Hey, Lets start some math discussion!

[Tim_Lou]

lets ask:

what is the meaning of numbers?

 

i personallly believe that humans create numbers.

back in the acient day, the early humans see that there is a sheep, a cow, an apple....they start to realize the meaning of one...

 

hmm....one thing weird is that why is it 9+1=10?

what not something like A? why do we use "10" as standard?

 

the nature is making some sort of sense out of numbers, hmmm....what if we all use another system like

oct, hex.....

how about this, make it that c=1, maybe the world would come out with some nice numbers!

 

think about this, does other animal know the meaning of numbers? do they have an idea of 1~10? i know some mammels do understand them.....so, is number "natural", or "artifical"?

(any comments?)

[Bo]

as for the pure linguistics: The meaning of a number is exactly what semantics tells you what it is. So we can easely from now on call all 10's A (or X to use the latin number system).

As to what a number means: in my opinion a number is just some property you can give to an object. For example: 1 apple is green (sort of). round (sort of) and we have 1; 2 apples have the same properties, but then we have 2 of them (that means:1 apple and 1 apple). So a numbering system is just a system to order the imaging of the world in our perception (or something like that)

i thnik the above definition is also unconsciously known by most intelligent mammals.

As for calling c=1 (i suppose you mean the speed of light): That is something which we in high energy physics do all the time (we also take h, planck's constant, equal to 2pi). You can easely do this, because physical observables (as c or h) are dependent on the scale you use. Take for example the difference between meters and inches, or Celsius, Fahrenheit and Kelvin. The main point is that the scale you choose doesn;t change your physics (as long as you're consistent).

There are however some numbers which we just *cannot* change. e.g. pi, e (the base of the so called 'natural logarithm'), gamma (the euler constant) and probably there are some more.

what these numbers 'mean' or where they come from is as far as i know still a mystery...

Bo

[Bo]

And also to say something about the formula you gave: i wondered how did you derive this? There is a general formula for these kind of problems (the so called Faulhaber's Formula see e.g: http://mathworld.wolfram.com/FaulhabersFormula.html) But that's quite hard to derive in general.

 

I tried to prove your formula myself and after some work (considering (n+1)^5 - n^5 for all n) i got an expression of the fourth power problem in 3, 2, and 1 power sum problems. These can eventually by the same technique be expressed in the well known (and easy to derive) 1st power sum sollution: [sigma( K=1...n)]K = 1/2(n*n+n)

substituting all gives your (and mathworld's ) formula

 

is this anything near your proof? or have you just done this by trial and error?

if you want the complete written out proof: just ask me and i'll write it down.

 

Bo

[Freethinker]

Originally posted by: Bo That is something which we in high energy physics

 

Bo

 

Really? You mean I have to stop faking it? lol

[Bo]

yups

 

This is something which is called 'natural unit's. in this way you save a lot of time (just writting E=m instead of E=mc^2) (the fact that you write h=2pi also saves time, because 'almost' always h is devided by 2pi in your formula)

Also a nice convenience of natural units is that all physical quantities are rescaled in such a way that they can be expressed in energy only: e.g. If we devide all our momenta by c (which we easely can doe, it's 1 after all) then the unit of momenta becomes [kg*m/sec --> Kg = Energy]

So this is a big simplification for a lot of formula's (that actually still look horrible...)

 

Bo

[Tim_Lou]

error

[Tim_Lou]

error

[Tim_Lou]

error

[Tim_Lou]

error!

(last one, phew..)

[Tim_Lou]

thx, bo...

 

all i want is to carry out a math discussion....thx

 

 

hmm.....how about this, i dont understand how pi is measured. is it simply use a ruler and stuff?

or formulas?

 

i come up with something to find the value of pi myself...

lim n-> infinity

sin(360/n)*n/2

im assuming a circle= infinite sides polygon.

 

hmm....but "sin".....how is sin measured (also tan, cos....)? just by some equipments? how do ppl know that

sinx= x-(x^3/3!)+(X^5/5!)-(x^7/7!)......

i fail to understand this..... any simple proof to this??

 

 

(ARH~~~ whats happening!!!! plz del all the repeated posts above, thx tormod...)

[Bo]

hmm.....how about this, i dont understand how pi is measured. is it simply use a ruler and stuff? or formulas?

Well pi is 'defined' as the ratio: circumference/diameter of a circle. So when pi was introduced, using rulers etc was for long the only way to determine the number. But nowadays there exist some more elegent ways, most of them use series. the simplest is probably: pi/4 =1 - 1/3 + 1/5 - 1/7 ...etc. For more see: http://mathworld.wolfram.com/PiFormulas.html

your formula would certainly go to pi (quite fast even; in radians it is proportional to: pi -pi^3/n^2 +O(1/n^4) [this last symbol is use to say: there are more terms but they are of order 1/n^4 or more] well 1/n^2 goes quite fast to 0 for n->inf. So you got pi left. Very nice technique

hmm....but "sin".....how is sin measured (also tan, cos....)? just by some equipments? how do ppl know that sinx= x-(x^3/3!)+(X^5/5!)-(x^7/7!)...... i fail to understand this..... any simple proof to this??

well you moght know that the sin and cos are defined as follows: if we take a point on a unitcircle, say (x,y) and we also look at the angle t of a vecor pointing from (0,0) to (x,y) with the x-axis. then sin(t)=y and cos(t)=x and tan(t)=y/x. So the sin and cos are basicly the 2 orthogonal (basicly means: the directions of the 2 projections are perpendicular to each other) projections of a point on the unit circle. Most numerical calculations are done with the series given above. to show where this comes from you need 2 concepts: namely complex numbers and taylors formula. I'll sketch it (BE WARNED! i cant do this without using explicit formula's... i'm sorry..)and you can look the details up on the internet or so (otherwise it would take me to much time and there is no formula editor on this board )

TAYLORS FORMULA: Basicly says that most functions can be written as follows:

f(x) = 1/(0!)*f(0) + (1/1!)*x*{df(x)/dx|x=0} + 1/(2!)*x^2 {(d/dx)*(df(x)/dx)|x=0} +...

where e.g df(x)/dx means: f(x) differentiated to x. the addition |x=0 means that in this differentiated function we have to take x=0. the (n!) means: "n faculty" = n*(n-1)*(n-2)* ... *1. We define 0!=1.

the sine and cosine functions also can be written in a defferent way: sin(x) = (exp(ix)-exp(-ix))/(2i). Here i is the complex number. this has the property that i^2 = -1. applying taylor's formula to the above expression for sin(x) gives the series given above.

ps i still wonder: how did you get the original formula with which you began this post? Bo

[Tim_Lou]

thx bo, it makes sense a little.....hmm...using complex numbers...

 

well, these are beyond my knowledge... TAYLORS FORMULA confounds me...

does the taylors formula have something to do with the binomial theorm?.....

 

i'll look for some more information, thx anyway.

[Bo]

ehm no there is no direct connection between the binomial theorem and taylors theorem. (except the fact they bove deal with polynomials...)

Bo

[Tim_Lou]

im trying to prove that

area of circle =pi r^2

and area of ellipse = ab pi

(is it ab pi?)

 

and i found out something interesting:

that pi =

 

lim n-->infinity (4/n^2) * (times)

 

n

sigma [ square root (n^2 - k^2) ]

k=1

 

 

is it right? or not?

if it is, any idea to prove it?

 

(tormod, how can i upload some images??)

[Turtle]

Yes even/odd exists in nature. If I have a pile of 31 beans & you & I wish to fairly share them, we find that using whole beans, no even divison is possible. What is more natural than that.

Further on Fibonacci sequence; in phylotaxy (the numerical arrangement of twigs on a palant) buds appear in a year as a ratio of number of times around the branch over the number of buds. Both numbers are always Fibonacci.

Please visit my site?

http://home.comcast.net/~turtlediable/wsb/index.html

[maddog]

Perhaps it does. It SEEMS that nature sets things up in "countable" groups. Even wave/ particle collapse seems oriented towards individual elements, not some continous stream. Matter seems to have a finite smallest size. there seems to be a discrete number of electrons/ protons/ ... in an atom and specific measurable quantities of atoms in different molecules, ...

 

Seems to be in the grouping Fermion/Bosons...

 

Or do all these things exist like this because we expect them to? Does the wave collapse into a singular particle because that is what we are measuring for?

 

This in essence is the Schroedinger's cat thought problem. Quantum entanglement is

here as well.

 

Do numbers exist because we want them to? Or did we discover numbers because nature is broken into discrete units?

 

The Greeks thought numbers were a discovery. Myself, I think of them as an invention to

keep track of all the sheep and goats. :hihi:

 

Maddog

[Turtle]

I love math. I have done some work on residues of powers; some graphs under Pictures on my page show this.

http://home.comcast.net/~turtlediable/wsb/index.html :hihi: