Mathematics with Minimum Raw Material, Part 2

Is Quark Boy same as Wasiq?

Anyway here is a much clearer proof of the Pythagorean theorem.
Let there be any right angled triangle with base b, perpendicular p and hypotenuse h and an acute angle theta. Drop a perpendicular from vertex to hypotenuse dividing the triangle into two parts.

Area of the triangle equals the sum of areas of two right angle parts. Hence,

0.5 h sin theta x h cos theta = 0.5 b sin theta x b cos theta + 0.5 p sin theta x p cos theta

Thus, hsquare = bsquare + psquare

QED

PS. It is interesting to note that this proof of the theorem was developed by Einstein at the age of eleven!!!!!

Some thing elegant that I found on the cover of a book on number theory is the so called golden rectangle. The ``note on the cover`` explains that if the largest square from a golden recangle is cut away the remaining rectangle is again a golden rectangle (a sort of a fractal). A golden rectangle is the one with length to width ratio of (1+SQRT(5))/2. It also says that the ancient Egyptian might have used this ratio to construct the pyramids. This ratio recures quite often in the number theory for example this is the ratio of n+1th and nth Fibonacci number as n approaches infinite.

shafqat:

Omar Khayyam is credied with geometrically finding solutions to cubic equations not quardratic as i said previously. I stand corrected.

In my last reply I should have said ‘sides’ instead of ‘areas’ . Oops!

Fozia

Re: Wasiq and the dissection of a square

Not sure if you`ll read this since this article is off the front page now, but I saw a dissection of the square today in the hallway leading to my office! Talk about coincidences :) It was posted on the wall by a colleague of mine who is away at present. He has a habit of posting such problems on the board for our amusement and must have done that before the semester break. I didn’t get a chance to look at it until today. Anyway, it is a clever dissection with a total of 24 squares. To get you started call the square abcd
and divide each side into 175 units.
Along side ab are squares of areas 64, 33, 35 and 43 ( in that order)
Along side bc ….43, 51 and 81
Along cd 81, 39 and 55
Along da 55, 56 and 64

The gap in the middle is filled by 15 other squares having areas 1, 2, 3, 4, 5, 8, 9, 14, 16, 18, 20, 29, 30, 31and 38

I don’t have the source, but I can check with my colleague once he comes back.

Fozia

RE: Goga.

I finally found my source for the statement about Al-Khwarizmi and the solution to quadratic equations. It is in a book by William Dunham entitled ``Journey Through Genius,`` (John Wiley & Sons, 1990), according to which Al-Khwarizimi is indeed credited with being the first one to propose the general solution to quadratics (minus b plus minus root b squared minus 4ac over 2a).

As to your other point, the term `algorithm` was not coined by Al-Khwarizmi. It is simply the English corruption of his name, and was employed by later mathematicians.

Interestingly, the term `algebra` is also indirectly attributable to Al-Khwarizimi. It is the English corruption (through Latin) of part of the name of his major treatise, Hisab al-jabr.

Saad

Re: Wasiq(16)

Very nice proof! Now I must find the first one.

Yes, I did check the pi algorithm. For me though, the digit hunt itself is not very interesting ( I lost interest in it after two decimals places :) ). What I find fascinating about pi is its unexpected occurrence in myriad mathematical situations.

Fozia

Shafqat:

My memory is that the general solution of quardratic equations was devised by Omar Khayyam by geometric means. Khwarizimi is known for introducing the numerals and most importantly the number ``zero`` which according to some has hindu origins and also the methods for division and multiplication. The term ``algorithm``, which is step by step application of operations to obtain a solution, was introduced by Khwarizimi.

In any case, he is deserves to be called a great mathematician.

Fozia(14):

The proof is due to R.L.Brooks and I find it to be very nice.

(First, in a square dissection obviously the smallest square cannot be on the edge otherwise one would require at least one other square of equal or smaller side).

Let us suppose that a cube dissection does exist. Then the cubes that stand on the bottom face induce a square dissection of that face. The smallest of the cubes itself will have its top and bottom faces (the bottom face induces the square dissection) surrounded by walls belonging to the cubes around it. The top face itself must therefore have other cubes that stand on top of it and whose bottom faces induce a square dissection of the top face of this smallest cube. Take the smallest of those cubes, the process continues indefinitely!

Did you get a chance to see the algorithm for the nth digit of pi?

Re: Goga(13)
Just read your query about women mathematicians. There are a few. I had already mentioned Emmy Noether Other names that come to mind are Hypatia, Sofia Kovalovskaya (sp?) and Sophie Germaine (amongst the dead mathematicians). Amongst the living - Ingrid Daubachies, Cathleen Morawetz and Mary Ellen Rudin to name a few.

Sophie Germaine’s story is quite interesting. As a child she was not allowed to pursue mathematical studies because of her gender, so, at night, she used to sneak math books from her father’s library into her bedroom to read. When her parents found out they took away her candles (and I believe her nightclothes as well so she wouldn’t venture out at night again!) But she persisted and became a very good mathematician. She had to initially publish her work under a male pseudonym but eventually found recognition. Hats off to her courage!

I remember reading somewhere that Pythagoras’ wife was also a mathematician and carried on his school after his death so she might be regarded as one the first known women mathematicians.

Fozia

Since we are naming names don’t forget Cantor and Noether and I agree Al-Khwarizimi also belongs to this club.There are two other Greek mathematicians who deserve mention – Eudoxes and Hippocrates (not the one related to medicine). Eudoxes’ method of exhaustion for finding areas can be regarded as the precursor of the notion of limits. He also rescued Greek mathematics from its ‘logical scandal’ by his theory of proportions. I would rank him higher than Pythagoras.

Re: Wasiq(9)
No, I have not come across any such proof but it sounds interesting. Do you have the proof for (2) ?
Re Goga(8)
Ah, but mathematicians are not concerned about such trivial ‘realities’.

Fozia

Wasiq:

It seem that the my message is not completely posted. May be because I used the ``hat`` character.

So I wanted to say that your proof works with:

Area of triangle = (1/4) x SQ(hypoteneus) x SIN(base_Angle)

I didn`t think of that the other day. My apologies.

Wasiq, Fozia, Khan:

Also can someone post few woman mathematicians. Our list containes all men. My choice is Daubechies. Her work on wavelets is revolutionary.

Shafqat:

For what I have read Ramanujan was a mathematical genuis. Because of his untimely death, he couldn`t produce lot of work. G.H. Hardy once ranked mathematicians. He gave himself 10, Hilbert 30, and Ramanujan 80.

Ramanujan is mainly know for his identities in the number theory. Some of them are extremely weird. That to some mathematicians may not be important.

However, in Kaku`s ``Hyperspace``, it is mentioned that some of Ramanujan identities have been useful in the formulation of string theories. Sometimes, unfortunately, the real wotrth of the work of a mathematician is not recongnized until sometime. Another example is Galios.

One of the three wishes of Hardy was to prove the non-existance of God. I think that was the reason he embarked in biology to show mathematically that life can originate by itself using, perhaps, the diffusion theory.

Goga(11);

I think you did not understand the proof. There is no restriction on a relationship between the area and the hypot. Let me state it again, hopefully more clearly this time:

Let us say that a right angled triangle ACB is divided into two right angled triangles ABD and BCD (let AC, AB and BC be the hypot of the three triangles respectively).

The ratio of areas of ABD and ACB is square(AB/AC) since they are similar triangles. Similarly the ratio of the areas of BCD and ACB is square(BC/AC). Since the sum of the two ratios must be one, the pythagoras theorem follows.

regards

Shafqat(10):

Yes, Al-Khwarzami should be included, but I really do not have a good feeling about the relative merit of mathematicians once one starts going back in time. My choices were rather obvious, for both Omar Khayyam and Ibn-Haytham are associated with important problems that they solved.

Leibniz is an interesting case. Though a co-inventor of calculus and a brilliant man, his contributions to mathematics are not as extensive say as that of Newton.

Russell, I am not sure either as a mathematician. His motivation behind Principia Mathematica was to execute the Hilbert program, which was frustrated by Godel`s incompleteness theorems. Most of his own contribution was to logic, and there he is overshadowed by greater figures. Russell is again a polymath like Leibniz with wide ranging interests and contributions.

I think Hilbert however should be added to the list of great mathematicians. And so should Dirichlet.

Wasiq:

Yes, your proof works out if:

Area = (1/2) x Hypotenuse

Wasiq, Goga, Fozia, Umair and others:

The greatest mathematician ? If you ask a biologist like me, why, it`s G.H. Hardy, of course. He`s the only mathematician immortalized in biology! :)

But more seriously.

Wouldn`t you say Al-Khwarizmi also deserves a spot in the Mathematics Hall of Fame ? I think he is the one who first put forward the general solution to quadratic equations in his major work, `Hisab al-Jabr`.

And while on that subject, how great was Ramanujan, really ? I mean, Kanigel`s ``The Man Who Knew Infinity`` is of course very compelling, but I thought that in ``A Mathematician`s Apology`` his own mentor, Hardy, was less generous (and that`s not even mentioning C.P. Snow`s excellent preface to that book, in which he effectively says that Ramanujan had nothing of the genius of the true greats). So what`s the real story here ?

Finally, I was surprised at the absence of two mathematicians in particular form all your lists - Leibniz and Russell. Surely, the independent co-discoverer of the calculus makes it to the hall of fame (or is that overstating his contribution ?). And Russell, I would have thought, was at least as important as Hardy and probably had greater intellectual stature.

Saad

Fozia(5):

Interesting application of the Pythagoras Theorem. Do you know of a simple proof of the following:

1. It is possible to dissect a square into a finite number of squares, all unequal.

2. It is impossible to dissect a cube into a finite number of such cubes.

I once read an explanation of a simple proof (2) but have not yet come across one for (1).

Khan(4):

I should have stated everything clearly but I was trying to be minimal. Indeed the proof proceeds by comparing the areas of the similar triangles that are formed and comparing the squares of the lengths of equivalent sides. I should have said that explicitly to make things clearer. Thanks.

Do you like the proof?

Goga(7):

Umair has clarified the proof, and as you would see it would hold for all proportions, not just for two equal sides.

Khan(3),Fozia(5),Goga(6):

Will have to agree with Goga that we`ve got to wait before we know who the Prince of Mathematicians is: (Euler and Gauss approach it the closest in my view)

In terms of great (dead) mathematicians, my pick would be Eodvos, Ramanujan, Poincare, Riemann, von Neumann, Gauss, Euler, Jacobi, Cauchy, Abel, Fermat, Khayyam, Archimedes, Euclid, Ibn-Haytham, Newton, Bernoulli, Godel, Lie. I would rank Ramanujan, Riemann, Gauss, Euler and Euclid above others. (I know of few Greek/Indian mathematicians ... but I do not know how to rank them, Hipparchus, Pythagoras, Ptolemy?)

For living mathematicians, Mandelbrot, Witten, Atiyah, Donaldson, Margulis ...

By the way do you know of the astonishing Bailey-Borwein-Plouffe Pi Algorithm which allows one to calculate ANY digit of pi (in hexadecimel) without calculating any prior digit?!!!

(See www.mathsoft.com/asolve/plouffe/plouffe.html)

Also, I think you will enjoy the parent site:

http://www.mathsoft.com/asolve/

(List of some unsolved math problems)

Fozia:

Interesting solution but I don`t know if the pizza people will let you do all those measurements before you give them money. :)

Wasiq:

``...the area is proportional to the square of a side of a triangle...``

This is not true in general.

Area of the triangle = 1/2 x Base x Height

What you are saying would only be true in the case of a right triangle with two equal sides. So your proof would work in the special case not in general.

Khan:

I think following also rightfully belong to the list of great mathematicians:

Galois, Riemann, Hilbert, Hardy, Ramanujan, Godel, Neumann

If you are including Newton then you shouldn`t forget Witten.

Also don`t forget Russians and Americans.

Gauss is indeed known as the ``Prince of Mathematics`` but I will hold my vote for now. There is so much to come.

Re: Khan

I don’t think there are any clear winners in either category but if I had to pick I’d choose Euler and Gauss amongst the mathematicians. In the second category I’d vote for the Greeks simply because their logical rigor provided the foundation on which other giants could build.

Re: Wasiq

Yes, Cantor and part III it should be.

Here is an appetizing application of Pythagoras theorem.

A large pizza costs the same as a medium and small combination. You have to decide whether to buy the large pizza or the medium & small combo. What should you do?

Answer: Cut each pizza in half (so that you have three semi-circles) and arrange them so that the diameters form a triangle. If it is an acute angled triangle then choose the medium & small combo, if one of the angles is obtuse then choose the large pizza and in the event that the triangle is right angled then either choice is equally good (or bad if you are into healthy food!).

Reason: Let us say the three diameters are dl, dm and ds with respective areas as Al, Am, and As

If the triangle is right angled, then

sq(dl)=sq(dm)+sq(ds)

or (pi/4) x sq(dl)= (pi/4) x sq(dm)+ (pi/4) x sq(ds)

and so Al=Am+As

If the triangle is acute angled, then

sq(dl) *is less than * sq(dm)+sq(ds)

etc.

Fozia

[The last post got ``shorted`` because I used special characters; so here goes again]

Re: On Pythagoras`s Proof

I think the fact that the two new triangles formed by dropping the perp. onto the hypotenuse are similar (all corresponding angles equal) to the large one needs to be stated. Otherwise the ``area of a triangle being proportional to the square of a side`` is an ambiguous statement [in general the area of a triangle is not of the form K.SQ(a), where ``a`` is any side and K, a constant].

If the original triangles area is X, the area of the smaller triangles (being similar to the original one) is:

(SQ{AB/AC}).X and (SQ{BC/AC}).X

These sum to the total area:

(SQ{AB/AC}).X + (SQ{BC/AC}).X = X

Which gives you SQ(AB) + SQ(BC) = SQ(AC)

Just curious who would you all think of as the greatest mathematician ever? Gauss, Euler, Archimedes, Euclid, Pythagoras? What about Newton, and the Frenchies (Laplace, Lagrange, etc.)? Which nation/race contributed the most to Math (is there a clear winner): Greeks, Germans, French, Italian, Arabs, Ancient Egyptian?

Re: On Pythagoras`s Proof

I think the fact that the two new triangles formed by dropping the perp. onto the hypotenuse are similar (all corresponding angles equal) to the large one needs to be stated. Otherwise the ``area of a triangle being proportional to the square of a side`` is an ambiguous statement [in general the area of a triangle is not of the form K.a

Re: Fozia

Yep, they are absolutely beautiful. Maybe we should present Cantor`s proof in the next article in this series.

Your suggestion about cutting a Mobius strip is very interesting, I am sure people will be surprised at what happens.

Talking of simple proofs, here`s one short elegant proof of the Pythagoras theorem.

Take a right angled triangle ABC. Drop a perpendicular to the longest side AC from B. Since the area of the two new triangles ADB and BDC must sum to the the area of ABC, and the area is proportional to the square of a side of a triangle, the Pythagoras theorem follows:

AB.AB + BC.BC = AC.AC

regards

Thanks for posting this. Euclid`s proof of the infinitude of primes and Cantor`s demonstration of the countability of rational numbers are two of the most elegant yet simple proofs in all of mathematics.

Interested folks may want to try the following...
.... cut a Mobius Strip lengthwise all the way around. You will be surprised with the result!

Fozia