Carl Friedrich Gauss

Gill Sahib,

A great article. After a long time on Chowk I found something I could read twice. This is al so, I believe, a rare case where the main article is far more interesting than the interacts. But then most Chowkies might not be so deep into maths.
Please accept my sincere thanx and congrats.

Gauss is every where. In mathematics I can still recall Gauss elimination method, Vector Calculus, probability distributions, number theory, complex integration. Also in Physics (electricity & magnetism). As such Gauss must have been a formidable intellect of his time.

I wonder what Masadi would have to say.

Gill Sahib, all of us math illiterates should be thankful to you for introducing us to this mathematics genius. Could you please explain how mathematics after Gauss became completely different from what it was before him? What is rigor to a mathematician?

the thereom states (algebra one) that given a polynomial in a single variable, and given coefficients (for simplicity assume that they are all real), it will have exactly the same number of roots as the degree of the polynomial. These roots are the solution to the equation you get when you equate the polynomial to zero. These roots can all be the same, then the polynomial can be reduced to the form

(x+a)^n=0; xi=-a, i=1...n

if they are not equal you can factorise the polynomial and it get reduced to

(x+a)^n X (x+b)^m=0 ==> order is n+m

or (x+a1)(x+a2)(x+a3)...(x+an)=0, roots are x=a1, a2, a3,....,an

however, the theorem also accomodates complex roots - here you will have to have a pair of roots of the form a+-jb (symmetric about the real axis)

(note for those familiar with differential equations- the roots are the poles of the equation (points at which everything goes to infinity).



You can also have complex coefficients in polynomial.

#5 GT

That was easy, my dear!

The whole world knows that ``beauty`` is in the eye of the beholder!

The ``I``!

#6
Thanks!

I believe the fundamental theorem of algebra states that an nth degree algebraic equation has exactly n roots, provided multiplicity of roots (if any) is taken into consideration, and complex roots are admitted.

Gill:

Here is a challenge for you:

E is equal to m c square is termed ``beautiful``, a sequence of measures tending to the dirac measure is ``beautiful``, many discovered physical phenomena are termed ``beautiful``.

So, how do Gaussian distributions contribute to the ``beauty`` of theoretical physics?

Regards.

Re: # 5
Gaussian distribution have a great property......a property which enables many solutions to have a so called ``optimal`` property. For example you can show that the Kalman filter is optimal with gaussian distributions, however for a random distribution, these are not optimal. The key appeears to be this

if we take a number say a in a gaussian distribution G1, and another b from G2, and perform a number of algebraic operation on them, you get number which are also in a gaussian distribution.

If you are able to generate n numbers from a gaussian distribution(s) and take their mean/afevrage etc this mean will also be in a gaussian distribution.

It is this property which akes it a great and beautiful distribution. Without it many of our current technology will not work.

Re: # 9 iron_mask

Your assertion (for large numbers) below is correct:

``If you are able to generate n numbers from a gaussian distribution(s) and take their mean/afevrage etc this mean will also be in a gaussian distribution.

It is this property which akes it a great and beautiful distribution. Without it many of our current technology will not work.``

I would like to reword (with your permission) the last sentence as:

``Without it we would not have CONFIDENCE in many of the beautiful theories that drive our current technology and observations``.

Recall the incident about M and M experiments in Bell Labs determining the speed of light ..... and the embarrasment of deleting the one ``outlier``!

I had forever associated Gauss with a measure of the magnetic field strength. It is very impressive to know that such crucial concepts as the binomial theorem, the bell-shaped curve, and the method of least squares all came from this person’s work!

When a scholar comes out from a family of scholars – that is no big deal! But being from poor, working class family, with little motivation or guidance – and still being able to accomplish the things this gentleman did – now that is truly something. (But Bell’s anecdote appears rather improbable.) I wonder how different his life and the state of math would have been had the kind teacher not helped the Gauss.

I am curious regarding the fundamental theorem of algebra – which does not ring a bell, or perhaps I have forgotten.

I sometimes worry that with the advent of advanced calculators and laptop computers, students make excessive use of those capabilities to solve math problems – even symbol-involving problems, and they do not spend much time digesting or grasping the underlying concepts.

Gauss was one of the fathers of the actuarial profession:

``In 1851, four years before his death, Gauss established one last set of scientific principles, this time in the field of actuarial mathematics. In his report for the University’s pension fund for widows, he introduced for the first time a computation of pension scheme contributions based on mortality rates and probability calculations.``

http://www.gaussjahr.de/gauss_genial_en.php?navid=2&supnavid=2

Tvyy, ncbybtvrf sbe #1 - gur grzcgngvba gb gnxr n fjvcr ng gur bns jnf whfg gbb zhpu

Yes, very true! If only we could all be like him.

No! to sing his praise is not my purpose here
Let me start the ball rollling by starting the degeneration process

Gill, sirjee, tell us whether this man Gauss was a believer and whether he found all of this in his book. You see zeemini maintains that everything is there in his little book, which is a complete description of life and how life is to be led and really there is nothing more that can be added to life. I see this gauss paaji lived 1000 years after the beloved prophet, so surely he must have read all of this in the book, and then saw the light and constructed his beautiful works as a result of seeing patterns there.

If not, how much more degenrate can you be. Please do not spoil our minds with this mental pollution.

Over to you zeemini!