weinburg equilibrium in mendelian genetics. I understood punnet squares intuitively, but recognising the equation as quadratic allowed me to fully rationlise the model and subsequently understand the implicit assumptions and caveats.

I will say one thing IG, I didn't understand this stuff in school. It was very much a black box of just-so stories; do this and this happens repeat twenty times kind of thing. I invested some money I earned in my final year of school on a Maths tutor, a really top bloke who had spent most of his adult life working in applied maths. It was only after this that the rather abstract concepts started to click into place. The only reason I got put in touch with this guy is because I approached my physics teacher, who knew clearly my level of understanding and was subsequently able to point me towards the correct person for additional tuition. Had I been lower or higher I have no doubt I probably would have been directed elsewhere.

Websites and books are helpful for sure, especially if you can find one directed to your level of understanding; but nothing beats one on one tuition particularly with Maths which can be quite abstract at times. The advice I could give is find the lecturer who is most likely to be empathetic of your situation and see if she/he can offer a practical solution for you bud.

Posted: Sat Jul 04, 2009 9:51 pm

I definitely know I need someone to help me. Who? I have no idea.

Posted: Sat Jul 04, 2009 9:58 pm

Reg,

You didn't seem to ask for an explanation of square_root(-1), but I'll throw one in anyway. The cause of the confusion is that there are two different number systems being used. In one of these systems, square_root(-1) is illegal, and in the other it is OK. So you will hear some people say square_root(-1) = i, while others will say square_root(-1) is illegal.

Everyone outside of the math department will assume the number system that works best for them and say you are wrong when you use the other system. Hopefully the people in the math department know the difference.

The two number systems (the proper name is sets) are real numbers and complex numbers.

The set of real numbers are what you consider most 'normal' numbers. 2, 6.8, pi, -4, etc. These can be represented as points on a line. All the standard arithmetic rules apply to real numbers. Standard addition, subtraction, multiplication and division all apply here. Multiplying two positive numbers produces a positive number and multiplying two negative numbers produces a positive number. Because of this, there is no square root of a negative number. When people say square_root(-1) doesn't exist, real numbers are what they are talking about. Squaring any real number yields either 0 or positive. Never negative.

Complex numbers are a completely different animal. Instead of a number line, they are points on a 2 dimensional plane. You can define all of the same operations on complex numbers as you can for real numbers. This is why it is so confusing. Addition, subtraction, multiplication, division, etc. are all defined for points on this plane. In fact, the operations are defined so that for points that are on the x axis (i.e. the imaginary or y component is 0), the operations act just as the operations on real numbers.

A couple of examples may help:

Real numbers:
2 + 4 = 6
4 * 6 = 24
-3 * -2 = 6

Complex numbers - they have two components: a real and imaginary component. They can be considered the x and y components. The vertical or y component is generally labelled with an i to indicate it is imaginary. i equals a distance of 1 in the vertical or imaginary direction.

So, for complex number examples:
(2, 4i) + (1, 3i) = (3, 7i). Addition simply adds the two components.
(2, 2i) * (3, 3i) = (0, 12i). Multiplication multiplies the two lengths and adds the two angles (don't worry about the details here).

To add to the confusion there is an implied 'upgrade' from real to complex. For any real number, you can just stick it on the complex plane by adding 0i to it and all the math operations work the same. So to 'upgrade' the real numbers in our previous examples we get:
(2 + 0i) + (4 + 0i) = (6 + 0i)
(4 + 0i) * (6 + 0i) = (24 + 0i)
(-3 + 0i) * (-2 + 0i) = (6 + 0i)

Because of the way multiplication works in the complex 2d plane, every point on the complex plane has a square root. Even the points that correspond to 'upgraded' real numbers. The positive upgraded numbers e.g. (4 + 0i) has the square root values (2 + 0i) and (-2 + 0i).

Now we are at the real beef of the confusion. Numbers on the complex plane such as (-1 + 0i) have square roots.
square_root( (-1 + 0i) ) = (0 + 1i) and (0 - 1i). However a complex number is not the same as a real number. In speech, people treat the real number -1 and the complex number (-1 + 0i) as the same. They are not the same, even though they have a lot of similarities.

When people talk, they forget to be explicit. The real number -1 does not have a square root. However, the corresponding value on the complex plane (-1 + 0i) does have a square root. People, even mathematicians, get loose in their speech and forget to be clear. So people just learning get all confused because they here two incompatible statements.

I hope this helps.

** edit **
I don't have any resources. I'm the hated person who finds math easy. Feel free to ask me anything if you think I can help.

Intern Joined: Feb 13, 2004 Posts: 297 Location: Belfast

I think thats the best way I've seen complex numbers described, thank gawd I don't have to deal with them everyday. What level of maths is giving you trouble reg? It might be that the resource is right here if you need it.

Just Arrived Joined: Jul 04, 2009 Posts: 1

For myself I couldn't understand math after calculus. I don't really feel I really understood much of anything from calculus but I was able to pass my classes. Last semester I took a class on Differential Equations and Linear Algebra. I didn't really understand the "why" for any of it and was really confused with vector spaces. I stopped going to class, I pretty much gave up. If anyone could explain anything about vector spaces in a way I could understand it I could really use the help. I did want to become a math major but because I dont feel like I will grasp any math beyond calculus I don't think it will happen.

Posted: Sun Jul 05, 2009 4:24 pm

Reg, I too suck at math.

But in my twenties, I began to see that really math is merely representations of pattern that exist in the world around us. It is an abstraction of concrete things, as Geneboy says above (I think).

I may not be as good a resource as it seems that GB and DB might be, but I'd like to throw in my 2¢ in regard to math and education.
It seems to me that if math was explained to younger (teenaged) students as a functional means to an end, then it may make more sense, and be more interesting to them. For example, here is the breakdown of how the computer monitor in front of you is working.

(Very simplified.)

The monitor is only 3 lamps, similtaneously shining onto one point (pixel) on your monitor. Red, green, and blue lamps create a convergence of their lights at the single pixel-point on the back of your screen.

Each lamp can shine at 256 different levels of intensity, and this is enough variance to re-create (most of) the colours that the human eye can see.

The math comes in when an programmer, using a computer, must instruct the monitor how intense to make each lamp to create, say, an orange pixel.

The computer has only two words in its language: "on" or "off" (ie. a burst of electrical current, or a lack of current, if you will.)

In math, we would represent these computer language "words" with a "1" or a "0". (aka a "bit")

How does a two-digit machine tell a 256-level lamp to shine at a specific intensity?
It needs to say eight words (aka a "byte")

10101010, or

2⁸ = 256

Voila! The math that makes the screen in front of you work.

It's a simple illustration (and a bit primitive, thanks to my limited math skills) but I think that it provides a window into the arcane world of mathematics. Math is a difficult subject for many of us to grasp. I often wonder if our math teachers had started the year off with an illustration like this, if more of us wouldn't have taken more interest in the subject, knowing that it had practical applications.

What do you folks think?

Posted: Sun Jul 05, 2009 5:43 pm

my two pennies ...

@ the original post

You asked why. Here is an example (-1) is a complex number. Why? to find the square root of a negative number. Why? To solve complex plane problems etc. Does this help? Probably not.

You might want to go to the libray and get basic simple Highschool math books. The Princeton Review set is very basic but helps my girl. Every University has at least some form of a tutoring program, real dialog with real flesh is still the best way to learn, I think, because you can ask questions like: What do you mean negative number?

Also, forget your logic. You will go nuts trying to logically comprehent a negative number, it does not exist, it can not be explaint in those terms. It's a tool invented by man like all other aspects of math.
btw. just thougth about your logic, yes, +1 is an abstract concept also, it does not exist.

Posted: Thu Jul 09, 2009 2:46 pm

Folks that wish to help me please shoot me an email off site: infidelguy@infidelguy.com

Thank you.

Posted: Mon Aug 23, 2010 12:23 pm

An interesting approach to the opening post here.

Posted: Mon Sep 27, 2010 12:07 am

Came across this recently, wow. It looks too good to be true, but unbelievably, isn't. Free.

Khan Academy
http://www.khanacademy.org/

His Youtube Channel
http://www.youtube.com/user/khanacademy

Intro video;
http://www.youtube.com/watch?v=dsFQ9kM1qDs&feature=player_embedded

Wiki page
http://en.wikipedia.org/wiki/Khan_Academy

It's amazing to watch this guy chart out an overview of his courses. The freehand nature of his presentation technique says a lot about how clunky the computers and software we currently use are.

Posted: Mon Sep 27, 2010 8:16 pm

iPondR wrote:

Came across this recently, wow. It looks too good to be true, but unbelievably, isn't. Free.

Khan Academy
http://www.khanacademy.org/

His Youtube Channel
http://www.youtube.com/user/khanacademy

Intro video;
http://www.youtube.com/watch?v=dsFQ9kM1qDs&feature=player_embedded

Wiki page
http://en.wikipedia.org/wiki/Khan_Academy

It's amazing to watch this guy chart out an overview of his courses. The freehand nature of his presentation technique says a lot about how clunky the computers and software we currently use are.

One should wonder how much further advanced our civilization might be if we hadn't commercialized the learning process. This Khan guy is great Very Happy

Posted: Mon Sep 26, 2011 1:10 am

Why is i the square root of -1 (rather than what you said)? Because it was helpful to say it is.

I read an article about i a few months ago and the writer made it very clear. But I was reading for fun. If I was reading for university, I'd figure that I was learning to answer their questions so they can continue with their mission of churning out robots for industry.

Universities also keep people out of the job market for a few more years.

I may sound kind of down on institutes of higher learning, but I'm not that impressed by them overall. There are some excellent things that they do, but I do sometimes wonder why people need to be taught those things in a formal educational setting.

When I was young (way back in the 70's) I trained myself to use computers because the college courses at that time had not yet come out of the stone age. I learned whatever I needed to learn and got a job in the computer industry without any problem. Eventually I became a computer consultant and I did that for about 25 years.

Now I fight crime.

No, wait, that's Batman. Anyway, whenever I want to know something I read up about it. I only have high school, but for my computer work I've learned about advanced statistics, solution of simultaneous equations, calculus, trigonometry, blah blah blah. If somebody wants a solution badly enough, they'll wait while I learn what I need to know.

However, the business world isn't set up for that kind of approach. They're in a big hurry for their robots. So there's university. And if they say i is the square root of -1, you don't need to ask why! Just think, "I can get a job if I pass this test!" It might even be true.


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