Math.

[Oblivious]

Need some help here =D

I have this:

cos x

Say I multiply x by 2 and get cos 2x, how does that affect the answer?

Of course you can just whip out a calculator and give me the answer, but when you do something simple like doubling the value, it seems it should be easy to derive the spewed out value without a calculator, but it's not.

Of course if you know what you're doing, you don't need one and that's what I want to know :o

And on another matter... regarding Binomials.

(a+b)^n

I found out that it is easier to just plot Pascal's triangle to the level of the required power of the binomial(n) and just remember that the powers of 'a' start out at n and decrease by 1 for every repetition, than to remember that long and tedious equation the exam paper is going to force out of me.

1) 11
2) 1 2 1
3) 1 3 3 1
4) 1 4 6 4 1
5) 1 5 10 10 5 1

In fact, that equation was just a generalisation of what I had just put forth just now. Yet the former was hardly considered to the point of nonexistance. Everyone seems hell-bent on just memorising the formula without caring about how it was derived no matter how shallow they care to explore.

Not only that, if I just use Pascal's triangle and remember how to handle the powers, thereby expanding the equation accurately at a fraction of the time it takes to do with the formula, I will probably be marked down. As opposed to the candidate who only cares about the gradings on his certificate, shoving the equation in his head for the duration of the exam, scoring full marks but forgeting it later.

The issue I have with the general equation, and equations in general, is that their making the students lazy. Just memorise this and that's Math.

No it is not.

Hell, the way it's being taught is lazy as well. Oh, and yes, this IS a rant.

[ObsessiveMathsFreak]

cos x

Say I multiply x by 2 and get cos 2x, how does that affect the answer?

There is a relationship between cos(x) and cos(2x). It's not a simple one. Strictly speaking:

cos(2x) = cos(x)^2 - sin(x)^2

It's not a simple linear relationship. Basically the value of cos(2x) depends on the value of cos(x) and sin(x). So you will need to know sin(x) as well. OR, you could compute sin(x) from cos(x) using the relationship sin(x) = sqrt( 1- cos(x)^2)

This might help, or maybe just confuse you more.
Your calculator calculates cos(x) using the following formula, known as a taylor series.(n! is n factorial)

cos(x) = 1 - x^2 /2! + x^4 /4! - x^6 /6! + x^8 /8! - .......

Of course if you know what you're doing, you don't need one and that's what I want to know :o

Yes and no. Yes there are formulae to help you calculate cos(2x) from cos(x), but no in the sense that even before calculators, anyone working with sines and cosines used log tables or slide rules. In fact Taylor type series were developed largely to precompute whole sets of log tables so people didn't have to compute cos(2x) from cos(x). Some work intensive computers still do this, i.e. instead of using the taylor series to compute cos(x) or various fiddling with formulae to get cos(2x), they have a precomputed lookup table of values.

In fact, that equation was just a generalisation of what I had just put forth just now. Yet the former was hardly considered to the point of nonexistance. Everyone seems hell-bent on just memorising the formula without caring about how it was derived no matter how shallow they care to explore.

Not only that, if I just use Pascal's triangle and remember how to handle the powers, thereby expanding the equation accurately at a fraction of the time it takes to do with the formula, I will probably be marked down. As opposed to the candidate who only cares about the gradings on his certificate, shoving the equation in his head for the duration of the exam, scoring full marks but forgeting it later.

What you mention is very unfortunate. Indeed, the student that did do as you suggest and write down Pascal's triangle, would be showing a much higher degree of understanding and proficiency in the question. The method is also as you say, faster, and is in fact more robust, i.e. less mistakes.

The issue I have with the general equation, and equations in general, is that their making the students lazy. Just memorise this and that's Math.

No it is not.

Hell, the way it's being taught is lazy as well. Oh, and yes, this IS a rant.

Your rant is justifiable in the context of ever decreasing proficiency in mathematics, I believe in most developed countries. This at a time when mathematics skills are needed more and more. I would put the blame on mathematics being learned by rote instead of by concept. I think the rot set in during the "New Math" debaucle, and a movement away from teaching basic numeracy to more abstract concepts. "Math" became "magic", instead of becoming understood.

Bless the offtopic forum.

Discuss this in relation to Neon Genesis Evangelion. (50 marks)

[Mr. Tines]

Your calculator calculates cos(x) using the following formula, known as a taylor series.

It's likely that it actually uses a different, iterative, approach (such as CORDIC); and in conjunction with range folding (at least so 0<=x<=π/2, and possibly further). I have in the past (~20 years ago) unpicked some microcode for such function evaluations which used a polynomial approximation and a couple of turns of Newton's method to refine the result.

While we're on the subject - do you follow John Baez (cousin of the more famous Joan)'s This Week's Finds in Mathematical Physics? There's some really fascinating stuff in there that makes me wish I'd persevered with abstract algebras rather than doing all the so-called applied courses I actually took in my third year.

[Oblivious]

Thanks for the help :o

The only thing rote learning is good at is achieving nice looking figures on sheets of paper. Whether or not true learning has been achieved is questionable. I guess learning itself falls on the individual, chiefly.

Too much is expected of teachers nowadays, we cannot expect them to teach us every single thing. Or perhaps what they are being told to teach is the problem.

Society has such strict laws regarding education(at least mine does), the only thing it has done so far is limit the growth of independant minds by forcing compliance to the system and killing the love of learning through rote memorisation. I have always been suspicious of man made rules, so I had no problems with this.:o

The effects of this can be seen especially in Mathematics, in which understanding and appreciation is highly important. When you are forced to memorise such abstract concepts you do not understand or have a need for, it is very easy to develop a hatred for it. It was only after a decade of suffering did I realise vaguely that Mathematics was very rooted in Science and that it's understanding is important to understanding our world.

Just what is our government trying to achieve by instituting such a system in which students are treated like robots? I feel like a mere pawn.

[The Eva Monkey]

My forum has been tainted by learning! NOOOOOO!!!

[Titus]

Whats worse, it's not english or history or anything, its math! I hate math! EM! DO sumtin! :shock: :cry:

[sachiel]

I pretty much failed my math final last year, and still passed, even though I rarely did my homework for that class.
Somebody must've liked me enough to pass me anyway. :o


...I really hate math.

[Mr. Tines]

Maths is more fun when there's not an exam at the end of it.

The most fascinating thing about it is quite how deeply things are intertwingled - there are correspondances between the most unusual things, such as between geodesics (the nearest thing to a straight line in a curved geometry) and prime numbers. Plus it generates the most amazing pictures

e.g. in http://math.ucr.edu/home/baez/week215.html

[sachiel]

The pictures can be pretty cool, but I'm still really really bad at math.

I'm more of an English/History person, I guess.

[Space Penis]

I didn't get into my university course because of my maths grade :x

[Oblivious]

My forum has been tainted by learning! NOOOOOO!!!

Why are the synchronisation graphs in Sine curves?

[ObsessiveMathsFreak]

Why are the synchronisation graphs in Sine curves?

Easier to generate? Amatuer bumbling follows.
[URL=http://imageshack.us][/URL]

[DatDude]

Math makes brain hurt !

[Mr. Tines]

Easier to generate?

Like the original second program after "Hello World!" always used to be printing out a sine-wave on the line printer.

[Zuggy]

Maths can be beautiful. They outta educate peeps using this

[Shnooks]

Yo, on the topic of school work...

Know any way to help me memorize the periodic table by Wednesday?

[ObsessiveMathsFreak]

Yo, on the topic of school work...

Know any way to help me memorize the periodic table by Wednesday?

Here's a mnemonic for the first 18 for a start.

Hell. Here Little Beatniks Brandish Countless Numbers Of Flick kNives.

Nagging Maggie Always Si-P-S Cl-Aret.

[Shnooks]

Yo, on the topic of school work...

Know any way to help me memorize the periodic table by Wednesday?

Here's a mnemonic for the first 18 for a start.

Hell. Here Little Beatniks Brandish Countless Numbers Of Flick kNives.

Nagging Maggie Always Si-P-S Cl-Aret.

Thanks.

I hope I can memorize that by wednesday. I dont know why I took this class.

[Mr. Tines]

Yo, on the topic of school work...

Know any way to help me memorize the periodic table by Wednesday?

Tom Lehrer to the rescue...

http://chemlab.pc.maricopa.edu/periodic/lyrics.html

and this has a post-1959 coda...

http://www.privatehand.com/flash/elements.html

Though why in this day and age anyone would want to rote-learn the table, I don't know